The Galois action on geometric lattices and the mod\(\ell \) I/OM
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Abstract
This paper studies the Galois action on a special lattice of geometric origin, which is related to mod\(\ell \) abelianbycentral quotients of geometric fundamental groups of varieties. As a consequence, we formulate and prove the mod\(\ell \) abelianbycentral variant/strengthening of a conjecture due to Ihara/OdaMatsumoto.
Mathematics Subject Classification
12F10 12G 12J10 19D451 Introduction
Our story begins with a question of Ihara from the 1980s, which asked for a combinatorial description of the absolute Galois group of \(\mathbb {Q}\). More precisely, this combinatorial description should be in the spirit of Grothendieck’s Esquisse d’un Programme [15], which suggested studying absolute Galois groups via their action on objects of “geometric origin,” and specifically the geometric fundamental group of algebraic varieties. Ihara asked whether the absolute Galois group of \(\mathbb {Q}\) is isomorphic to the automorphism group of the geometric fundamental group functor on \(\mathbb {Q}\)varieties, and OdaMatsumoto [20] later conjectured that the answer is affirmative, based on motivic evidence. We will henceforth refer to this question/conjecture (and its various variants) as the “I/OM.”
The original I/OM conjecture, which deals with the full geometric fundamental group, and which we call “the absolute I/OM” below, was proven by Pop in an unpublished manuscript from the 1990s. A variant of the I/OM over padic fields, using tempered fundamental groups, was then developed and proved by André [1]. Later on, Pop formulated and proved a strengthening of the absolute I/OM, which instead deals with the maximal pro\(\ell \) abelianbycentral quotient of the geometric fundamental group. The pro\(\ell \) abelianbycentral I/OM implies the absolute I/OM, and both contexts are treated by Pop in [29].
In this paper, we develop and prove a further strengthening of I/OM, which deals with the mod\(\ell \) abelianbycentral quotient of the geometric fundamental group. This mod\(\ell \) context strengthens both the pro\(\ell \) abelianbycentral and the absolute situations. Furthermore, the mod\(\ell \) abelianbycentral quotient is the smallest possible functorial (pro\(\ell \)) quotient which remains nonabelian. In this sense, the mod\(\ell \) context yields the strongest possible results that one could hope for.
Most importantly however, the mod\(\ell \) abelianbycentral context gets much closer to the spirit of Ihara’s original question of finding a combinatorial description of absolute Galois groups. Indeed, the geometric fundamental group of a variety and its pro\(\ell \) abelianbycentral quotient are both finitelygenerated profinite resp. pro\(\ell \) groups, and the topology of such groups plays a crucial role in both situations. In contrast to this, the mod\(\ell \) abelianbycentral quotient can be seen as a (discrete) finitedimensional \(\mathbb {Z}/\ell \)vector space endowed with some extra linear structure. In other words, the mod\(\ell \) abelianbycentral quotient of a geometric fundamental group is an object of a purely combinatorial nature, being a finitedimensional linear object over \(\mathbb {Z}/\ell \).
 (1)
Open image in new window for the geometric fundamental group of X, i.e. the étale fundamental group of the basechange \(\bar{X}\) of X to \(\bar{k}_0\), with respect to some geometric point \(\bar{x}\).
 (2)
\(\pi ^a(X)\) for the maximal mod\(\ell \) abelian quotient of \(\bar{\pi }_1(X)\).
 (3)
\(\pi ^c(X)\) for the maximal mod\(\ell \) abelianbycentral quotient of \(\bar{\pi }_1(X)\).
For an essentially small category \(\mathcal {V}\) of normal geometricallyintegral \(k_0\)varieties, we consider the group \({{\mathrm{Aut}}}^\mathrm{c}(\pi ^a_\mathcal {V})\) which consists of systems \((\phi _X)_{X \in \mathcal {V}}\), where X varies over the objects of \(\mathcal {V}\), and the \(\phi _X \in {{\mathrm{Aut}}}^\mathrm{c}(\pi ^a(X))\) are compatible with morphisms arising from \(\mathcal {V}\). Since \(\pi ^a(X)\) is a \(\mathbb {Z}/\ell \)vector space for all \(X \in \mathcal {V}\), and the morphisms \(\pi ^a(X) \rightarrow \pi ^a(Y)\) arising from morphisms \(X \rightarrow Y\) in \(\mathcal {V}\) are \(\mathbb {Z}/\ell \)linear, we obtain a canonical action of \((\mathbb {Z}/\ell )^\times \) on \({{\mathrm{Aut}}}^\mathrm{c}(\pi ^a_\mathcal {V})\) by leftmultiplication. We write \(\underline{{{\mathrm{Aut}}}}^\mathrm{c}(\pi ^a_\mathcal {V}) {:}{=}\, {{\mathrm{Aut}}}^\mathrm{c}(\pi ^a_\mathcal {V})/(\mathbb {Z}/\ell )^\times \) for the quotient by this canonical action of \((\mathbb {Z}/\ell )^\times \).
Main Theorem
We will define precisely what we mean by “sufficiently large” in Sect. 1.7, where the precise assumption/terminology is that \(\mathcal {V}\) should be “5connected.” However, we note here that both the full category of all normal geometricallyintegral \(k_0\)varieties, and the full category of geometricallyintegral smooth quasiprojective \(k_0\)varieties, are “sufficiently large” in this sense. The above Main Theorem is proved for such 5connected categories in Theorem B. In fact, our approach is birational, and we obtain the above Main Theorem for many more categories \(\mathcal {V}\). The following is a concrete example which follows directly from Theorem A.
Example 1.1
The four main theorems of the paper (Theorems A, B, C and D) all require some variant of a “dimension \(\ge 5\)” assumption; this also leads to the “5connectedness” condition mentioned above. It is natural to ask whether these results hold true for dimension d, with \(1< d < 5\). However, the lower bound of 5 seems to be the best that one can presently hope for, in the mod\(\ell \) context. As explained below, this assumption first arises in the results of Evans–Hrushovski [8, 9] and Gismatullin [14], which play a crucial role in our proofs. In these works, this assumption is required in a technical step arising from the use of the group configuration theorem. The same assumption also arises, for different reasons, in two key Lemmas 10.11, 10.12 of the present paper. Nevertheless, the main large categories of geometricallyintegral normal \(k_0\)varieties, which are of interest for the I/OM, are indeed “5connected,” as we have mentioned above.
1.1 Automorphism groups of functors
We will frequently consider several different target categories \({\mathcal {D}}\). To keep the notation consistent throughout, if some symbol/notation is used to denote the automorphism groups of objects in \({\mathcal {D}}\), then we will use the same symbol/notation to denote the automorphism group of a functor whose values are in \({\mathcal {D}}\).
1.2 The absolute I/OM
Throughout the paper, we will work with a fixed infinite perfect field \(k_0\), and \(k = \bar{k}_0\) will denote the algebraic closure of \(k_0\). Furthermore, \({{{\mathrm{Gal}}}_{k_0}}{:}{=}\, {{\mathrm{Gal}}}(kk_0)\) will denote the absolute Galois group of \(k_0\). We will only consider normal geometricallyintegral \(k_0\)varieties, and we denote the category of all such varieties by \(\mathbf {Var}_{k_0}\). Namely, the objects of \(\mathbf {Var}_{k_0}\) are schemes which are normal, geometricallyintegral, separated and of finitetype over \(k_0\). The morphisms in \(\mathbf {Var}_{k_0}\) are just morphisms of \(k_0\)schemes.
The nature of the map \(\rho _{k_0,\mathcal {V}}\) in general is still quite mysterious. Nevertheless, for \(k_0 = \mathbb {Q}\), the injectivity of this morphism has been extensively studied. For instance Drinfeld [6] observed that Belyi’s theorem [2] implies \(\rho _{\mathbb {Q},\mathcal {V}}\) is injective as soon as \(\mathcal {V}\) contains the tripod, \(\mathbb {P}^1_\mathbb {Q}{{\smallsetminus }} \{0,1,\infty \}\). More generally, it follows from the work of Voevodsky [38] and Matsumoto [19] in the affine case, and Hoshi–Mochizuki [16] in general, that \(\rho _{\mathbb {Q},\mathcal {V}}\) is injective as soon as \(\mathcal {V}\) contains a (possibly affine) hyperbolic curve.
The surjectivity of \(\rho _{k_0,\mathcal {V}}\) is much less understood, even in the case \(k_0 = \mathbb {Q}\). For instance, if one takes \(\mathcal {V}= \{{\mathcal {M}}_{0,n}\}_n\) with the “connecting morphisms” (or certain smaller subcategories) then \({{\mathrm{Out}}}(\bar{\pi }_1_\mathcal {V})\) is the intensively studied Grothendieck–Teichmüller group. The surjectivity of \(\rho _{\mathbb {Q},\mathcal {V}}\) in this case (and for other subcategories of the Teichmüller modular tower) is still a major open question in modern Galois theory.
In any case, the original question of Ihara, and the subsequent conjecture of OdaMatsumoto [20], predict that \(\rho _{\mathbb {Q},\mathcal {V}}\) is an isomorphism in the case where \(\mathcal {V}= \mathbf {Var}_\mathbb {Q}\). In 1999, Pop proved in an unpublished manuscript that \(\rho _{k_0,\mathcal {V}}\) is an isomorphism for more general fields \(k_0\) in the case where \(\mathcal {V}= \mathbf {Var}_{k_0}\). Pop’s proof was eventually released in [29], along with a stronger pro\(\ell \) abelianbycentral variant which we now summarize.
1.3 The pro\(\ell \) abelianbycentral I/OM
In order to get closer to the spirit of Ihara’s original question of finding a combinatorial description of the absolute Galois group of \(\mathbb {Q}\), it makes sense to replace the geometric fundamental group by certain smaller functorial quotients. The first such strengthening was formulated and proved by Pop [29] who uses the maximal pro\(\ell \) abelianbycentral quotient of \(\bar{\pi }_1\). We will give only a very brief summary of the pro\(\ell \) abelianbycentral context, since the purpose of this paper is to develop a mod\(\ell \) variant/strengthening of loc. cit.
Pop shows in [29] that \(\rho ^\mathrm{C}_{k_0,\mathcal {V}}\) is an isomorphism for certain categories \(\mathcal {V}= \mathcal {V}_X\) which are similar to the categories of the form \(\mathcal {U}_\mathbf {a}\) (see Sect. 1.6 below). Loc. cit. also shows that \(\rho ^\mathrm{C}_{k_2,\mathcal {V}}\) is an isomorphism for socalled “connected” subcategories \(\mathcal {V}\) of \(\mathbf {Var}_{k_0}\). This “connectedness” condition holds, in particular, for \(\mathbf {Var}_{k_0}\) itself, as well as for the category of geometricallyintegral smooth quasiprojective \(k_0\)varieties. Although this notion of connectedness is somewhat technical, we note that it is similar to what we call “2connected” in Sect. 1.7.
The pro\(\ell \) abelianbycentral context gets closer to a truly combinatorial description of absolute Galois groups than the absolute context. However, the groups considered in this context, \(\Pi ^A(X)\) and \(\Pi ^C(X)\), are still quite large, as they still have a nontrivial profinite topology which plays a very significant role.
In this paper, we develop a further strengthening of the I/OM, by considering the mod\(\ell \) abelianbycentral quotient of \(\bar{\pi }_1(X)\). As mentioned before, this quotient of \(\bar{\pi }_1\) which we consider is the smallest functorial pro\(\ell \) quotient of \(\bar{\pi }_1\) which remains nonabelian; in particular, it is a quotient of \(\Pi ^C(X)\), and it can be seen as a purely combinatorial (i.e. finite and discrete) object. Thus, the mod\(\ell \) abelianbycentral context is essentially the best one could hope for, in the profinite context. In more broad terms, considering the I/OM with other variants of the geometric fundamental group could lead to further substantial developments in various facets of Galois theory.
In this paper, we will prove that the mod\(\ell \) abelianbycentral I/OM holds for socalled “5connected” subcategories \(\mathcal {V}\) of \(\mathbf {Var}_{k_0}\). Similar to Pop’s notion of connectedness, our notion of 5connectedness applies to \(\mathbf {Var}_{k_0}\) itself, as well as to the full category of geometricallyintegral smooth quasiprojective \(k_0\)varieties. However, for the time being, it is unclear whether the mod\(\ell \) I/OM holds true for dconnected categories with \(1< d < 5\).
It is particularly important to note that Pop [29] uses ideas related to Bogomolov’s Program [3] in birational anabelian geometry, which considers pro\(\ell \) abelianbycentral Galois groups of higherdimensional function fields over k. In a few words, the proof of the pro\(\ell \) abelianbycentral I/OM first reduces to a birational context, and eventually uses both the local theory [4, 25] and global theory [28] from pro\(\ell \) abelianbycentral birational anabelian geometry.
The initial step in our proof of the mod\(\ell \) abelianbycentral I/OM is moreorless the same as the pro\(\ell \) context, in the sense that we will reduce the mod\(\ell \) I/OM to a birational context. We will then use techniques from the mod\(\ell \) abelianbycentral variant of Bogomolov’s Program, including both the mod\(\ell \) local theory [26, 33, 36] and the mod\(\ell \) global theory [34].
Because of this strategy, we run into precisely the same problems/difficulties that arise when one passes from the pro\(\ell \) to the mod\(\ell \) abelianbycentral variants of Bogomolov’s Program. These fundamental differences between the pro\(\ell \) and mod\(\ell \) context were described in detail in the introduction of [34], and we refer the reader there for these details. Nevertheless, we mention here that, in the pro\(\ell \) context, one eventually uses the Fundamental Theorem of Projective Geometry applied to an infinitedimensional kprojective space embedded in \({\text {H}}^1(K,\mathbb {Z}_\ell (1))\), where K is a function field over k. The main difficulty in the mod\(\ell \) context is that \({\text {H}}^1(K,\mathbb {Z}/\ell (1))\) contains no such kprojective space. Therefore, our proof of the mod\(\ell \) I/OM is fundamentally different than the proof of the pro\(\ell \) variant. See Sect. 2 for a detailed summary of the proof of the mod\(\ell \) I/OM, and see the introduction of [34] for more on the comparison between the pro\(\ell \) and mod\(\ell \) contexts. We now introduce the mod\(\ell \) abelianbycentral context in detail.
1.4 The mod\(\ell \) abelianbycentral quotients
 (1)
\(\mathcal {G}^{(2)} {:}{=}\, [\mathcal {G},\mathcal {G}] \cdot \mathcal {G}^\ell \).
 (2)
\(\mathcal {G}^{(3)} {:}{=}\, [\mathcal {G},\mathcal {G}^{(2)}] \cdot \mathcal {G}^{\delta \cdot \ell }\) where \(\delta = 1\) if \(\ell \ne 2\) and \(\delta = 2\) if \(\ell = 2\).
Remark 1.2
The primary reason for the distinction between odd/even \(\ell \) in the definition of \(\mathcal {G}^{(3)}\) is that we require \(\mathcal {G}^c = \mathcal {G}/\mathcal {G}^{(3)}\) to be nonabelian. Nevertheless, \(\mathcal {G}^c\) is the smallest pro\(\ell \) quotient of \(\mathcal {G}\) which is functorial in \(\mathcal {G}\), and which is nonabelian. In cohomological terms, this distinction between odd/even \(\ell \) is related to the fact that the mod2 Bockstein morphism agrees with the Steenrod square \({\mathrm{Sq}}^1\), whereas no such relationship exists for odd \(\ell \). For more on the mod\(\ell \) Zassenhaus filtration and its connection with mod\(\ell \) (group) cohomology, we refer the reader to [7] and [12].

\({{\mathrm{Hom}}}^\mathrm{c}(\mathcal {G}_1^a,\mathcal {G}_2^a) = {{\mathrm{Hom}}}_{\mathbf {AbC}_{\ell }}(\mathcal {G}_1^\mathrm{ac},\mathcal {G}_2^\mathrm{ac})\).

\({{\mathrm{Aut}}}^\mathrm{c}(\mathcal {G}^a) = {{\mathrm{Aut}}}_{\mathbf {AbC}_{\ell }}(\mathcal {G}^\mathrm{ac})\).
1.5 The mod\(\ell \) abelianbycentral I/OM
Remark 1.3
1.6 The main result: birational systems
With the preparation above, we will now introduce the subcategories of \(\mathbf {Var}_{k_0}\) which we consider in this paper. Let \(X \in \mathbf {Var}_{k_0}\) have dimension \(\ge 1\), and let \(\mathcal {U}_X\) be a basis of open neighborhoods of the generic point of X. We will always consider \(\mathcal {U}_X\) as a subcategory of \(\mathbf {Var}_{k_0}\) whose objects are the elements of \(\mathcal {U}_X\) and whose morphisms are the inclusions among them as open subsets of X. Moreover, we write \(\mathcal {U}_X^+ = \mathcal {U}_X \cup \{X\}\) for the basis of open neighborhoods of the generic point of X which also includes X as a terminal object.
Let \(X \in \mathbf {Var}_{k_0}\) be an object. To simplify the exposition, we will say that a subcategory \(\mathcal {U}_X\) of \(\mathbf {Var}_{k_0}\) is a birational system of X if \(\mathcal {U}_X\) is a basis of open neighborhoods of the generic point of X. We will use the notation \(\mathcal {U}_X^+\) as above to denote the existence of X as a terminal object. In other words, while a birational system \(\mathcal {U}_X\) of X need not have a terminal object, the birational system \(\mathcal {U}^+_X\) always has X as a terminal object. We say that \(\mathcal {U}\) is a birational system if \(\mathcal {U}\) is a birational system of X for some \(X \in \mathbf {Var}_{k_0}\). The dimension of a birational system \(\mathcal {U}\), denoted \(\dim \mathcal {U}\), is defined to be the dimension of one (hence all) of the objects in \(\mathcal {U}\). In particular, \(\dim \mathcal {U}_X = \dim X\).
 (1)
The objects of \({\mathcal {U}_\mathbf {a}}\) are given by \({\mathcal {U}\cup \{\mathrm{U}_\mathbf {a}\}}\).
 (2)
The morphisms in \(\mathcal {U}_\mathbf {a}\) are the inclusions among the objects in \(\mathcal {U}\), the identity on \({\mathrm{U}_\mathbf {a}}\), and all of the dominant morphisms \(U \rightarrow \mathrm{U}_\mathbf {a}\) for \(U \in \mathcal {U}\).
Theorem A
1.7 The main result: connected categories

If \(\dim \mathcal {U}_2 > 1\): For all \(V \in \mathcal {U}_2\), there exists some \(U \in \mathcal {U}_1\) such that \(\mathcal {V}\) contains a dominant morphism \(U \rightarrow V\).

If \(\dim \mathcal {U}_2 = 1\): For all \(V \in \mathcal {U}_2\), there exists some \(U \in \mathcal {U}_1\) such that \(\mathcal {V}\) contains a dominant morphism \(U \rightarrow V\) with geometrically integral fibers.
 (1)
The category \(\mathcal {V}\) contains \(\mathcal {U}_\mathbf {a}\) for some finite tuple \(\mathbf {a}\) of elements of \(k_0^\times \).
 (2)
The birational system \(\mathcal {U}\) dominates both \(\mathcal {U}_1\) and \(\mathcal {U}_2\) in \(\mathcal {V}\).
 (1)
One has \(\dim \mathcal {U}_{2i+1} \ge d\).
 (2)
The birational system \(\mathcal {U}_{2i+1}\) attaches \(\mathcal {U}_{2i}\) to \(\mathcal {U}_{2i+2}\) in \(\mathcal {V}\).
 (1)
\(\mathcal {V}\) is essentially small and it contains a positivedimensional object.
 (2)
For every object X of \(\mathcal {V}\), there exists some birational system \(\mathcal {U}_X^+\) of X which contains X as the terminal object, such that \(\mathcal {U}_X^+\) is contained in \(\mathcal {V}\).
 (3)
Any two birational systems \(\mathcal {U}_0,\mathcal {U}_{2r}\) which are contained in \(\mathcal {V}\) are dconnected in \(\mathcal {V}\).
Theorem B
Theorems A and B together form the mod\(\ell \) variant/strengthening (in dimension \(\ge 5\)) of the main results from [29], where the absolute and pro\(\ell \) I/OM are proven. And as mentioned above, this mod\(\ell \) context is optimal with respect to functorial pro\(\ell \) quotients of \(\bar{\pi }_1\) which remain nonabelian. The recent work [30] proves yet another refinement of [29] by considering coarser categories of varieties, but still necessarily remaining in the pro\(\ell \) context. Therefore, the present paper and [30] both refine the results of [29], while these two refinements seem to be in orthogonal directions.
1.8 BirationalGalois variant
Let \(X \in \mathbf {Var}_{k_0}\) be given and let \(\mathcal {U}= \mathcal {U}_X\) be a birational system for X, as defined above. Recall that elements of \({{\mathrm{Aut}}}^\mathrm{c}(\pi ^a_\mathcal {U})\) consist of systems of elements \((\phi _U)_{U \in \mathcal {U}}\), where \(\phi _U \in {{\mathrm{Aut}}}^\mathrm{c}(\pi ^a(U))\) are compatible with the morphisms arising from \(\mathcal {U}\). By taking the projective limit over \(\mathcal {U}\), one obtains an element of \({{\mathrm{Aut}}}^\mathrm{c}(({{\mathrm{Gal}}}_{k(X)})^a)\). Moreover, since X is geometrically normal, it follows that the induced canonical map \({{\mathrm{Aut}}}^\mathrm{c}(\pi ^a(X)) \rightarrow {{\mathrm{Aut}}}^\mathrm{c}(({{\mathrm{Gal}}}_{k(X)})^a)\) is injective. Therefore, in order to prove Theorem A, it makes sense to first develop a birational variant of that theorem, which deals with quotients of absolute Galois groups of function fields as opposed to quotients of fundamental groups of varieties. Therefore, the main focus of this paper is to develop and prove birational variants of our main theorems, and we now introduce the appropriate notation and terminology.
Theorem C
1.9 BirationalMilnor variant
It turns out that it will be more convenient to work with the Kummer Dual of Theorem C. While the Kummer dual of \(\mathcal {G}_K^a\) is \(K^\times /\ell \), it will be a consequence of the Merkurjev–Suslin Theorem [22] that the “dual” of the object \(\mathcal {G}_K^\mathrm{ac}\) can be considered as the mod\(\ell \) Milnor Kring of K, which we denote by \({\text {k}}^\mathrm{M}_*(K)\). Thus, the primary focus of this paper will be to prove a Milnor variant of Theorem C, which deals with the mod\(\ell \) Milnor Kring of the function field K.
Theorem D
We give a brief description of the proof of Theorem D, as it pertains to the mod\(\ell \) anabelian tools mentioned above. A much more detailed summary is given in Sect. 2. In the above context, let \(\sigma \in {{\mathrm{Aut}}}^\mathrm{M}_\mathbf {a}({\text {k}}^\mathrm{M}_1(K))\) be given. First, we will use the mod\(\ell \) local theory from [33, 36], along with the compatibility of \(\sigma \) with \(\mathbf {a}\), to show that \(\sigma \) is compatible with certain special onedimensional geometric subgroups (see Sect. 2 for this terminology). The majority of the work is then devoted to showing that \(\sigma \) is compatible with all such onedimensional geometric subgroups, and these include the rational subgroups considered in [34]. One then concludes, along similar lines to the mod\(\ell \) global theory from loc. cit., that \(\sigma \) arises from some automorphism of K, while some additional arguments show that this automorphism fixes \(K_0\) pointwise. In other words, \(\sigma \) arises from an element of \({{\mathrm{Gal}}}_{k_0} = {{\mathrm{Gal}}}(KK_0)\).
1.10 A guide through the paper
This paper contains a total of 11 sections, including Sect. 1 which is the introduction, and Sect. 2 which introduces some notation, and includes a summary of the proof of the main theorems.
Sections 3, 4 and 5 contain mostly generalities, appropriately translated to our context. More specifically, in Sect. 3, we recall some basic facts about the mod\(\ell \) Milnor Kring of fields. In Sect. 4, we recall the cohomological framework which allows us to translate back and forth between mod\(\ell \) abelianbycentral Galois groups and mod\(\ell \) Milnor Krings—this can be seen as a grouptheoretical formulation of the Merkurjev–Suslin Theorem [22]. Such cohomological results have seen a recent resurgence in [5, 10, 12, 35], especially in connection with the Merkurjev–Suslin Theorem [22] and/or the Bloch–Kato conjecture, which is now a highlycelebrated theorem due to VoevodskyRost et al. [31, 37, 39]. Nevertheless, the Merkurjev–Suslin Theorem is sufficient for the considerations in Sect. 4, as we summarize the appropriate results for our context in Theorem 4.2.
In Sect. 5, we recall the required results concerning the local theory in mod\(\ell \) abelianbycentral birational anabelian geometry. These results have been developed incrementally over the last several years by [4, 11, 21, 25, 26, 33, 36]. We summarize the applicable results for our context in Theorem 5.4.
The core of the paper begins in §6, where we discuss the mod\(\ell \) Milnor Ktheory of function fields. The ideas in this section are similar to [34, §3], although the results themselves refine loc. cit. somewhat. Perhaps the most important result in §6 is Corollary 6.4 which shows how to reconstruct a geometric subgroup given sufficiently many of its elements.
In Sect. 7, we summarize (see Theorem 7.1) the main results from Evans–Hrushovski [8, 9] and Gismatullin [14], translated appropriately to the context of the present paper. In Sect. 7 we also prove Corollary 7.4, which shows that the absolute Galois group \({{{\mathrm{Gal}}}_{k_0}}\) can be canonically identified with a Galois group of certain geometric lattices associated to Kk and \(K_0k_0\); this corollary will be used in a fundamental way in the proof of Theorem D.
In Sect. 8, we introduce the socalled essential branch locus, and the notion of an essentially unramified point. We use this concept of essential ramification in several technical results in coordination with the local theory, in order to ensure that certain divisorial valuations can be “detected” in the mod\(\ell \) setting.
In Sect. 9, we recall the notion of a general element, and introduce the notion of a stronglygeneral element. In this section we also recall the socalled BirationalBertini theorem for general elements. We also prove a BirationalBertini theorem for stronglygeneral elements, which uses the “yoga” of essential ramification in a fundamental way.
In Sect. 10, we give the detailed proof of Theorem D, and note that Theorem C follows from this by applying Theorem 4.2 from Sect. 4. Finally, in Sect. 11, we conclude the proofs of Theorems A and B by using Theorem C.
2 Notation and a summary
When we discuss other fields which are potentially unrelated to \(K_0k_0\) and/or Kk and which might have characteristic \(\ell \), we will use letters such as F, L, M, etc. The perfect closure of a field F will be denoted by \(F^i\). If \({{\mathrm{Char}}}F = p > 0\), then we will write \({\text {Frob}}_F\) for the usual Frobenius map on F, and we note that \({\text {Frob}}_{F^i}\) is an automorphism of \(F^i\). In order to keep the notation consistent, if \({{\mathrm{Char}}}F = 0\), then \({\text {Frob}}_F\) is defined to be the identity on F. Finally, the absolute Galois group of a field F will be denoted by \({{\mathrm{Gal}}}_F\), and the maximal pro\(\ell \) Galois group of F will be denoted by \(\mathcal {G}_F\).
For a valuation v of F, we will use the following standard notation associated with v. We denote the valuation ring of v by \(\mathcal {O}_v\) and the maximal ideal of \(\mathcal {O}_v\) is denoted by \(\mathfrak {m}_v\). We will also write \({\mathrm{U}_v {:}{=}\, \mathcal {O}_v^\times }\) for the vunits and \({\mathrm{U}_v^1 {:}{=}\, (1+\mathfrak {m}_v)}\) for the principal vunits. Finally, we write vF for the value group of v and Fv for the residue field of v. The residue map \(\mathcal {O}_v \twoheadrightarrow Fv\) will usually be denoted by \(t \mapsto \bar{t}\). If L is a subfield of F, we will abuse the notation and write vL resp. Lv for the value group resp. residue field of the restriction \(v_L\) of v to L.
The majority of this paper deals with \({\text {k}}^\mathrm{M}_*(K)\), the mod\(\ell \) Milnor Kring of K, the definition of which is recalled in Sect. 3. We note now that for a field F, one has \({\text {k}}^\mathrm{M}_1(F) = F^\times /\ell \), and that the canonical projection \(F^\times \twoheadrightarrow F^\times /\ell = {\text {k}}^\mathrm{M}_1(F)\) is denoted by \(x \mapsto \{x\}_F\).
 (1)
\(\mathbb {K}(S) {:}{=}\, \overline{k(S)} \cap K\) for the relative algebraicclosure of k(S) in K.
 (2)
\(\mathfrak {K}(S) {:}{=}\, \{\mathbb {K}(S)^\times \}_K\) for the image of \(\mathbb {K}(S)^\times \) in \({\text {k}}^\mathrm{M}_1(K)\).
 (1)
\({\mathfrak {U}_v {:}{=}\, \{\mathrm{U}_v\}_K}\) for the image of the vunits in \({\text {k}}^\mathrm{M}_1(K)\).
 (2)
\({\mathfrak {U}_v^1 {:}{=}\, \{\mathrm{U}_v^1\}_K}\) for the image of the principal vunits in \({\text {k}}^\mathrm{M}_1(K)\).
We will also identify the closed points of \(\mathbb {A}^r_\mathbf {t}\) resp. \(\mathbb {P}^r_\mathbf {t}\) with the set \(\mathbb {A}^r_\mathbf {t}(k) = k^r\) resp. \(\mathbb {P}^r_\mathbf {t}(k) = (k^{r+1} {{\smallsetminus }} \{0\})/k^\times \) of krational points. We will use affine coordinates \((a_1,\ldots ,a_r)\) to denote elements of \(\mathbb {A}^r_\mathbf {t}(k) = k^r\), and we will use homogeneous coordinates \((a_0:\cdots :a_r)\) to denote elements of \(\mathbb {P}^r_\mathbf {t}(k) = (k^{r+1} {{\smallsetminus }} \{0\})/k^\times \). In particular, we identify \(\mathbb {A}^r_\mathbf {t}(k) = k^r\) with the elements of the form \((1:a_1:\cdots :a_r)\) in \(\mathbb {P}^r_\mathbf {t}(k) = (k^{r+1} {{\smallsetminus }} \{0\})/k^\times \).
 (1)
First, reduce all the main theorems to Theorem D.
 (2)
Second, prove that any element of \(\underline{{{\mathrm{Aut}}}}^\mathrm{M}_\mathbf {a}({\text {k}}^\mathrm{M}_1(K))\) induces an automorphism of a certain lattice \({\mathbb {G}^*(Kk)}\) which is of geometric origin. This step is the most difficult and takes up the majority of the paper.
 (3)
Finally, we use an analogue of the Fundamental Theorem of Projective Geometry for this lattice \(\mathbb {G}^*(Kk)\), to deduce that any element of \(\underline{{{\mathrm{Aut}}}}^\mathrm{M}_\mathbf {a}({\text {k}}^\mathrm{M}_1(K))\) arises in a unique way from \({{{\mathrm{Gal}}}_{k_0}}\). This analogue of the fundamental theorem of projective geometry comes from the work of Evans–Hrushovski [8, 9] and Gismatullin [14], and it relies on the socalled groupconfiguration theorem from geometric stability theory.
2.1 Reduction to Theorem A
In the terminology introduced above, suppose that \(\mathcal {U}_X\) and \(\mathcal {U}_Y\) are birational systems for X resp. Y. Furthermore, suppose that \(\mathcal {U}_X\) dominates \(\mathcal {U}_Y\) in \(\mathcal {V}\). Note that any element \(\phi \) of \({{\mathrm{Aut}}}^\mathrm{c}(\pi ^a_\mathcal {V})\) defines an element of \(\phi _{\mathcal {U}_X} \in {{\mathrm{Aut}}}^\mathrm{c}(\pi ^a_{\mathcal {U}_X})\) and an element of \(\phi _{\mathcal {U}_Y} \in {{\mathrm{Aut}}}^\mathrm{c}(\pi ^a_{\mathcal {U}_Y})\) by restriction. The condition that \(\mathcal {U}_X\) dominates \(\mathcal {U}_Y\) implies the following property: If \(\phi _{\mathcal {U}_X}\) is defined by \(\tau \in {{{\mathrm{Gal}}}_{k_0}}\), then \(\phi _{\mathcal {U}_Y}\) is defined by \(\tau \) as well. The “5connectedness” assumption on \(\mathcal {V}\) is then used to reduce Theorem B to Theorem A.
2.2 Reduction to Theorem C
2.3 Reduction to Theorem D
2.4 The mod\(\ell \) geometric lattice
We show in Proposition 7.3 that the map \(\mathbb {K}(S) \mapsto \mathfrak {K}(S)\) (see the notation introduced above) induces an isomorphism of graded lattices \(\mathbb {G}^*(Kk) \cong \mathfrak {G}^*(Kk)\), where \(\mathbb {G}^*(Kk)\) is the lattice of relativelyalgebraically closed subextensions of Kk graded by transcendence degree. Thus, any element of \(\underline{{{\mathrm{Aut}}}}^\mathrm{M}_\mathbf {a}({\text {k}}^\mathrm{M}_1(K))\) will define an automorphism of \(\mathbb {G}^*(Kk)\) which fixes subextensions arising from \(K_0k_0\). We then use the results of Evans–Hrushovski [8, 9] and Gismatullin [14] to show that any such automorphism of \(\mathbb {G}^*(Kk)\) arises from some element of \({{{\mathrm{Gal}}}_{k_0}}\). See Theorem 7.1, Proposition 7.2 and Corollary 7.4 for more details.
2.5 Generic generators of \(\mathfrak {G}^*(Kk)\)
The idea of the proof is to “produce” elements of \(\mathfrak {G}^*(Kk)\), i.e. geometric subgroups of \({\text {k}}^\mathrm{M}_1(K)\), using the “given” data of the mod\(\ell \) Milnor Kring \({\text {k}}^\mathrm{M}_*(K)\) endowed with some extra structure which is compatible with all automorphisms in \(\underline{{{\mathrm{Aut}}}}^\mathrm{M}_\mathbf {a}({\text {k}}^\mathrm{M}_1(K))\), and also to ensure that this process is compatible with such automorphisms. The reconstruction process of \(\mathfrak {G}^*(Kk)\) relies on a certain “closure operation” called the Milnor Supremum, which takes place entirely in the ring \({\text {k}}^\mathrm{M}_*(K)\), and which takes in a set of geometric subgroups as an input and returns a geometric subgroup as an output.
The fact that this closure operation produces geometric subgroups follows from some explicit vanishing and nonvanishing results in \({\text {k}}^\mathrm{M}_*(K)\). The vanishing results say that \({\text {k}}^\mathrm{M}_*(K) = 0\) for \(* > {{\mathrm{tr.deg}}}(Kk)\), and this follows from wellknown cohomological dimension calculations of K and the Bloch–Kato conjecture, which is now a theorem of VoevodskyRost et al. [31, 37, 39]. The nonvanishing results say that there are “many” elements of \({\text {k}}^\mathrm{M}_1(K)\) which have nontrivial products. The “many” above refers to the fact that these nonvanishing results all involve some open condition on some model of Kk (or some subextension of Kk) which is usually the complement of some branch locus.
2.6 Fixing elements of \(\mathfrak {G}^*(Kk)\) which arise from \(K_0\)
The fact that an automorphism \(\sigma \in \underline{{{\mathrm{Aut}}}}^\mathrm{M}_\mathbf {a}({\text {k}}^\mathrm{M}_1(K))\) is compatible with the tuple \(\mathbf {a}\) implies that \(\sigma \) fixes the elements of \(\mathfrak {G}^1(Kk)\) which come from \(K_0\). Applying the “closure operation” described above shows that \(\sigma \) fixes all of the elements of \(\mathfrak {G}^*(Kk)\) which come from \(K_0\). However, this is still very far from what we need, because at this point there is absolutely nothing we can “construct/produce” which is moved around by the action of \({{{\mathrm{Gal}}}_{k_0}}\).
2.7 Fixing elements of \(K_0^\times \)
A key step in the proof is to show that any element of \(\underline{{{\mathrm{Aut}}}}^\mathrm{M}_\mathbf {a}({\text {k}}^\mathrm{M}_1(K))\) has a representative \(\sigma \in {{\mathrm{Aut}}}^\mathrm{M}({\text {k}}^\mathrm{M}_1(K))\) such that \(\sigma \{x\}_K = \{x\}_K\) for all \(x \in K_0^\times \). To show this, we introduce the concept of a stronglygeneral element of Kk, which is related to the concept of a general element from [27, 28] but has further assumptions. Another key input comes from the local theory in abelianbycentral birational anabelian geometry for function fields over algebraically closed fields. In this context, the local theory says that \(\sigma \) is compatible with quasidivisorial valuations. But using the previous step, one can show that \(\sigma \) is actually compatible with divisorial valuations. The literature concerning the local theory in abelianbycentral birational anabelian geometry is quite rich, and it includes the following works among others [4, 21, 25, 26, 33, 36]. See the introduction of [36] for a detailed overview of the history of the local theory.
To show that \(\sigma \in \underline{{{\mathrm{Aut}}}}^\mathrm{M}_\mathbf {a}({\text {k}}^\mathrm{M}_1(K))\) fixes elements from \(K_0^\times \), we first show this for elements of \(K_0^\times \) which are stronglygeneral in Kk, and this uses the local theory in an essential way. To deduce that \(\sigma \) fixes all elements arising from \(K_0^\times \), we prove a BirationalBertini type result for stronglygeneral elements, which shows that there are “sufficiently many” stronglygeneral elements in higherdimensional function fields.
2.8 The base case
By using our “closure operation” described above, in order to show that \(\sigma \in \underline{{{\mathrm{Aut}}}}^\mathrm{M}_\mathbf {a}({\text {k}}^\mathrm{M}_1(K))\) induces an automorphism of the lattice \(\mathfrak {G}^*(Kk)\), it suffices to show that \(\sigma \) induces a permutation of \(\mathfrak {G}^1(Kk)\), the set of 1dimensional geometric subgroups. Using the notation introduced above, a onedimensional geometric subgroup is a subgroup of \({\text {k}}^\mathrm{M}_1(K)\) which is of the form \(\mathfrak {K}(t)\) for some \(t \in K^\times {{\smallsetminus }} k^\times \). Note that every element \(t \in K = K_0 \otimes _{k_0} k\) can be written as a sum \(a_0 x_0 + \cdots + a_r x_r\) for some \(x_i \in K_0\) and \(a_i \in k\). The proof now proceeds by induction on the length r of such an expression. The case \(r = 0\) was discussed above, and so the base case for the induction is \(r = 1\).
The base case works as follows. Using the concept of essential ramification, we show that there are “many” elements of the form \(x_0 + a_1 x_1\) with \(x_i \in K_0\) and \(a_i \in k\) which are “acceptable” with respect to \(\sigma \). The term “acceptable” means that there exists some \(t \in K\) such that \(\sigma \) sends the geometric subgroup associated to \(x_0 + a_1 x_1\) to the geometric subgroup associated to t. As before, the term “many” is related to a precise open condition on a certain model of a subextension of Kk, with the condition being related to the essential branch locus.
Once we have “many” acceptable elements of the form \(x_0 + a_1 x_1\), we use the “yoga” of the “generic generators” mentioned above to show that all pairs \((t_0,t_1)\), with \(t_0 \in K_0\) and \(t_1 = a_0 x_0 + a_1 x_1\), \(x_i \in K_0\), \(a_i \in k\), are acceptable with respect to \(\sigma \) (acceptability is defined similarly for pairs as it was for elements of K). We then take appropriate intersections of certain twodimensional geometric subgroups to deduce that every element of the form \(a_0 x_0 + a_1 x_1\) is acceptable.
2.9 Inductive case
To conclude the proof, one proceeds by induction on r as above, with the inductive hypothesis being that every element of K of the form \(a_0 x_0 + \cdots + a_s x_s\) with \(s < r\), \(x_i \in K_0\), \(a_i \in k\), is acceptable with respect to \(\sigma \). The proof is now similar in nature to the proof of the base case. Indeed, first we show that pairs of certain elements are acceptable, then take intersections of certain twodimensional geometric subgroups to deduce that all elements of the form above are acceptable.
2.10 Concluding the proof
Indeed, any element \(\sigma \) in the kernel of the map above must fix all geometric subgroups of \({\text {k}}^\mathrm{M}_1(K)\). First, we show this implies that the restriction of \(\sigma \) to any stronglygeneral geometric subgroup looks like some element of \((\mathbb {Z}/\ell )^\times \cdot \mathbf {1}\). Finally, one uses a BirationalBertini type argument again to deduce that \(\sigma \) is indeed an element of \((\mathbb {Z}/\ell )^\times \cdot \mathbf {1}_{{\text {k}}^\mathrm{M}_1(K)}\). This thereby proves the injectivity of the map above, hence concluding the proof of Theorem D.
3 Milnor Ktheory
Note that \({\text {k}}^\mathrm{M}_*(F)\) is functorial in F. The notation \(\{\bullet ,\ldots ,\bullet \}_F\) will also be used to indicate this functoriality. Namely, if \(b_1,\ldots ,b_r \in F^\times \) are given, and \(F \hookrightarrow L\) is a field extension, then \(\{b_1,\ldots ,b_r\}_L\) denotes the image of \(\{b_1,\ldots ,b_r\}_F\) under the canonical map \({\text {k}}^\mathrm{M}_r(F) \rightarrow {\text {k}}^\mathrm{M}_r(L)\).
3.1 Purely inseparable extensions
We will frequently reduce some arguments concerning finite field extensions to the case where the extension is separable. This will be possible because a purelyinseparable extension of fields of characteristic \(\ne \ell \) induces an isomorphism on the mod\(\ell \) Milnor Kring which is also compatible with valuations, as the following two lemmas show.
Lemma 3.1
Let LF be a finite and purely inseparable extension of fields, such that \({{\mathrm{Char}}}F \ne \ell \). Then the canonical map \({\text {k}}^\mathrm{M}_*(F) \rightarrow {\text {k}}^\mathrm{M}_*(L)\) is an isomorphism for all \(* \ge 0\).
Proof
Put \(p = {{\mathrm{Char}}}F\) and assume that \(p > 0\). Since L is finite and purely inseparable over F, one has \(L \subset F^{1/p^n}\) for sufficiently large n. Since p is invertible in \(\mathbb {Z}/\ell \), the canonical map \({\text {k}}^\mathrm{M}_*(F) \rightarrow {\text {k}}^\mathrm{M}_*(F^{1/p^n})\) is an isomorphism. As this map factors through \({\text {k}}^\mathrm{M}_*(L)\), we deduce that the map \({\text {k}}^\mathrm{M}_*(F) \rightarrow {\text {k}}^\mathrm{M}_*(L)\) is injective. To deduce that \({\text {k}}^\mathrm{M}_*(F) \rightarrow {\text {k}}^\mathrm{M}_*(L)\) is also surjective, it suffices to prove that \({\text {k}}^\mathrm{M}_*(L) \rightarrow {\text {k}}^\mathrm{M}_*(F^{1/p^n})\) is injective.
For any \(\eta \) in the kernel of \({\text {k}}^\mathrm{M}_*(L) \rightarrow {\text {k}}^\mathrm{M}_*(F^{1/p^n})\), there exists some intermediate extension M of \(F^{1/p^n}L\) such that ML is finite and such that \(\eta \) is in the kernel of \({\text {k}}^\mathrm{M}_*(L) \rightarrow {\text {k}}^\mathrm{M}_*(M)\). But such a subextension ML is purely inseparable, so the argument above shows that \(\eta = 0\). Thus \({\text {k}}^\mathrm{M}_*(L) \rightarrow {\text {k}}^\mathrm{M}_*(F^{1/p^n})\) is injective, as required. \(\square \)
Lemma 3.2
Let (L, w)(F, v) be a finite and purely inseparable extension of valued fields, such that \({{\mathrm{Char}}}F \ne \ell \). Then the canonical map \({\text {k}}^\mathrm{M}_1(F) \rightarrow {\text {k}}^\mathrm{M}_1(L)\) restricts to an isomorphism \({\{\mathrm{U}_v\}_F \xrightarrow {\cong } \{\mathrm{U}_w\}_L}\).
Proof
3.2 Tame symbols
We will primarily use tame symbols to prove that the mod\(\ell \) Milnor Kring of a function field contains many nontrivial elements. Most such “nonvanishing” results will essentially follow from the following fact concerning the field of Laurent series.
Fact 3.3
Proof
4 Galois cohomology
In this section, we recall the basic framework which allows us to translate back and forth between the “Galois” context (i.e. mod\(\ell \) abelianbycentral Galois groups) and the “Milnor” context (i.e. mod\(\ell \) Milnor Krings) by using Kummer theory. This theory is moreorless well known, as it follows from the fact that \({\text {H}}^2\) of a pro\(\ell \) group is “dual” to the relations in a minimal free presentation of the group, while the cup product in \({\text {H}}^*\) is “dual” to the commutator \([\bullet ,\bullet ]\) as defined in Sect. 1.4.
The essential calculations concerning cup products and commutators were first carried out by Labute [17] (see also [23, §3.9]). These calculations have seen a recent resurgence of interest in [5, 10, 12, 35], especially in connection with the Merkurjev–Suslin Theorem [22] and the Bloch–Kato conjecture, which is now a theorem of VoevodskyRost et al. [31, 37, 39]. The Merkurjev–Suslin Theorem is sufficient for our considerations here. In fact, the discussion in this section can be seen as a summary of [34, §8], appropriately translated to the context of this paper.
Finally, recall that for a field F, we let \(\mathcal {G}_F\) denote the maximal pro\(\ell \) quotient of \({{\mathrm{Gal}}}_F\). Most of the general results we say in this section hold true for general fields F such that \({{\mathrm{Char}}}F \ne \ell \) and \(\mu _\ell \subset F\). However, in order to simplify the discussion, we will restrict our attention to the function field Kk which is the main focus of the paper.
4.1 Generalities on cohomology of pro\(\ell \) groups
Fact 4.1
 (1)
The preimage of \({\text {H}}^2(\mathcal {G}^a)_{\mathrm{dec}}\) under the map \(d_2 : {\text {H}}^1(\mathcal {G}^{(2)})^\mathcal {G}\rightarrow {\text {H}}^2(\mathcal {G}^a)\) is precisely \({\text {H}}^1(\mathcal {G}^{(2)}/\mathcal {G}^{(3)})\).
 (2)
The cup product induces a canonical isomorphism \(\wedge ^2({\text {H}}^1(\mathcal {G}^a)) \rightarrow {\text {H}}^2(\mathcal {G}^a)/\beta {\text {H}}^1(\mathcal {G}^a)\).
 (3)Identify \({\text {H}}^2(\mathcal {G}^a)/\beta {\text {H}}^1(\mathcal {G}^a)\) with \(\wedge ^2({\text {H}}^1(\mathcal {G}^a))\) using the isomorphism from (2) above. Then the mapis \(\mathbb {Z}/\ell \)dual (hence Pontryagindual) to the map$$\begin{aligned}{}[\bullet ,\bullet ] : \widehat{\wedge }^2(\mathcal {G}^a) \rightarrow \mathcal {G}^{(2)}/\mathcal {G}^{(3)} \end{aligned}$$$$\begin{aligned} {\text {H}}^1(\mathcal {G}^{(2)}/\mathcal {G}^{(3)}) \xrightarrow {d_2} {\text {H}}^2(\mathcal {G}^a) \twoheadrightarrow {\text {H}}^2(\mathcal {G}^a)/\beta {\text {H}}^1(\mathcal {G}^a) = \wedge ^2({\text {H}}^1(\mathcal {G}^a)). \end{aligned}$$
Proof
See [34, Lemma 8.2] for assertion (1). Assertion (2) follows from the Künneth formula along with the fact that \(\mathcal {G}^a\) is isomorphic to a direct power of \(\mathbb {Z}/\ell \); see [34, Fact 8.1] and the surrounding discussion for more details.
Assertion (3) is the standard “duality” between the commutator and the cupproduct. This “duality” has been wellknown for some time (see e.g. [23, Proposition 3.9.13]), but [35, Theorem 2] can also be used as a reference. See [34, Fact 8.3] for more details. \(\square \)
4.2 Kummer theory
4.3 Galois versus Milnor
With this discussion, we may now present the main theorem of this section which allows us to pass back and forth between the Galoissetting and the Milnorsetting. The following theorem is essentially a grouptheoretical interpretation of the Merkurjev–Suslin Theorem [22].
Theorem 4.2
 (1)
Let \(\phi \in {{\mathrm{Aut}}}({\text {k}}^\mathrm{M}_1(K))\) be given and consider \(\phi ^* \in {{\mathrm{Aut}}}(\mathcal {G}_K^a)\) the Kummerdual of \(\phi \). Then one has \(\phi \in {{\mathrm{Aut}}}^\mathrm{M}({\text {k}}^\mathrm{M}_1(K))\) if and only if \(\phi ^* \in {{\mathrm{Aut}}}^\mathrm{c}(\mathcal {G}_K^a)\).
 (2)
The canonical isomorphism \(\mathcal {K}: {{\mathrm{Aut}}}(\mathcal {G}_K^a) \xrightarrow {\cong } {{\mathrm{Aut}}}({\text {k}}^\mathrm{M}_1(K))\) restricts to an isomorphism \(\mathcal {K}: {{\mathrm{Aut}}}^\mathrm{c}(\mathcal {G}_K^a) \xrightarrow {\cong } {{\mathrm{Aut}}}^\mathrm{M}({\text {k}}^\mathrm{M}_1(K))\).
 (3)
The canonical isomorphism \(\mathcal {K}: \underline{{{\mathrm{Aut}}}}(\mathcal {G}_K^a) \xrightarrow {\cong } \underline{{{\mathrm{Aut}}}}({\text {k}}^\mathrm{M}_1(K))\) restricts to an isomorphism \(\mathcal {K}: \underline{{{\mathrm{Aut}}}}^\mathrm{c}(\mathcal {G}_K^a) \xrightarrow {\cong } \underline{{{\mathrm{Aut}}}}^\mathrm{M}({\text {k}}^\mathrm{M}_1(K))\) which is compatible with \(\rho _{k_0}\).
Concerning the proof of Theorem 4.2, the implication \((1) \Rightarrow (2)\) is clear, while the implication \((2) \Rightarrow (3)\) follows from the observations about cyclotomic twists made above. It therefore remains to prove assertion (1), which was essentially already proven in [34, Theorem 8.6]; alternatively, one can deduce (1) from the main results of [35]. We give a summary of the proof of Theorem 4.2 below. First, we need to recall some calculations in Galois cohomology.
Fact 4.3
 (1)
The Bockstein morphism \(\beta : {\text {H}}^1({{\mathrm{Gal}}}_K) \rightarrow {\text {H}}^2({{\mathrm{Gal}}}_K)\) is trivial.
 (2)
The inflation map \({\text {H}}^*(\mathcal {G}_K) \rightarrow {\text {H}}^*({{\mathrm{Gal}}}_K)\) is an isomorphism for \(* = 0,1,2\).
Proof
Proof of Theorem 4.2
First, since \(\mu _\ell \subset K\), we may simplify the notation and choose a fixed isomorphism \(\mu _\ell \cong \mathbb {Z}/\ell \) of \({{\mathrm{Gal}}}_K\)modules. In particular, we may identify \({\text {k}}^\mathrm{M}_1(K)\) with \({\text {H}}^1(\mathcal {G}_K^a) = {\text {H}}^1(\mathcal {G}_K)\) via the Kummer pairing, and \({\text {k}}^\mathrm{M}_2(K)\) with \({\text {H}}^2(\mathcal {G}_K)\) via the Merkurjev–Suslin Theorem [22] and Fact 4.3(2).
Next, recall from Fact 4.3 that the Bockstein morphism \(\beta : {\text {H}}^1(\mathcal {G}_K) \rightarrow {\text {H}}^2(\mathcal {G}_K)\) is trivial. Hence, the inflation map \({\text {H}}^2(\mathcal {G}_K^a) \rightarrow {\text {H}}^2(\mathcal {G}_K)\) factors through \({\text {H}}^2(\mathcal {G}_K^a)/\beta {\text {H}}^1(\mathcal {G}_K^a)\), which is isomorphic to \(\wedge ^2({\text {k}}^\mathrm{M}_1(K))\) by Fact 4.1 and our identification \({\text {H}}^1(\mathcal {G}_K^a) = {\text {k}}^\mathrm{M}_1(K)\) above.
5 The local theory
In this section, we recall the required results from the Local Theory in “almostabelian” anabelian geometry for function fields over algebraically closed fields. All such results have been generally stated for abelianbycentral Galois groups in the literature. But since we work primarily with the “Kummer dual” of this context, i.e. with the mod\(\ell \) Milnor Kring, we will need to translate these results to the context of Milnor Ktheory via Theorem 4.2.
5.1 Minimized Galois theory and Cpairs
 (1)
\(\sigma ,\tau \) forms a Cpair in \(\mathfrak {g}(F_1)\).
 (2)
\(\phi \sigma ,\phi \tau \) forms a Cpair in \(\mathfrak {g}(F_2)\).
Fact 5.1
 (1)
\(\sigma ,\tau \) form a Cpair.
 (2)
One has \([\sigma ,\tau ] = 0\).
5.2 Minimized decomposition theory
For \(\sigma \in {{\mathrm{D}}}_v\), considered as a homomorphism \({\sigma : F^\times /\mathrm{U}_v^1 \rightarrow \mathbb {Z}/\ell }\), we write \(\sigma _v\) for the restriction of \(\sigma \) to \(\mathrm{U}_v/\mathrm{U}_v^1 = Fv^\times \). In particular, the map \(\sigma \mapsto \sigma _v\) yields a canonical morphism \({{\mathrm{D}}}_v \rightarrow \mathfrak {g}(Fv)\).
Fact 5.2
 (1)
The canonical map \(\sigma \mapsto \sigma _v : {{\mathrm{D}}}_v \rightarrow \mathfrak {g}(Fv)\) induces an isomorphism \({{\mathrm{D}}}_v/{{\mathrm{I}}}_v \cong \mathfrak {g}(Fv)\).
 (2)
One has \({{\mathrm{I}}}_v \subset {{\mathrm{I}}}_{w \circ v} \subset {{\mathrm{D}}}_{w \circ v} \subset {{\mathrm{D}}}_v\).
 (3)
Identifying \({{\mathrm{D}}}_v/{{\mathrm{I}}}_v\) with \(\mathfrak {g}(Fv)\) as in (1), one has \({{\mathrm{D}}}_{w \circ v}/{{\mathrm{I}}}_v = {{\mathrm{D}}}_w\) and \({{\mathrm{I}}}_{w \circ v}/{{\mathrm{I}}}_v = {{\mathrm{I}}}_w\).
Proof
Assertion (2) is clear, while assertions (1) and (3) are [33, Lemma 2.1(1)(2)]. \(\square \)
It turns out that the Cpair condition in the minimized decomposition group of a valuation can actually be completely determined from the residue field. We summarize this property in the following fact.
Fact 5.3
 (1)
\(\sigma ,\tau \) form a Cpair in \(\mathfrak {g}(F)\).
 (2)
\(\sigma _v,\tau _v\) form a Cpair in \(\mathfrak {g}(Fv)\).
Proof
See [33, Lemma 2.1(3)]. \(\square \)
5.3 Quasidivisorial valuations
 (1)
vK contains no nontrivial \(\ell \)divisible convex subgroups.
 (2)
vK / vk is isomorphic to \(\mathbb {Z}\) as an abstract group.
 (3)
\({{\mathrm{tr.deg}}}(Kk)1 = {{\mathrm{tr.deg}}}(Kvkv)\).
The main results concerning the local theory (in the abelianbycentral setting) state that the minimized inertia and decomposition groups of rquasidivisorial valuations are preserved under the action of elements of \({{\mathrm{Aut}}}^\mathrm{c}(\mathcal {G}_K^a)\), and that the partially ordered structure of the associated valuations is preserved as well. We present the following theorem merely as a translation of these results which replaces \({{\mathrm{I}}}_v\) resp. \({{\mathrm{D}}}_v\) with \(\mathfrak {U}_v\) resp. \(\mathfrak {U}_v^1\), and \({{\mathrm{Aut}}}^\mathrm{c}(\mathcal {G}_K^a)\) with \({{\mathrm{Aut}}}^\mathrm{M}({\text {k}}^\mathrm{M}_1(K))\). The fact that elements of \({{\mathrm{Aut}}}^\mathrm{c}(\mathcal {G}_K^a)\) preserve minimized decomposition/inertia groups of quasidivisorial valuations was first proven by Pop in [26], which uses the Galois context and the usual inertia/decomposition groups. Nevertheless, the key arguments from loc. cit. can be made to work in the minimized context as well. In any case, in order to keep things precise, we will instead use the reference [33] which uses the minimized context exclusively.
Theorem 5.4
 (1)Let v be an rquasidivisorial valuation of Kk, with \(r < {{\mathrm{tr.deg}}}(Kk)\). Then there exists a unique rquasidivisorial valuation \(v^\sigma \) of Kk such that$$\begin{aligned} \sigma \mathfrak {U}_v = \mathfrak {U}_{v^\sigma }, \ \ \sigma \mathfrak {U}_v^1 = \mathfrak {U}_{v^\sigma }^1. \end{aligned}$$
 (2)
Let v be an rquasidivisorial valuation of Kk and w an squasidivisorial valuation of Kk, with \(r,s < {{\mathrm{tr.deg}}}(Kk)\), and let \(v^\sigma ,w^\sigma \) be as in (1) above. Then v is a coarsening of w if and only if \(v^\sigma \) is a coarsening of \(w^\sigma \).
Proof
Before we begin the proof, we choose an isomorphism \(\mu _\ell \cong \mathbb {Z}/\ell \) which will be fixed throughout, and which allows us to identify \(\mathcal {G}_K^a\) with \(\mathfrak {g}(K)\) by Kummer theory.
Proof of (1):
Recall from Theorem 4.2(1) that \(\sigma ^*\) is compatible with \(\mathcal {R}\), hence \(\sigma ^*\) is compatible with Cpairs by Fact 5.1. The proof now proceeds by induction on r. The base case \(r = 1\) follows immediately from [33, Theorem D] and/or [26, Theorem 1].
Proof of (2):
Since vK and wK contain no nontrivial \(\ell \)divisible convex subgroups, it follows from [33, Lemma 3.1] or [36, Lemma 3.4] that v is a coarsening of w if and only if \({{\mathrm{I}}}_v \subset {{\mathrm{I}}}_w\). By assertion (1), this is true if and only if \({{\mathrm{I}}}_{v^\sigma } \subset {{\mathrm{I}}}_{w^\sigma }\), which is similarly equivalent to \(v^\sigma \) being a coarsening of \(w^\sigma \). This observation also implies that \(v^\sigma \) is uniquely determined by v and by \(\sigma \). \(\square \)
5.4 Divisorial valuations
It turns out (using the valuative criterion for properness) that any divisorial valuation indeed arises from some Weilprimedivisor on some normal model of Kk. More generally, any rdivisorial valuation arises from a flag of Weilprimedivisors on some normal model. See [24, §5] for more details on this.
In general, it is still a major open question to determine which rquasidivisorial valuations are actually rdivisorial, using the grouptheoretical structure of \(\mathcal {G}_K^c\) resp. the ring structure of \({\text {k}}^\mathrm{M}_*(K)\). Nevertheless, in our context, we can use geometric subgroups (as defined in Sect. 2) which arise from \(K_0\) to distinguish the rdivisorial valuations among the rquasidivisorial ones. See also [26, Theorem 19] for a related result which uses a similar argument.
Proposition 5.5
 (1)
The valuation v is an rdivisorial valuation of Kk.
 (2)
There exists some \(t \in K_0 {\smallsetminus } k_0\) such that the intersection \(\mathfrak {K}(t) \cap \mathfrak {U}_v^1\) is finite.
Proof
First of all, note that \(KK_0\) and \(kk_0\) are both algebraic extensions. Thus, the associated residue extensions, \(KvK_0v\) and \(kvk_0v\), are also algebraic. Moreover, since k is algebraically closed, the residue field kv is also algebraically closed. In particular, if \({{\mathrm{tr.deg}}}(Kvkv) \ge 1\), then \(K_0v\) cannot be contained in kv, and therefore there exist (many) elements t in \({\mathrm{U}_v \cap K_0^\times }\) whose image \(\bar{t}\) in Kv is transcendental over kv.
Conversely, assume that v is not rdivisorial, hence v is nontrivial on k, and let \(t \in K_0 {\smallsetminus } k_0\) be given. We must show that \(\mathfrak {K}(t) \cap \mathfrak {U}_v^1\) is infinite. By replacing t with \(t^{1}\) if needed, we may assume, without loss of generality, that \(v(t) \ge 0\).
6 Milnor Ktheory of function fields
In this section, we will prove some important vanishing and nonvanishing results for the mod\(\ell \) Milnor Kring of a function field. In particular, we will show that a significant portion of the “algebraicindependence” structure of a function field K over an algebraically closed field k is encoded in the mod\(\ell \) Milnor Kring of K.
6.1 Vanishing
We begin by recalling the following fact which follows from some wellknown cohomological dimension bounds, combined with the highly celebrated VoevodskyRost Theorem [31, 37, 39].
Fact 6.1
One has \({\text {k}}^\mathrm{M}_s(K) = 0\) for all \(s > {{\mathrm{tr.deg}}}(Kk)\).
Proof
6.2 Nonvanishing
The nonvanishing results in mod\(\ell \) Milnor Ktheory of function fields follow the “yoga” that algebraically independent elements should have nontrivial products in Milnor Ktheory. This turns out to be true, but with various exceptions which arise from modding out by \(\ell \)th powers. Nevertheless, it turns out that these exceptions can be avoided because they are related to some ramification phenomena which are concentrated in codimension one.
Lemma 6.2
Proof
By extending \(t_1,\ldots ,t_r\) to a transcendence base \(\mathbf {t}= (t_1,\ldots ,t_d)\) for Kk, we may assume without loss of generality that \(r = d = {{\mathrm{tr.deg}}}(Kk)\). Let \(X = \mathbb {A}^d_\mathbf {t}\) denote affine dspace over k with parameters \(\mathbf {t}\), so that \(k(\mathbf {t})\) is canonically identified with the function field of X. Let \(K'\) be the maximal separable subextension of \(Kk(\mathbf {t})\), and let Y denote the normalization of X in \(K'\).
Then \(Y \rightarrow X\) is a finite separable (possibly branched) cover of kvarieties. Since the branch locus of \(Y \rightarrow X\) has codimension one, there exists a nonempty open subset U of X such that Y is unramified over U.
The following proposition is our second main “nonvanishing” result, and it will be crucial in describing geometric subgroups of \({\text {k}}^\mathrm{M}_1(K)\). This will play a primary role in the reconstruction of the “geometric lattice” in the proof of the main theorems.
Proposition 6.3
Proof
Put \(d = {{\mathrm{tr.deg}}}(Kk)\) and \(s {:}{=}\, dr\). Let \(\mathbf {t}= (t_1,\ldots ,t_r,y_1,\ldots ,y_s)\) be a transcendence base of Kk extending \(t_1,\ldots ,t_r\). Let \(K'\) denote the maximal separable subextension of \(Kk(\mathbf {t})\). By Lemma 3.1, there exists some \(z' \in K'\) such that \(\{z'\}_K = \{z\}_K\). By replacing z with such a \(z'\), we may assume without loss of generality that \(z \in K'\).
Let \(F'\) denote the relative algebraic closure of \(k(t_1,\ldots ,t_r)\) in \(K'\). In particular, note that \(F'\) is separable over \(k(t_1,\ldots ,t_r)\) and that \(F {:}{=}\, \mathbb {K}(t_1,\ldots ,t_r)\) is purely inseparable over \(F'\). Therefore, one has \(\{z\}_{K'} \notin \{(F')^\times \}_{K'}\) by Lemma 3.1. This implies that the field \(F'\) is relatively algebraically closed also in \(K'(\root \ell \of {z}) =: L\). And since L is separable over \(k(\mathbf {t})\), it follows that L is regular over \(F'\).
Let B denote the normalization of \(\mathbb {A}^r_{t_1,\ldots ,t_r}\) in \(F'\). Moreover, we let \(X_0\) denote \(\mathbb {A}^s_{B;y_1,\ldots ,y_s}\), an affine sspace over B with coordinates \((y_1,\ldots ,y_s)\). Finally, let \(X_1\) denote the normalization of \(X_0\) in \(K'\), and let \(X_2\) denote the normalization of \(X_0\) in L.
 (1)
The fibers of \(X_i \rightarrow B\), \(i = 0,1,2\), over points of V are all geometrically integral. This is an open condition on B since \(k(X_i)\) is regular over \(F' = k(B)\), hence \(X_i \rightarrow B\) has generically geometrically integral fibers.
 (2)
Any point of V is unramified over \(\mathbb {A}^r_{t_1,\ldots ,t_r}\). This is an open condition on B because the extension \(F'k(t_1,\ldots ,t_r)\) is finite and separable, hence the ramification locus of \(B \rightarrow \mathbb {A}^r_{t_1,\ldots ,t_r}\) has codimension one.
 (3)
For all \(x \in V\), the function z is regular and nonzero on the generic point of \((X_1)_x\), the fiber of \(X_1 \rightarrow B\) over x. This is clearly an open condition on B, since the support of z has codimension one in \(X_1\), and \(X_0\) is flat over B of relative dimension s.
 (4)For all \(x \in V\), letting \(\bar{z}\) denote the image of z in \(k((X_1)_x)\), one hasThis is an open condition on B because \(L = K'[\root \ell \of {z}]\), hence for a sufficiently small nonempty affine open subset of \(X_1\), say \({{\mathrm{Spec}}}A\), one has \(z \in A\) and the normalization of A in L is precisely \(A[\root \ell \of {z}]\).$$\begin{aligned} k((X_2)_x) = k((X_1)_x)[\root \ell \of {\bar{z}}]. \end{aligned}$$
 (5)
For all \(x \in V\), one has \([k(X_2):k(X_0)] = [k((X_2)_x):k((X_0)_x)]\), and therefore one also has \(\ell = [k((X_2)_x):k((X_1)_x)]\) and \([k(X_1):k(X_0)] = [k((X_1)_x):k((X_0)_x)]\). This is an open condition on B by a standard application of the BertiniNoether theorem (cf. [13, Proposition 9.4.3]).
 (6)
For all \(x \in V\), letting \(\eta ^i_x\) denote the generic point of the fiber \((X_i)_x\) over x, the point \(\eta ^1_x\) is unramified over \(\eta ^0_x\). This is clearly an open condition on B, since the branch locus of \(X_1 \rightarrow X_0\) is a proper closed subset of \(X_0\), and \(X_0\) is flat over B.
Corollary 6.4
Proof
6.3 Milnor dimension
Fact 6.5
Let S be a subset of K. Then one has \(\dim ^\mathrm{M}(\mathfrak {K}(S)) = {{\mathrm{tr.deg}}}(\mathbb {K}(S)k)\).
6.4 Milnorsupremum
Lemma 6.6
Proof
Put \(d' {:}{=}\, {{\mathrm{tr.deg}}}(\mathbb {K}(S)k)\) and \(d {:}{=}\, \dim ^\mathrm{M}(\mathfrak {K})\). First we show that \(d = d'\). The inequality \(d \le d'\) follows from Fact 6.1 since \(\mathfrak {K}\subset \mathfrak {K}(S)\). Conversely, we note that there exist \(s_1,\ldots ,s_{d'} \in S\) which are algebraically independent over k. Thus, \(d \ge d'\) by Lemma 6.2.
7 Geometric lattices
 (1)
Every subset S of \(\mathcal {L}^*\) has a greatest lower bound \(\wedge S\) and a least upper bound \(\vee S\) in \(\mathcal {L}^*\). Namely, the partially ordered set \((\mathcal {L}^*,\le )\) is a complete lattice.
 (2)
If \(A \in \mathcal {L}^s\) and \(B \in \mathcal {L}^t\) are such that \(A < B\), then one has \(s < t\). Namely, the partial ordering \(\le \) is strictly compatible with the grading of \(\mathcal {L}^*\).
7.1 Relative algebraic closure
Assume now that F is a perfect field and that L is an extension of finite transcendence degree over F, such that F is relatively algebraically closed in L. We denote by \({\mathbb {G}^*(LF)}\) the graded lattice of relatively algebraically closed subextensions of LF, graded by transcendence degree.
More precisely, we let \({\mathbb {G}^s(LF)}\) denote the collection of relatively algebraically closed subextensions of LF of transcendence degree s over F. Then one has \({\mathbb {G}^*(LF) = \coprod _{s \ge 0} \mathbb {G}^s(LF)}\), and the ordering on \({\mathbb {G}^*(LF)}\) is the partial ordering given by inclusion of subextensions of LF.
Theorem 7.1
Proof
The only part of this theorem which doesn’t follow directly from [14, Theorem 4.2] is that loc. cit. uses the combinatorial geometry associated to \(\mathbb {G}^*(LF)\) as its basic structure, whereas we use the whole lattice \(\mathbb {G}^*(LF)\). These two formulations are easily seen to be equivalent, as follows.
7.2 The Galois action on the geometric lattice
We now consider the canonical Galois action of \({{{\mathrm{Gal}}}_{k_0}}\) on \({\mathbb {G}^*(Kk)}\). First, recall that \({{{\mathrm{Gal}}}_{k_0}}\) can be canonically identified with the Galois group \({{\mathrm{Gal}}}(KK_0)\), since \(K_0k_0\) is a regular function field. In particular, \({{{\mathrm{Gal}}}_{k_0}}\) acts on the (relatively algebraically closed) subextensions of Kk in the obvious way, and this action preserves transcendence degree. In other words, \({{{\mathrm{Gal}}}_{k_0}}\) acts on the graded lattice \(\mathbb {G}^*(Kk)\).
Proposition 7.2
Proof
We first use Theorem 7.1 to make the following observation: Any automorphism \(\Phi \) of \(\mathbb {G}^*(Kk)\) arises from an automorphism \(\tilde{\Phi }\) of \(K^i\) which is unique upto composition with some power of \({\text {Frob}}_{K^i}\). Assume moreover that \(\Phi \) restricts to an automorphism \(\Phi _0\) of \(\mathbb {G}^*(K_0k_0)\). Then Theorem 7.1 implies that \(\Phi _0\) arises from an automorphism \(\tilde{\Phi }_0\) of \(K_0^i\) which is unique upto composition with some power of \({\text {Frob}}_{K_0^i}\). It therefore follows from the functoriality of the situation that this automorphism \(\tilde{\Phi }\) restricts to an automorphism of \(K_0^i\), which is precisely \(\tilde{\Phi }_0\), upto composition with a power of \({\text {Frob}}_{K_0^i}\).
7.3 The mod\(\ell \) geometric lattice
Proposition 7.3
 (1)
The set \(\mathfrak {G}^*(Kk)\), endowed with the ordering by inclusion in \({\text {k}}^\mathrm{M}_1(K)\) and the grading by \(\dim ^\mathrm{M}\), is a graded lattice.
 (2)
The map \(\mathbb {K}(S) \mapsto \mathfrak {K}(S)\) induces a canonical isomorphism of graded lattices \(\mathbb {G}^*(Kk) \rightarrow \mathfrak {G}^*(Kk)\).
 (3)
For a collection \((\mathfrak {K}_i)_i\) of elements of \(\mathfrak {G}^*(Kk)\), the leastupperbound of \((\mathfrak {K}_i)_i\) in the lattice \(\mathfrak {G}^*(Kk)\) is precisely the Milnorsupremum \({\sup }^\mathrm{M}(\bigcup _i \mathfrak {K}_i)\).
Proof
This proves, in particular, that the map \(\mathbb {G}^*(Kk) \rightarrow \mathfrak {G}^*(Kk)\) is an isomorphism of partially ordered sets, while the compatibility with the grading was already noted above. Assertion (3) follows immediately from assertion (2) and Lemma 6.6. \(\square \)
7.4 Galois action on the mod\(\ell \) lattice
Corollary 7.4
8 Essentially unramified points and fibers
A key difficulty which arises by working in the mod\(\ell \) context is that the presence of ramification can make certain valuations “invisible.” More precisely, suppose that t is a nonconstant element of K. While every divisorial valuation of \(\mathbb {K}(t)k\) is the restriction of some divisorial valuation of Kk, there may be some divisorial valuations v of \(\mathbb {K}(t)k\) such that \(\{\mathrm{U}_v\}_K\) is not of the form \(\mathfrak {U}_w \cap \mathfrak {K}(t)\) for any divisorial valuation w of Kk. Dualizing using the Kummer pairing, there may be some minimized inertia subgroups of \(\mathcal {G}_{\mathbb {K}(t)}^a\) arising from divisorial valuations of \(\mathbb {K}(t)k\) which are not the image of any minimized inertia subgroup of \(\mathcal {G}_K^a\) that arise from a divisorial valuation of Kk. This “difficulty” is clearly intimately tied to ramification (specifically, ramification indices which are divisible by \(\ell \)). In this section, we introduce some general conditions which will suffice to prevent such problems.
8.1 The flag associated to regular parameters
Let \((f_1,\ldots ,f_r)\) be a system of regular parameters for the (maximal ideal of the) regular local ring \(\mathcal {O}_{X,x}\), and put \(L {:}{=}\, k(X)\). We will abuse the terminology and also say that \((f_1,\ldots ,f_r)\) is a system of regular parameters at x in X.
Fact 8.1
 (1)
For all \(i = 1,\ldots ,r\), one has \(\mathrm{U}_{v_{i1}}/\mathrm{U}_{v_{i}} \cong \mathbb {Z}\). Moreover, one has \(f_i \in \mathrm{U}_{v_{i1}}\) and its image in \(\mathrm{U}_{v_{i1}}/\mathrm{U}_{v_i}\) is a generator.
 (2)
For all \(i = 1,\ldots ,r\), one has \(\{\mathrm{U}_{v_{i1}}\}_{L}/\{\mathrm{U}_{v_{i}}\}_{L} \cong \mathbb {Z}/\ell \). Moreover, one has \(\{f_i\}_{L} \in \{\mathrm{U}_{v_{i1}}\}_{L}\) and its image in \(\{\mathrm{U}_{v_{i1}}\}_{L}/\{\mathrm{U}_{v_{i}}\}_{L}\) is a generator.
8.2 \(\ell \)unramified prolongations
The notion of an \(\ell \)unramified prolongation and Fact 8.1 lead to the following particularly useful observation. If the flag \((v_1,\ldots ,v_r)\) arises from a system of regular parameters \((f_1,\ldots ,f_r)\) at a regular point \(x \in X\) as described in Sect. 8.1, and \((w_1,\ldots ,w_r)\) is an \(\ell \)unramified prolongation of \((v_1,\ldots ,v_r)\) to K, then \(\{f_i\}_K\) is a generator of \(\mathfrak {U}_{w_{i1}}/\mathfrak {U}_{w_i} \cong \mathbb {Z}/\ell \) for all \(i = 1,\ldots ,r\) by Fact 8.1(2).
8.3 The essential branch locus
Let X be a regular kvariety, and suppose that K is a finite extension of k(X). Let \(K'\) denote the maximal separable subextension of Kk(X), and let \(Y \rightarrow X\) denote the normalization of X in \(K'\). Thus, \(Y \rightarrow X\) is a finite separable (possibly branched) cover of kvarieties. The branch locus of this finite separable cover \(Y \rightarrow X\) will be called the essential branch locus of X in K. Recall that the essential branch locus of X in K is actually a divisor of X. However, we will only be interested in the support of this divisor. Namely, we will consider the essential branch locus of X in K only as a closed subvariety of X which has codimension one.
Along the same lines, if x is a (schemetheoretic) point of X, we say that x is essentially unramified in K if x is not contained in the essential branch locus of X in K. Similarly, if Z is an integral closed subvariety of X, we say that Z is essentially unramified in K if the generic point of Z is essentially unramified in K. Since the essential branch locus of X in K is closed in X, we note that Z is essentially unramified in K if and only if Z is not contained in the essential branch locus of X in K.
Finally, let \(\mathbf {w}= (w_1,\ldots ,w_r)\) be any prolongation of \(\mathbf {w}'\) to K. Then \(\mathbf {w}\) is a flag of divisorial valuations, and it follows from Lemma 3.2 that \(\mathbf {w}\) is an \(\ell \)unramified prolongation of \(\mathbf {v}\) as defined in Sect. 8.2. We summarize this discussion for later use in the following fact.
Fact 8.2
 (1)
The system \((f_1,\ldots ,f_r)\) is a system of regular parameters at y.
 (2)
Any prolongation \(\mathbf {w}\) to K of the flag of divisorial valuations of \(k(Y) = K'\) associated to \((f_1,\ldots ,f_r)\), considered as a system of regular parameters at y, is an \(\ell \)unramified prolongation of \(\mathbf {v}\).
8.4 Essentially unramified fibers
We will primarily use Fact 8.2 in the case where \(x \in X\) is the generic point of a fiber of some smooth morphism. More precisely, suppose that \(X \rightarrow S\) is a dominant smooth morphism of regular kvarieties with geometrically integral fibers, and let \(s \in S\) be a closed point in the image of \(X \rightarrow S\). Let \(\eta _s \in X\) be the generic point of the fiber of \(X \rightarrow S\) over s.
With the setup above, if \((f_1,\ldots ,f_r)\) is a system of regular parameters at \(s \in S\), then \((f_1,\ldots ,f_r)\) is also a system of regular parameters at \(\eta _s \in X\). Thus, if K is a finite extension of k(X) and \(\eta _s\) is essentially unramified in K, then we may apply Fact 8.2 to \(\eta _s \in X\) endowed with a system of regular parameters \((f_1,\ldots ,f_r)\), which arises from \(s \in S\).
8.5 Mod\(\ell \) divisors
In this subsection, we will use the concept of essentially unramified fibers, as discussed in Sect. 8.4, to compare the divisorial valuations on a field of the form \(\mathbb {K}(t)\) with the divisorial valuations which can be detected in the mod\(\ell \) setting from K.
Note that one has \(\mathbb {K}(t)/\mathrm{U}_v \cong \mathbb {Z}\) for all \(v \in \mathscr {D}_t\), and that \(\mathfrak {K}(t)/\mathfrak {V}\cong \mathbb {Z}/\ell \) for all \(\mathfrak {V}\in \mathfrak {D}_t\). Our primary goal in this section is to compare \(\mathscr {D}_t\) and \(\mathfrak {D}_t\). First, we show that \(\mathfrak {D}_t\) can be embedded canonically in \(\mathscr {D}_t\).
Lemma 8.3
Let \(t \in K {\smallsetminus } k\) be given. For every \(\mathfrak {V}\in \mathfrak {D}_t\), there exists a unique \(v \in \mathscr {D}_t\) such that \(\mathfrak {V}= \{\mathrm{U}_v\}_K\). In particular, one has a canonical injective map \(\mathfrak {D}_t \hookrightarrow \mathscr {D}_t\) which satisfies \(\mathfrak {V}\mapsto v\) if and only if \(\mathfrak {V}= \{\mathrm{U}_v\}_K\).
Proof
Let w be a divisorial valuation of Kk such that \(\mathfrak {V}= \mathfrak {U}_w \cap \mathfrak {K}(t)\). Since \(\mathfrak {K}(t) \not \subset \mathfrak {U}_w\), we deduce that w is nontrivial on \(\mathbb {K}(t)\), hence \(w_{\mathbb {K}(t)}\) is a divisorial valuation on \(\mathbb {K}(t)\). Denote this divisorial valuation of \(\mathbb {K}(t)\) by v. Then clearly one has \(\{\mathrm{U}_v\}_K \subset \mathfrak {U}_w \cap \mathfrak {K}(t) = \mathfrak {V}\). On the other hand, both \(\mathfrak {K}(t)/\mathfrak {V}\) and \(\mathfrak {K}(t)/\{\mathrm{U}_v\}_K\) are isomorphic to \(\mathbb {Z}/\ell \), hence \(\mathfrak {V}= \{\mathrm{U}_v\}_K\).
Concerning the uniqueness of v, suppose that \(v'\) is another divisorial valuation of \(\mathbb {K}(t)\) such that \(\mathfrak {V}= \{\mathrm{U}_{v'}\}_K\). Then one has \(\{\mathrm{U}_v\}_K = \{\mathrm{U}_{v'}\}_K\), and since \({\text {k}}^\mathrm{M}_1(\mathbb {K}(t)) \rightarrow {\text {k}}^\mathrm{M}_1(K)\) is injective, we deduce that \(\{\mathrm{U}_v\}_{\mathbb {K}(t)} = \{\mathrm{U}_{v'}\}_{\mathbb {K}(t)}\). In particular, v and \(v'\) must be dependent as valuations of \(\mathbb {K}(t)\), for otherwise \(\mathrm{U}_v \cdot \mathrm{U}_{v'} = \mathbb {K}(t)^\times \) by the approximation theorem for independent valuations. Since both v and \(v'\) have value groups isomorphic to \(\mathbb {Z}\), it follows that \(v = v'\). This proves the uniqueness of v, as required. \(\square \)
A primary difficuly which arises in the mod\(\ell \) case is that the canonical map \(\mathfrak {D}_t \hookrightarrow \mathscr {D}_t\) described in Lemma 8.3 need not be surjective in general. Nevertheless, we can use the notion of essentiallyunramified fibers to give a sufficient condition for an element of \(\mathscr {D}_t\) to arise from \(\mathfrak {D}_t\). Although we will not need it later, we note that an argument similar to the proof of Lemma 8.4 below shows that all but finitely many of the elements of \(\mathscr {D}_t\) arise from some element of \(\mathfrak {D}_t\) via this injection.
Lemma 8.4
 (1)
K is a finite extension of k(X).
 (2)
s is in the image of \(X \rightarrow S\).
 (3)
The fiber \(X_s\) of \(X \rightarrow S\) over s is essentially unramified in K.
Proof
Let \(\pi \in \mathbb {K}(t)\) be a uniformizer for v, and let \(s \in S\) be the (unique) center of v on S. Furthermore, let \(\eta \in X\) be the generic point of the fiber of \(X \rightarrow S\) over s. Following the discussion of Sect. 8.4 with the morphism \(X \rightarrow S\), it follows that \(\eta \) is a regular point of codimension one, and that \(\pi \) is a local parameter at \(\eta \). Let \(w_0\) be the divisorial valuation of k(X) associated to \(\pi \) at \(\eta \). By Fact 8.2, there exists an \(\ell \)unramified prolongation w of \(w_0\) to Kk. The lemma follows by taking \(\mathfrak {V}= \mathfrak {U}_w \cap \mathfrak {K}(t)\) and noting that the image of \(\{\pi \}_K\) is a generator of \({\text {k}}^\mathrm{M}_1(K)/\mathfrak {U}_w \cong \mathbb {Z}/\ell \). \(\square \)
9 Strongly general elements and birational Bertini
In this section, we recall the notion of a general element of a regular function field. We also introduce the notion of a stronglygeneral element which is related to the notion of a general element, but has further assumptions which are motivated by the discussion of Sect. 8. The primary goal of this section is to prove socalled Birational Bertini results for both general and stronglygeneral elements, which will show that there are “many” such elements in higherdimensional function fields.
9.1 General elements
Let LF be a regular field extension and let \(x \in L {\smallsetminus } F\) be given. We say that x is separable in LF if \(x \notin F \cdot L^p\) where \(p = {{\mathrm{Char}}}F\). If \({{\mathrm{Char}}}F = 0\), then every element of L is separable in LF by convention. We say that x is general in LF provided that L is a regular extension of F(x). In particular, if x is general in LF then F(x) is relatively algebraically closed in L.
The following lemma is our socalled Birational Bertini result for general elements. The first assertion of this lemma can be found in [18, Ch. VIII, p. 213]. The second assertion of this lemma has essentially the same proof as in loc. cit., but since it hasn’t explicitly appeared in the literature, we provide a detailed proof below.
Lemma 9.1
 (1)
For all but finitely many \(a \in F\), the element \(x+ay\) is general in LF.
 (2)
There exists a nonempty open subset U of \(\mathbb {A}^2_F\) such that for all \((a,b) \in U(F)\), the element \((xa)/(yb)\) is general in LF.
Proof
For each \(P \in F^2\), write \(E_P {:}{=}\, F(t_P)\) and \(E'_P\) for the relative algebraic closure of \(E_P\) in L. If P, Q are two different points of \(F^2\), note that one has \(E_P \cdot E_Q = F(x,y)\). Note that \(E_P'\) and \(E_Q'\) are regular of transcendence degree 1 over F, since they are contained in L which is regular over F. This implies that \(E_P'\) and \(E_Q'\) are linearly disjoint over F, since LF is regular.
Let M denote the relative algebraic closure of F(x, y) in L, and note that there are only finitely many intermediate subextensions of MF(x, y). In particular, there are finitely many subextensions of MF(x, y) of the form \(E'_P(x,y)\). Let \(P_1,\ldots ,P_n\) be finitely many points in \(F^2\) such that \(E'_{P_i}(x,y)\) exhaust all such subextensions. Now suppose that Q is any point of \(F^2\) which is different from \(P_0,\ldots ,P_n\). Since \(Q \ne P_0\), we see that \(E'_Q\) is the separable algebraic closure of \(E_Q\) in M. Moreover, by the discussion above, \(E'_Q(x,y)\) must be linearly disjoint from \(E'_{P_i}(x,y)\) over F(x, y), for all \(i = 1,\ldots ,n\). This forces \(E'_Q(x,y)\) to be precisely F(x, y), and therefore \(E'_Q = E_Q\). In other words, \(E_Q = F(t_Q)\) is algebraically closed in L, and since \(Q \ne P_0\), we see that L is regular over \(E_Q = F(t_Q)\), as required. \(\square \)
9.2 Stronglygeneral elements
 (1)
The element t is general in Kk.
 (2)
The canonical injective map \(\mathfrak {D}_t \rightarrow \mathscr {D}_t\) from Lemma 8.3 is surjective (hence bijective).
Proposition 9.2
Let \(x,y \in K\) be algebraically independent over k such that y is separable in Kk. Then there exists a nonempty open subset U of \(\mathbb {A}^2_k\) such that, for all \((a,b) \in U(k)\), the element \((xa)/(yb)\) is stronglygeneral in Kk.
Proof
By Lemma 9.1(2), there exists a nonempty open subset \(U_1\) of \(\mathbb {A}^2_k\) such that for all \((a,b) \in U_1(k)\), the element \(t_{a,b} = (xa)/(yb)\) is general in Kk. We must therefore prove that the second condition for a stronglygeneral element is also an open condition on \((a,b) \in \mathbb {A}^2_k(k)\) as above.
Put \(x = t_1\), \(y = t_2\) and extend \(t_1,t_2\) to a transcendence base \(\mathbf {t}= (t_1,\ldots ,t_d)\) for Kk. Furthermore, consider the canonical coordinate projection \(\mathbb {A}^d_\mathbf {t}\rightarrow \mathbb {A}^2_{t_1,t_2}\). Also, identify \(\mathbb {A}^2_k\) as above with \(\mathbb {A}^2_{t_1,t_2}\), so that we may consider \(U_1\) as an open subset of \(\mathbb {A}^2_{t_1,t_2}\). Finally, note that K is a finite extension of \(k(\mathbf {t}) = k(\mathbb {A}^d_\mathbf {t})\). Thus, there exists a nonempty open subset U of \(U_1\) such that for all \((a,b) \in U(k)\), the fiber of \(\mathbb {A}^d_\mathbf {t}\rightarrow \mathbb {A}^2_{t_1,t_2}\) over the point (a, b) is essentially unramified in K. We will show that this open set U satisfies the required assertion.
With this setup, the fact that \(\mathfrak {D}_t \rightarrow \mathscr {D}_t\) is surjective follows from Lemma 8.4, by taking X as above, \(S = \mathbb {P}^1_t\), and \(X \rightarrow S\) the map defined above. \(\square \)
The main benefit of Proposition 9.2 is that it can be used to show that stronglygeneral elements multiplicatively generate higherdimensional function fields. The following lemma is a precise formulation of this fact which we will use later.
Lemma 9.3
Let L be a relativelyalgebraically closed subfield of K, such that \(k_0 \subset L\) and \({{\mathrm{tr.deg}}}(Lk_0) \ge 2\). Then the multiplicative group \(L^\times \) is generated by elements \(t \in L^\times \) which are stronglygeneral in Kk.
Proof
It suffices to prove that every transcendental element x of L is a product of finitely many elements of L which are stronglygeneral in Kk. Since L is relatively algebraically closed in K, it follows that every nonconstant element of L is a power of some element of L which is separable in Kk. Thus, we may assume without loss of generality that x is separable. Moreover, since \({{\mathrm{tr.deg}}}(Lk \cap L) = {{\mathrm{tr.deg}}}(Lk_0) \ge 2\), there exists another element \(y \in L\) which is algebraically independent from x over k, and such that y is separable in Kk.
9.3 Rationallike collections
Lemma 9.4
Let \(t \in K {\smallsetminus } k\) be strongly general in Kk, and let \(\Psi = (\Psi _\mathfrak {V})_{\mathfrak {V}\in \mathfrak {D}_t}\) be the canonical rationallike collection associated to \(\mathbb {K}(t) = k(t)\). Also, let \((\Phi _\mathfrak {V})_{\mathfrak {V}\in \mathfrak {D}_t}\) be another rationallike collection. Then there exists a unique \(\epsilon \in (\mathbb {Z}/\ell )^\times \), such that \(\Phi _\mathfrak {V}= \epsilon \cdot \Psi _\mathfrak {V}\) for all \(\mathfrak {V}\in \mathfrak {D}_t\).
Proof
We now use the notions of stronglygeneral elements and rationallike collections to prove a proposition which will be useful in several steps of the proof of Theorem D.
Proposition 9.5
 (1)
There exists an \(\epsilon \in (\mathbb {Z}/\ell )^\times \) such that \(\sigma = \epsilon \cdot \mathbf {1}_{{\text {k}}^\mathrm{M}_1(K)}\).
 (2)
For all \(\mathfrak {K}\in \mathfrak {G}^1(Kk)\), one has \(\sigma \mathfrak {K}= \mathfrak {K}\).
Proof
 (1)
First, we show that for all \(t \in K {\smallsetminus } k\) and for all \(\mathfrak {V}\in \mathfrak {D}_t\), one has \(\sigma \mathfrak {V}= \mathfrak {V}\).
 (2)
Second, we show that for all t which is stronglygeneral in Kk, the restriction of \(\sigma \) to \(\mathfrak {K}{:}{=}\, \mathfrak {K}(t)\) is of the form \(\epsilon _\mathfrak {K}\cdot \mathbf {1}_\mathfrak {K}\), for some \(\epsilon _\mathfrak {K}\in (\mathbb {Z}/\ell )^\times \), which a priori might depend on \(\mathfrak {K}\).
 (3)
Finally, we show that \(\epsilon _\mathfrak {K}\) from step (2) doesn’t actually depend on \(\mathfrak {K}\), and then conclude the proof of the proposition by using Lemma 9.3.
Step (3): Let \(t_1,t_2\) be stronglygeneral in Kk, and put \(\mathfrak {K}_i {:}{=}\, \mathfrak {K}(t_i)\) and \(\epsilon _i {:}{=}\, \epsilon _{\mathfrak {K}_i}\) for \(i = 1,2\). If \(t_1,t_2\) are algebraically dependent over k, then \(\mathfrak {K}(t_1) = \mathfrak {K}(t_2)\), so that \(\epsilon _1 = \epsilon _2\).
Assume, on the other hand, that \(t_1,t_2\) are algebraically independent over k. Since \(t_i\) is general in Kk, we see that \(\{\{t_ia\}_K \ : \ a \in k\}\) is a linearlyindependent subset of \({\text {k}}^\mathrm{M}_1(K)\). Thus, by Proposition 9.2, we may choose \(a,b \in k\) such that \(\{t_1a\}_K\) and \(\{t_2b\}_K\) are \(\mathbb {Z}/\ell \)independent in \({\text {k}}^\mathrm{M}_1(K)\) and such that \(t_0 {:}{=}\, (t_1a)/(t_2b)\) is stronglygeneral in Kk. Put \(\mathfrak {K}_0 = \mathfrak {K}(t_0)\) and \(\epsilon _0 = \epsilon _{\mathfrak {K}_0}\).
Letting \(\epsilon = \epsilon _\mathfrak {K}\) for some (hence any) \(\mathfrak {K}= \mathfrak {K}(t)\) with t stronglygeneral, we deduce that \(\sigma _\mathfrak {K}= \epsilon \cdot \mathbf {1}_\mathfrak {K}\) for all \(\mathfrak {K}= \mathfrak {K}(t)\) with t stronglygeneral. By Lemma 9.3, we see that \({\text {k}}^\mathrm{M}_1(K)\) is generated by its subgroups of the form \(\mathfrak {K}(t)\) for t stronglygeneral in Kk. Hence \(\sigma = \epsilon \cdot \mathbf {1}_{{\text {k}}^\mathrm{M}_1(K)}\), as required. \(\square \)
9.4 Faithfulness of the Galois action
We conclude this section by proving that the Galois action of \({{{\mathrm{Gal}}}_{k_0}}\) on the mod\(\ell \) Milnor Kring of K is faithful. Although there are many ways to prove this fact, we can use geometric subgroups and the BirationalBertini results to prove this for function fields of dimension \(\ge 2\). The result also holds for function fields of dimension 1, but a different argument is needed in that case.
Lemma 9.6
Proof
Next, if K has transcendence degree \(\ge 2\), we note that the action of \(\tau \in {{{\mathrm{Gal}}}_{k_0}}\) on \({\text {k}}^\mathrm{M}_1(K)\) restricts to an automorphism on any geometric subgroup of the form \(\mathfrak {K}(S)\) for \(S \subset K_0\). By Lemma 9.1, there exists some \(t \in K_0\) which is general in Kk. In this case, the map \({\text {k}}^\mathrm{M}_1(k(t)) \rightarrow {\text {k}}^\mathrm{M}_1(K)\) is injective with image \(\mathfrak {K}(t)\). This injection is compatible with the action of \({{{\mathrm{Gal}}}_{k_0}}\), so the assertion follows from the argument above.
We now choose some \(t \in K_0\) such that K is finite and separable over k(t). Let C denote the complete normal model of Kk, and consider the finite (possibly branched) separable cover \(C \rightarrow \mathbb {P}^1_t\) induced by the inclusion \(k(t) \hookrightarrow K\).
Since the morphism \(\rho _{k_0} : {{{\mathrm{Gal}}}_{k_0}}\rightarrow \underline{{{\mathrm{Aut}}}}^\mathrm{M}({\text {k}}^\mathrm{M}_1(K))\) factors through \({{\mathrm{Aut}}}^\mathrm{M}({\text {k}}^\mathrm{M}_1(K))\), we deduce that the morphism \(\rho _{k_0} : {{{\mathrm{Gal}}}_{k_0}}\rightarrow {{\mathrm{Aut}}}^\mathrm{M}({\text {k}}^\mathrm{M}_1(K))\) is injective as well. \(\square \)
10 The main proof
We now turn to the proof of Theorems C and D, which is the main focus of this paper. The primary focus will be on Theorem D, since we have been primarily working with mod\(\ell \) Milnor Ktheory, while Theorem C will follow by applying Theorem 4.2.
Using the notation from Theorem D, recall that \(\mathbf {a}= (a_1,\ldots ,a_r)\) is an arbitrary (possibly empty) finite tuple of elements of \(k_0^\times \). We start off the proof by working with a fixed element \(\tau \in {{\mathrm{Aut}}}^\mathrm{M}_\mathbf {a}({\text {k}}^\mathrm{M}_1(K))\), although we will eventually replace \(\tau \) by another element \(\sigma \in {{\mathrm{Aut}}}^\mathrm{M}_\mathbf {a}({\text {k}}^\mathrm{M}_1(K))\) of the form \(\epsilon \cdot \tau \) for some \(\epsilon \in (\mathbb {Z}/\ell )^\times \). In particular, \(\sigma \) and \(\tau \) represent the same element of \(\underline{{{\mathrm{Aut}}}}^\mathrm{M}_\mathbf {a}({\text {k}}^\mathrm{M}_1(K))\), but this \(\sigma \) will have some further special properties which we will need. In any case, if A is any subgroup of \({\text {k}}^\mathrm{M}_1(K)\) and \(\tau ,\sigma \) are as above, then one has \(\sigma A = \tau A\). Since the primary goal of the proof is to show that \(\tau \) induces an automorphism of the lattice of geometric subgroups of \({\text {k}}^\mathrm{M}_1(K)\), this observation shows that it doesn’t actually matter if we replace \(\tau \) with \(\sigma = \epsilon \cdot \tau \). We now fix this initial element \(\tau \in {{\mathrm{Aut}}}^\mathrm{M}_\mathbf {a}({\text {k}}^\mathrm{M}_1(K))\).
 (1)
\(\tau \) extends to an automorphism of \({\text {k}}^\mathrm{M}_*(K)\).
 (2)For all \(t \in K_0 {\smallsetminus } k_0\), \(\tau \) restricts to an automorphism of the subgroup$$\begin{aligned} \langle \{t\}_K,\{ta_1\}_K,\ldots ,\{ta_r\}_K \rangle . \end{aligned}$$
Fact 10.1
 (1)
Let \(x_1,\ldots ,x_r \in {\text {k}}^\mathrm{M}_1(K)\) be given. Then one has \(\{x_1,\ldots ,x_r\}_K = 0\) if and only if one has \(\{\tau x_1,\ldots ,\tau x_r\}_K = 0\).
 (2)
For all \(t \in K_0 {\smallsetminus } k_0\), one has \(\tau \{t\}_K \in \langle \{t\}_K,\{ta_1\}_K,\ldots ,\{ta_r\}_K \rangle \).
10.1 Acceptable subsets
Fact 10.2
10.2 Fixing \(K_0\)
We begin by showing that every subset of \(K_0\) is \(\tau \)acceptable. In fact, since our goal will be to apply Corollary 7.4, we must show the stronger property that \(\tau \mathfrak {K}(S) = \mathfrak {K}(S)\) for all subsets \(S \subset K_0\). This assertion is the starting point of the proof and is accomplished in the following lemma.
Lemma 10.3
Let \(S \subset K_0\) be a subset. Then one has \(\tau \mathfrak {K}(S) = \mathfrak {K}(S)\). In particular, every subset of \(K_0\) is \(\tau \)acceptable.
Proof
Although Lemma 10.3 will be used in the final steps of the proof, we will need a stronger variant of this result. Namely, we will need to prove that there exists a \(\sigma \) of the form \(\sigma = \epsilon \cdot \tau \) for some \(\epsilon \in (\mathbb {Z}/\ell )^\times \) such that \(\sigma \{t\}_K = \{t\}_K\) for all \(t \in K_0^\times \). This stronger variant appears in Proposition 10.6 below, and proving this proposition is the main goal of this subsection. Naturally, this property can be seen as a crude approximation to our goal of proving that \(\tau \) arises from some element of \({{{\mathrm{Gal}}}_{k_0}}\).
Lemma 10.4
Let \((v_1,\ldots ,v_r)\) be a flag of divisorial valuations of Kk, with \(r < {{\mathrm{tr.deg}}}(Kk)\). Then \((v_1^\tau ,\ldots ,v_r^\tau )\) is a flag of divisorial valuations of Kk.
Proof
The following lemma shows that \(\tau \) fixes subgroups of the form \(\mathfrak {V}[t;c]\) which arise from \(K_0\), i.e. such that \(t \in K_0 {\smallsetminus } k_0\) and \(c \in \mathbb {P}^1(k_0)\). This is a key step towards proving Proposition 10.6 below. The proof of this lemma essentially follows by considering the \(\mathfrak {D}_t\)supports of various elements associated to t and using Fact 10.1(2).
Lemma 10.5
Let \(t \in K_0 {\smallsetminus } k_0\) be strongly general in Kk, and let \(c \in \mathbb {P}^1(k_0)\) be given. Then one has \(\tau \mathfrak {V}[t;c] = \mathfrak {V}[t;c]\).
Proof
As mentioned above, the following proposition is the primary goal of this subsection, and it can be seen as the first major step towards the proof of Theorem D.
Proposition 10.6
 (1)
For all \(t \in K_0^\times \), one has \(\tau \{t\}_K = \epsilon \cdot \{t\}_K\).
 (2)
For all \(t \in K_0\) which is strongly general in Kk and for all \(b \in k\), there exists a unique \(c \in k\) such that \(\tau \{tb\}_K = \epsilon \cdot \{tc\}_K\).
Proof
 (1)
First, \(\tau \mathfrak {K}(t) = \mathfrak {K}(t)\) by Lemma 10.3.
 (2)
Second, one has \(\tau \mathfrak {V}\in \mathfrak {D}_t\) for all \(\mathfrak {V}\in \mathfrak {D}_t\) by Lemma 10.4 and the definition of \(\mathfrak {D}_t\).
 (3)
Third, for all \(c \in k_0 \cup \{\infty \} = \mathbb {P}^1(k_0)\), one has \(\tau \mathfrak {V}[t;c] = \mathfrak {V}[t;c]\) by Lemma 10.5.
Proof of (1):
Now let \(c \in k_0\) be given. Then one has \({{\mathrm{div}}}_\Psi (\{tc\}_K) = [\mathfrak {V}[t;c]]  [\mathfrak {V}[t;\infty ]]\) since \(\Psi \) is the canonical rationallike collection associated to \(\mathbb {K}(t)\).
In order to conclude the proof of assertion (1), we will apply Lemma 9.3 to the subfield \(K_0\) of K. In light of this, it suffices to show that the \(\epsilon _t\) which appears in the argument above is independent of \(\mathfrak {K}(t)\) for \(t \in K_0\) which is stronglygeneral in Kk.
With this in mind, suppose that \(u,t \in K_0\) are both stronglygeneral in Kk. Let \(\epsilon _t\) resp. \(\epsilon _u\) be as above, and assume for a contradiction that \(\epsilon _t \ne \epsilon _u\). Since the \(\epsilon _t\) resp. \(\epsilon _u\) depends only on \(\mathfrak {K}(t)\) resp. \(\mathfrak {K}(u)\) and \(\tau \), this implies that \(\mathfrak {K}(t) \ne \mathfrak {K}(u)\), hence \(\mathbb {K}(t) \ne \mathbb {K}(u)\) by Proposition 7.3. In particular, u, t are algebraically independent over k.
By Proposition 9.2, there exists a nonempty open subset U of \(\mathbb {A}^2_k\) such that \((a,b) \in U(k)\) implies that \((ta)/(ub)\) is stronglygeneral in Kk. Moreover, since t, u are general in Kk, it follows that the sets \(\{\{ta\}_K\}_{a \in k_0}\), \(\{\{ua\}_K\}_{a \in k_0}\) are \(\mathbb {Z}/\ell \)independent in \({\text {k}}^\mathrm{M}_1(K)\). Since \(k_0\) is infinite, there exists some \((a,b) \in U(k_0)\) such that \(\{ta\}_K\) and \(\{ub\}_K\) are \(\mathbb {Z}/\ell \)independent in \({\text {k}}^\mathrm{M}_1(K)\).
Proof of (2):
 (1)
\(\sigma \) and \(\tau \) represent the same element of \(\underline{{{\mathrm{Aut}}}}^\mathrm{M}_\mathbf {a}({\text {k}}^\mathrm{M}_1(K))\). In particular \(\sigma A = \tau A\) for all subgroups A of \({\text {k}}^\mathrm{M}_1(K)\). Thus, one has \(\sigma \mathfrak {K}(S) = \mathfrak {K}(S)\) for all \(S \subset K_0\) by Lemma 10.3.
 (2)
One has \(\sigma \{t\}_K = \{t\}_K\) for all \(t \in K_0^\times \) by Proposition 10.6(1).
 (3)
If \(t \in K_0\) is stronglygeneral in Kk, and \(b \in k\) is arbitrary, then there exists a unique element \(c \in k\) such that \(\sigma \{tb\}_K = \{tc\}_K\), by Proposition 10.6(2).
Proposition 10.7
 (1)
The closed point \((0:b:1) \in \mathbb {P}^2_{x,y}\) is in the image of \(X \rightarrow \mathbb {P}^2_{x,y}\).
 (2)
K is a finite extension of k(X).
 (3)
The fiber of \(X \rightarrow \mathbb {P}^2_{x,y}\) over (0 : b : 1) is essentially unramified in K.
Proof
For \(f_1,\ldots ,f_s \in k[x,y]\), we denote by \(V(f_1,\ldots ,f_s)\) the (reduced) closed subvariety of \(\mathbb {A}^2_{x,y}\) given by the ideal \(\sqrt{(f_1,\ldots ,f_s)}\). We will abuse the notation, and also write \(V(f_1,\ldots ,f_r)\) for the closure of this subvariety in \(\mathbb {P}^2_{x,y}\), also considered as a (reduced) subvariety. Following our notational conventions, we denote the closed points of \(\mathbb {P}^2_{x,y}\) by their associated krational point, written in homogeneous coordinates. In particular, the closed points of \(\mathbb {A}^2_{x,y}\) are written as \((1:a_1:a_2)\), \(a_1,a_2 \in k\), and the points of \(\mathbb {P}^2_{x,y} {\smallsetminus } \mathbb {A}^2_{x,y}\) are written as \((0:a_1:a_2)\), \(a_1,a_2 \in k\), \((a_1,a_2) \ne (0,0)\).
Let v denote the divisorial valuation of k(x, y)k associated to the prime Weildivisor \(Z := \mathbb {P}^2_{x,y} {\smallsetminus } \mathbb {A}^2_{x,y}\) on \(\mathbb {P}^2_{x,y}\). For any \(a \in k\), let \(v_a\) be the 2divisorial valuation of k(x, y)k refining v which is associated to the point (0 : a : 1) on Z. Thus \((v,v_a)\) is a flag of divisorial valuations of k(x, y)k of length 2.
Let \(\eta \) denote the generic point of Z. To simplify the exposition later in the proof, we will say that \(a \in k\) is allowable if the fiber of \(X \rightarrow \mathbb {P}^2_{x,y}\) over the point (0 : a : 1) is nonempty and essentially unramified in K. Since (0 : b : 1) lies on Z, assumptions (1) and (3) imply that the fiber \(X_\eta \) of \(X \rightarrow \mathbb {P}^2_{x,y}\) over \(\eta \) is nonempty and essentially unramified in K. Therefore, we see that all but finitely many \(a \in k\) (including b) are allowable.
 (1)
One has \(\{1/x\}_K \in {\text {k}}^\mathrm{M}_1(K) {\smallsetminus } \mathfrak {U}_w\) hence \(\{x\}_K \in {\text {k}}^\mathrm{M}_1(K) {\smallsetminus } \mathfrak {U}_w\).
 (2)
For all \(h \in k\), one has \(\{t_ha\}_K \in \mathfrak {U}_w {\smallsetminus } \mathfrak {U}_{w_a}\).
 (1)
\(w_a^\sigma \) refines \(w^\sigma \).
 (2)
\(\sigma \mathfrak {U}_w = \mathfrak {U}_{w^\sigma }\) and \(\sigma \mathfrak {U}_w^1 = \mathfrak {U}_{w^\sigma }^1\).
 (3)
\(\sigma \mathfrak {U}_{w_a} = \mathfrak {U}_{w_a^\sigma }\) and \(\sigma \mathfrak {U}_{w_a}^1 = \mathfrak {U}_{w_a^\sigma }^1\).
Claim
One has \(v^\sigma = v\).
Proof
Recall that \(\eta \) denotes the generic point of Z in \(\mathbb {P}^2_{x,y}\). As \(\mathbb {P}^2_{x,y}\) is proper over k, the valuation \(v^\sigma \) has a unique center on \(\mathbb {P}^2_{x,y}\). Since \(\eta \) is a regular codimension 1 point of \(\mathbb {P}^2_{x,y}\), it suffices to show that the center of \(v^\sigma \) on \(\mathbb {P}^2_{x,y}\) is \(\eta \).
On the other hand, note that one also has \({\{x1\}_{k(x,y)} \notin \{\mathrm{U}_v\}_{k(x,y)}}\), hence \(\{x1\}_K\) is also a generator of \({\text {k}}^\mathrm{M}_1(K)/\mathfrak {U}_w\). By repeating the same argument with \(x1\) in place of x, and noting that \(\sigma \{x1\}_K = \{x1\}_K\) by Proposition 10.6, we deduce that the center of \(v^\sigma \) on \(\mathbb {P}^2_{x,y}\) must actually be contained in Z, since the support of \(x1\) on \(\mathbb {P}^2_{x,y}\) is \(V(x1) \cup Z\).
Now assume for a contradiction that the center of \(v^\sigma \) is a closed point P on Z. For an allowable \(a \in k\), recall that \(v_a^\sigma \) refines v. It therefore follows that the center of \(v_a^\sigma \) on \(\mathbb {P}^2_{x,y}\) must also be this closed point P.
Now let \(a_0 \in k_0\) be allowable—such an \(a_0\) exists since \(k_0\) is infinite. By Proposition 10.6, we have \(\{ta_0\}_K = \sigma \{ta_0\}_K\). Therefore \(\{ta_0\}_K \notin \mathfrak {U}_{w_{a_0}^\sigma }\), hence \(v_{a_0}^\sigma (ta_0) \ne 0\). By considering the support of \(ta_0\) in \(\mathbb {P}^2_{x,y}\), we see that the point P must be either the point \((0:a_0:1)\) or the point (0 : 1 : 0). In fact, it follows that P must be the point (0 : 1 : 0) since we can repeat this argument with a different allowable \(a_1 \in k_0\), \(a_1 \ne a_0\), which exists since \(k_0\) is infinite.
Now consider the function \((ta_0)/(ta_1)\) for \(a_1 \in k_0\) distinct from \(a_0\). By Proposition 10.6, we have \(\{(ta_0)/(ta_1)\}_K = \sigma \{(ta_0)/(ta_1)\}_K\), while \(\{ta_1\}_K \in \mathfrak {U}_{w_{a_0}}\). Therefore, \(\{(ta_0)/(ta_1)\}_K \notin \mathfrak {U}_{w_{a_0}^\sigma }\), hence \(v_{a_0}^\sigma ((ta_0)/(ta_1)) \ne 0\). By looking at the support of \((ta_0)/(ta_1)\) on \(\mathbb {P}^2_{x,y}\), we obtain a contradiction to the fact that \(P = (0:1:0)\).
Having obtained our contradiction, it follows that the center of \(v^\sigma \) cannot be a closed point on Z. This means that the center of \(v^\sigma \) must be the generic point of Z, which concludes the proof of the claim. \(\square \)
Recall from Proposition 10.6(2) that there exist unique constants \(b_e,b_f \in k\) such that \(\sigma \{t_eb\}_K = \{t_eb_e\}_K\) and \(\sigma \{t_fb\}_K = \{t_fb_f\}_K\). We must show that \(b_e = b_f\).
10.3 Intersections
We now turn to the second main part of the proof. In this part, we develop a condition for detecting onedimensional geometric subgroups as intersections of certain twodimensional geometric subgroups. The first lemma in this direction shows that any onedimensional geometric subgroup can actually be realized as the intersection of two twodimensional geometric subgroups. The second lemma in this direction gives a sufficient condition for the intersection of two twodimensional geometric subgroups to be geometric.
Lemma 10.8
Proof
Extend \(t_1,t_2,t_3\) to a transcendence base \(\mathbf {t}= (t_1,\ldots ,t_r)\) for KF, and let \(K'\) denote the maximal separable subextension of \(KF(\mathbf {t})\). Note that \(K'\) is then a regular extension of F. By Lemma 9.1(1), for all but finitely many \(a \in k\), the field \(K'\) is regular over \(F(t_1+a \cdot t_3)\); since \(k_0\) is infinite, we may choose such an a which lies in \(k_0\). Applying Lemma 9.1(1) again, for all but finitely many \(b \in k\), the field \(K'\) is regular over \(F(t_1 + a \cdot t_3,t_2 + b \cdot t_3)\); again, we may choose such a b which lies in \(k_0\). We put \(x = t_1 + a \cdot t_3\) and \(y = t_2 + b \cdot t_3\) for \(a,b \in k_0\) as above.
Next, put \(F_1 = F(x)\), \(F_2 = F(y)\) and \(F_{12} = F(x,y)\), and note that \(F_*\) is relatively algebraically closed in \(K'\) for \(* = 1,2,12\). Also, for \(* = 1,2,12\), let \(M_*\) denote the relative algebraic closure of \(F_*\) in K. Since \(KK'\) is purelyinseparable, the extension \(M_*F_*\) is also purelyinseparable.
Lemma 10.9

\(A \in \mathfrak {G}^1(Kk)\)

\(B_1,B_2,C \in \mathfrak {G}^2(Kk)\)

\(B_1',B_2',D \in \mathfrak {G}^3(Kk)\)
 (1)
\(B_1 \cup B_2 \subset D\), \(B_1 \ne B_2\), and \(A \not \subset D\).
 (2)
\(B_1 \cup A \subset B_1'\) and \(B_2 \cup A \subset B_2'\).
 (3)
\(C \subset B_1' \cap B_2'\) and \(\mathfrak {K}= B_1 \cap B_2\).
Proof
First suppose that \(t \in K {\smallsetminus } k\) is given, and consider the geometric subgroup \(\mathfrak {K}(t)\). Then for all \(a,b \in \mathfrak {K}(t)\), one has \(\{a,b\}_K = 0\) by Fact 6.1. Moreover, by Proposition 6.3, if \(c \in {\text {k}}^\mathrm{M}_1(K) {\smallsetminus } \mathfrak {K}(t)\), then there exist (many) elements \(d \in \mathfrak {K}(t)\) such that \(\{c,d\}_K \ne 0\). In particular, \(\mathfrak {K}(t)\) is maximal among subgroups \(\Delta \) of \({\text {k}}^\mathrm{M}_1(K)\) such that \(\dim ^\mathrm{M}(\Delta ) = 1\).
Now suppose that \(\mathfrak {K}\) satisfies the assumptions on the lemma. The goal of this proof will be to show that there exists some \(t \in K {\smallsetminus } k\) such that \(\mathfrak {K}(t) \subset \mathfrak {K}\). Then the “maximality” in the observation above would imply that \(\mathfrak {K}(t) = \mathfrak {K}\). We will tacitly use Proposition 7.3(1), which says that for \(E_1,E_2 \in \mathbb {G}^*(Kk)\), one has \(E_1 \subset E_2\) if and only if \(\mathfrak {K}(E_1) \subset \mathfrak {K}(E_2)\) as subgroups of \({\text {k}}^\mathrm{M}_1(K)\).
Let \(F_1,F_2 \in \mathbb {G}^2(Kk)\) be such that \(\mathfrak {K}(F_i) = B_i\) and put \(F = F_1 \cap F_2\). Then condition (1) implies that \(F_1 \ne F_2\) hence \({{\mathrm{tr.deg}}}(Fk) \le 1\). Let \(F_{12} \in \mathbb {G}^3(Kk)\) be such that \(\mathfrak {K}(F_{12}) = D\), then condition (1) implies that \(F_1 \cdot F_2 \subset F_{12}\). Let \(x \in K {\smallsetminus } k\) be such that \(A = \mathfrak {K}(x)\). Condition (1) implies that x is transcendental over \(F_{12}\), hence it is also transcendental over \(F_1\) and \(F_2\). Thus, one has \(B_i' = \mathfrak {K}(F_i,x)\) by condition (2).
10.4 The base case
We begin by proving that “many” elements of the form \(a_0 x_0 + a_1 x_1\) are \(\sigma \)acceptable, and we will then use the “intersection” results proved in the previous subsection to deduce the full base case.
Lemma 10.10
Proof
In particular, if \(t_e = (xe)/y\) is general in Kk, then we see that \(t_e\) is automatically stronglygeneral in Kk by applying Lemma 8.4 to the morphism \(X_e \rightarrow \mathbb {P}^1_{t_e}\). Moreover, as 1 / y is separable in Kk, it follows from Lemma 9.1(1) that \(t_e\) is general (hence stronglygeneral) for all but finitely many \(e \in k_0\).
The following lemma concludes the base case for our induction.
Lemma 10.11
Assume that \({{\mathrm{tr.deg}}}(Kk) \ge 5\) and let \(x_0,y_0 \in K_0\) be given. Then for all \(d \in k\), the element \(x_0+d\cdot y_0\) is \(\sigma \)acceptable.
Proof
We may assume without loss of generality that \(x_0,y_0\) are algebraically independent over k, for otherwise the claim is either trivial (if \(x_0+d \cdot y_0 \in k\)) or it follows from Lemma 10.3.
 (1)
\(A = \sigma \mathfrak {K}(t_3)\) and \(C = \sigma \mathfrak {K}(t,t_3)\).
 (2)
\(B_1 = \sigma \mathfrak {K}(t,x)\), \(B_2 = \sigma \mathfrak {K}(t,y)\) and \(D = \sigma \mathfrak {K}(t,x,y)\).
 (3)
\(B_1' = \sigma \mathfrak {K}(t,x,t_3)\) and \(B_2' = \sigma \mathfrak {K}(t,y,t_3)\).
10.5 The general case
Lemma 10.12
Assume that \({{\mathrm{tr.deg}}}(Kk) \ge 5\). Then every element of K is \(\sigma \)acceptable.
Proof
 (1)
\(A = \sigma \mathfrak {K}(x_r) = \mathfrak {K}(x_r)\) and \(C = \sigma \mathfrak {K}(t,x_r) = \sigma \mathfrak {K}(ta_r x_r,x_r)\).
 (2)
\(B_1 = \sigma \mathfrak {K}(t,x) = \sigma \mathfrak {K}(tx,x)\), \(B_2 = \sigma \mathfrak {K}(t,y) = \sigma \mathfrak {K}(ty,y)\).
 (3)
\(D = \sigma \mathfrak {K}(t,x,y) = \sigma \mathfrak {K}(tx,x,y)\).
 (4)
\(B_1' = \sigma \mathfrak {K}(t,x,x_r) = \sigma \mathfrak {K}(tx,x,x_r)\).
 (5)
\(B_2' = \sigma \mathfrak {K}(t,y,x_r) = \sigma \mathfrak {K}(ty,y,x_r)\).
10.6 Concluding the Proofs of Theorems C and D
For the rest of this section, we assume that \({{\mathrm{tr.deg}}}(Kk) \ge 5\) so that we can use Corollary 7.4 and Lemma 10.12.
Finally, it easily follows from Kummer theory that the isomorphism \(\underline{{{\mathrm{Aut}}}}^\mathrm{c}(\mathcal {G}_K^a) \cong \underline{{{\mathrm{Aut}}}}^\mathrm{M}({\text {k}}^\mathrm{M}_1(K))\) of Theorem 4.2 restricts to an isomorphism \(\underline{{{\mathrm{Aut}}}}^\mathrm{c}_\mathbf {a}(\mathcal {G}_K^a) \cong \underline{{{\mathrm{Aut}}}}^\mathrm{M}_\mathbf {a}({\text {k}}^\mathrm{M}_1(K))\). Thus, Theorem C follows immediately from Theorem D by applying Theorem 4.2.
11 Concluding the proof of the mod\(\ell \) I/OM
We now turn to the proof of Theorems A and B. As we will see, Theorem A follows rather easily from Theorem C. On the other hand, Theorem B follows from Theorem A moreorless because of our definition of a 5connected subcategory of \(\mathbf {Var}_{k_0}\).
11.1 Proof of Theorem A
11.2 Proof of Theorem B
Before we conclude the proof of Theorem B, we need a small lemma concerning the domination condition between two birational systems, as defined in Sect. 1.7.
Lemma 11.1
 (1)
The restriction \(\phi _{\mathcal {U}_2} \in {{\mathrm{Aut}}}^\mathrm{c}(\pi ^a_{\mathcal {U}_2})\) is also defined by \(\tau \).
 (2)
If there exists some \(\epsilon \in (\mathbb {Z}/\ell )^\times \) and some \(\tau ' \in {{{\mathrm{Gal}}}_{k_0}}\) such that \(\epsilon \cdot \phi _{\mathcal {U}_2}\) is defined by \(\tau '\), then \(\tau ' = \tau \) and \(\epsilon = 1\).
Proof
Put \(K = k(X)\) for some \(X \in \mathcal {U}_2\) and \(L = k(Y)\) for some \(Y \in \mathcal {U}_1\). Since \(\mathcal {U}_1\) dominates \(\mathcal {U}_2\) in \(\mathcal {V}\), we see that K is a subfield of L.
Proof of (1):
\(\underline{\hbox {Case }\dim \mathcal {U}_2 = 1:}\) In this case, our assumptions ensure that \(\mathcal {G}_L^a \rightarrow \mathcal {G}_K^a\) is surjective. Since \(\phi _L\) is defined by \(\tau \), and since \(\phi _L\) and \(\phi _K\) are compatible with the projection \(\mathcal {G}_L^a \twoheadrightarrow \mathcal {G}_K^a\), it follows that \(\phi _K\) must also be defined by \(\tau \).
\(\underline{\hbox {Case }\dim \mathcal {U}_2 > 1:}\) Let \(\psi = \tau ^{1} \cdot \phi _K\) denote the composition of \(\phi _K\) with the element of \({{\mathrm{Aut}}}^\mathrm{c}(\mathcal {G}_K^a)\) induced by \(\tau ^{1}\). Furthermore, let \(\psi ^* \in {{\mathrm{Aut}}}^\mathrm{M}({\text {k}}^\mathrm{M}_1(K))\) be the element associated to \(\psi \) via the Kummer pairing (see Theorem 4.2). Finally, note that the image of \(\mathcal {G}_L^a \rightarrow \mathcal {G}_K^a\) is an open subgroup of \(\mathcal {G}_K^a\), on which \(\psi \) acts as the identity. Thus, there exists a finite subgroup \(H_0\) of \({\text {k}}^\mathrm{M}_1(K)\) such that for every \(x \in {\text {k}}^\mathrm{M}_1(K)\), one has \(\psi ^* x \in x + H_0\). Therefore, there is a subgroup H of \({\text {k}}^\mathrm{M}_1(K)\) such that \(\psi ^*\) acts as the identity on H, and such that H has finite index in \({\text {k}}^\mathrm{M}_1(K)\).
We will show that \(\psi ^*\) is some \((\mathbb {Z}/\ell )^\times \)multiple of the identity on \({\text {k}}^\mathrm{M}_1(K)\). First, suppose that \(t \in K {\smallsetminus } k\) is given and consider the onedimensional geometric subgroup \(\mathfrak {K}(t)\). The inclusion \(k(t) \hookrightarrow K\) induces a (possibly noninjective) map \({\text {k}}^\mathrm{M}_1(k(t)) \rightarrow {\text {k}}^\mathrm{M}_1(K)\) and we let M denote the preimage of H in \({\text {k}}^\mathrm{M}_1(k(t))\). Since H has finite index in \({\text {k}}^\mathrm{M}_1(K)\), it follows that M must have finite index in \({\text {k}}^\mathrm{M}_1(k(t))\).
Claim
There exists some \(x \in k(t) {\smallsetminus } k\) such that \(\{xc\}_{k(t)} \in M\) for infinitely many \(c \in k\).
Proof
Let \(x \in k(t) {\smallsetminus } k\) be as in the claim above. Then one has \(\psi ^*\{xc\}_K = \{xc\}_K\) for infinitely many \(c \in k\). As \(\mathfrak {K}(t) = \mathfrak {K}(x)\), it follows from Corollary 6.4 that \(\psi ^* \mathfrak {K}(t) = \mathfrak {K}(t)\). Finally, since \(t \in K {\smallsetminus } k\) was arbitrary, Proposition 9.5 shows that \(\psi ^* \in (\mathbb {Z}/\ell )^\times \cdot \mathbf {1}_{{\text {k}}^\mathrm{M}_1(K)}\). In particular, \(\psi = \tau ^{1} \cdot \phi _K\) is contained in \((\mathbb {Z}/\ell )^\times \cdot \mathbf {1}_{\mathcal {G}_K^a}\).
Proof of (2):
The proof of the surjectivity of this map is more difficult, but it essentially follows from the technical definition of \(\mathcal {V}\) being 5connected and Theorem A, as follows. First, let us fix an element \(\phi \) of \({{\mathrm{Aut}}}^\mathrm{c}(\pi ^a_\mathcal {V})\). Recall that this element \(\phi \) is represented by a system \((\phi _X)_{X \in \mathcal {V}}\) with \(\phi _X \in {{\mathrm{Aut}}}^\mathrm{c}(\pi ^a(X))\) which is compatible with the morphisms from \(\mathcal {V}\). To prove surjectivity, we must show that there exists some \(\tau \in {{{\mathrm{Gal}}}_{k_0}}\) and some \(\epsilon \in (\mathbb {Z}/\ell )^\times \), such that for all positivedimensional \(X \in \mathcal {V}\), \(\epsilon \cdot \phi _X\) is defined by \(\tau \).
By the definition of \(\mathcal {V}\) being 5connected, we see that there exists some birational system \(\mathcal {U}\) and some finite tuple \(\mathbf {a}\) of elements of \(k_0^\times \) such that \(\dim \mathcal {U}\ge 5\) and such that \(\mathcal {U}_\mathbf {a}\) is contained in \(\mathcal {V}\). Thus, by Theorem A, there exists a unique \(\tau \in {{{\mathrm{Gal}}}_{k_0}}\) and an \(\epsilon \in (\mathbb {Z}/\ell )^\times \) such that \(\epsilon \cdot \phi _{\mathcal {U}_\mathbf {a}}\) is defined by \(\tau \), and thus \(\epsilon \cdot \phi _\mathcal {U}\) is defined by \(\tau \) as well. To simplify the notation, we replace \(\phi \) by \(\epsilon \cdot \phi \), so we must prove that \(\phi \) itself is defined by \(\tau \).
 (1)
One has \(\dim \mathcal {U}_{2i+1} \ge 5\).
 (2)
The birational system \(\mathcal {U}_{2i+1}\) attaches \(\mathcal {U}_{2i}\) to \(\mathcal {U}_{2i+2}\) in \(\mathcal {V}\).
Footnotes
Notes
Acknowledgements
First and foremost, the author would like to thank Florian Pop and Thomas Scanlon for numerous technical discussions concerning the topics in this paper. The author also thanks all who expressed interest in this paper, and especially Y. Hoshi, E. Hrushovski, M. Kim, J. Mináč, A. Obus, A. Silberstein and A. Tamagawa. Finally, the author thanks the anonymous referee, whose thoughtful comments helped improve this paper in various ways.
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