Abstract
In previous work, the authors defined a category \(\text {SMod}_F\) of finite Galois modules decorated with local conditions for each global field F. In this paper, given an extension K/F of global fields, we define a restriction of scalars functor from \(\text {SMod}_K\) to \(\text {SMod}_F\) and show that it behaves well with respect to the Cassels–Tate pairing. We apply this work to study the class groups of global fields in the context of the Cohen–Lenstra heuristics.
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1 Introduction
The aim of this paper is twofold. First, we wish to apply the theory of Selmer groups and the Cassels–Tate pairing developed in [19] to study class groups of global fields. The idea of using the Cassels–Tate pairing in the study of class groups is due to Flach [8, p. 122], but much of the power of this idea comes from the flexibility of the framework introduced by the authors in [19].
In that paper, we defined for each global field F a category \(\text {SMod}_F\) of “Selmerable modules”, or finite \(G_F\) modules decorated with suitable local conditions. To each object in \(\text {SMod}_F\) we defined a corresponding Selmer group, resulting in a functor \(\text {Sel}\) from \(\text {SMod}_F\) to the category of abelian groups. In this framework, exact sequences in \(\text {SMod}_F\) give rise to instances of the Cassels–Tate pairing on Selmer groups.
However, for our applications to class groups, the theory in [19] is incomplete. Specifically, given an extension of global fields K/F, our applications require us to move between the categories \(\text {SMod}_K\) and \(\text {SMod}_F\), and we did not develop the theory to do this in [19]. With this in mind, the second goal of this paper is to explore the relationship between the Cassels–Tate pairings defined for these two categories. Our main result here shows that the Cassels–Tate pairing behaves well under a certain restriction of scalars functor from \(\text {SMod}_K\) to \(\text {SMod}_F\).
We now go through our work towards these two goals in more detail, using the language of [19]. In an effort to be self-contained, we have provided a brief recap of what we need from that paper in Sect. 2.2.
1.1 The behavior of the Cassels–Tate pairing in field extensions
Take K/F to be an extension of global fields. Given a \(G_{K}\)-module M, we can consider the corresponding induced \(G_{F}\)-module
Shapiro’s lemma gives a natural isomorphism
and we also have a local Shapiro isomorphism
We discuss this latter map in Sect. 4.1. With this set, we obtain an exact functor
via the assignment
Using a subscript to record which of the categories \(\text {SMod}_F\) or \(\text {SMod}_K\) we are working in, the global Shapiro isomorphism then gives a natural isomorphism
Now given a short exact sequence
in \(\text {SMod}_K\), we can combine (1.1) with the explicit identification between \(\text {Ind}_{K/F}M_1^\vee \) and \((\text {Ind}_{K/F}M_1)^\vee \) detailed in Definition 4.3, to view the Cassels–Tate pairing associated to the induced sequence \(\text {Ind}_{K/F}E\) as a pairing
We then have the following, which constitutes the final basic result on our version of the Cassels–Tate pairing. For a more precise statement, see Theorem 4.5.
Theorem 1.1
Take any short exact sequence E in \(\text {SMod}_K\), and let \(\text {Ind}_{K/F}E\) denote the corresponding exact sequence in \(\text {SMod}_F\). Then, with the identifications above, we have
According to Theorem 1.1, the functor \(\text {Ind}_{K/F}\) preserves the structure on the category \(\text {SMod}_K\) provided by the Cassels–Tate pairing. Since \(\text {Ind}_{K/F} M\) is typically a richer object than M, we can often prove extra identities for pairings over K by changing fields to F. Furthermore, as we detail in Corollary 4.9, a consequence of Theorem 1.1 is that corestriction is an adjoint to restriction with respect to the Cassels–Tate pairing, assuming local conditions are handled appropriately. This generalizes a result of Fisher for the pairing associated to an elliptic curve [7].
1.2 Results on class groups
As mentioned above, after proving Theorem 1.1 the rest of the paper considers how the theory may be applied, in particular to the study of class groups. We go through a couple of these applications in detail now.
1.2.1 Class groups of number fields with many roots of unity
Choose a prime \(\ell \), and take F to be a global field of characteristic not equal to \(\ell \). Take \(\text {Cl}\,\, F\) to be the class group of F, take \(H_F\) to be its Hilbert class field, and define
Taking
to be the Artin reciprocity map, we define the reciprocity pairing
by the rule \(\text {RP}(\phi ,I) = \phi (\text {rec}_F(I))\). As Flach observed [8, p. 122], this pairing may, in the terminology of [19, Sect. 4.2], be packaged as the Cassels–Tate pairing associated to the exact sequence
decorated with the unramified local conditions.
From this repackaging, we can correctly predict that the duality identity for the Cassels–Tate pairing, which we recall explicitly as (2.3), might give interesting information about the reciprocity pairing on class groups of fields with many roots of unity; this is the case where a portion of the above sequence is self-dual. The following notation will be useful.
Notation 1.2
Let m be a positive integer indivisible by the characteristic of F, and suppose that F contains \(\mu _m\). Given \(\alpha \) in \(F^{\times }\), take \(\alpha ^{1/m}\) to be an \(m^{th}\) root of \(\alpha \). We then denote by \(\chi _{m, \,\alpha }\) the homomorphism
defined by
Proposition 1.3
Let a and b be positive integers indivisible by the characteristic of F such that F contains \(\mu _{ab}\). Choose fractional ideals I, J of F and \(\alpha \), \(\beta \) in \(F^{\times }\) satisfying
Then, if \(F(\alpha ^{1/a})/F\) and \(F(\beta ^{1/b})/F\) are everywhere unramified, we have
In the case where a and b are powers of a given odd prime, this explicit reciprocity law is a consequence of work by Lipnowski, Sawin, and Tsimerman [15], who proved it while developing a variant of the Cohen–Lenstra–Martinet heuristics for the class groups of quadratic extensions of number fields containing many roots of unity. We will elaborate further on the connection between our work and [15] in Sect. 6.
By combining the results on base change with our theory of theta groups [19, Sect. 5], we are also able to prove the following result.
Proposition 1.4
Choose a global field F of characteristic other than 2, and take m to be a positive integer indivisible by the characteristic of F. Choose a positive divisor d of m such that F contains \(\mu _{m^2/d}\). We further suppose that there is a field automorphism \(\kappa :F\rightarrow F\) that satisfies \(\kappa ^2 = 1\) and which restricts to the map \(\zeta \mapsto \zeta ^{-1}\) on \(\mu _{m^2/d}\).
Take I to be a fractional ideal of F such that \(I^m\) is principal, and choose \(\alpha \in F^{\times }\) so
Assume that \(F(\alpha ^{1/m})/F\) is unramified at all places of F and splits completely at all primes dividing 2. We then have
Applying Proposition 1.3 in the above situation with \(a = m\), \(b = m/d\), and \(J = \kappa I^d\) gives
where the second equality follows from the equivariance of the Artin reciprocity map and the third follows from the assumptions on \(\kappa \). We may conclude that
In this way, Proposition 1.4 may be thought of as a supplement to Proposition 1.3. Both of these propositions will be proved in Sect. 5.
It is our hope that Proposition 1.4 and other results like it (see Proposition 6.6, for example) will prove to be useful in extending the Cohen–Lenstra–Martinet heuristics to handle 2-primary parts of class groups.
1.2.2 Decomposing class groups
In [4], Cohen and Lenstra gave a conjectural heuristic describing how the class group \(\text {Cl}\,\, K\) varies as K moves through the quadratic extensions of \({\mathbb {Q}}\). These heuristics omitted the 2-primary part of the class group, and Gerth [10] later supplemented that work with a prediction for the distribution of the groups \(2(\text {Cl}\,\, K) [2^\infty ]\). For imaginary quadratic fields, Gerth’s prediction is now known to be correct [27]. By considering \(2(\text {Cl}\,\, K)\) rather than \(\text {Cl}\,\, K\), Gerth avoided the subgroup \((\text {Cl}\,\, K)[2]\), which can be directly computed using genus theory and is typically large.
A related question concerns the behavior of Selmer groups of elliptic curves in quadratic twist families, where one fixes an elliptic curve \(E/{\mathbb {Q}}\) and studies the behavior of the Selmer groups of the quadratic twists \(E^\chi /{\mathbb {Q}}\) as \(\chi \) varies over quadratic characters. See e.g. [12, 13, 27].
These are analogous problems, but they belong to different genres of arithmetic statistical questions. Both kinds of problem start with a fixed \(G_{F}\)-module M, where F is a fixed global field. In the first, one asks how \(\text {Sel}_K\, M\) varies as K moves through a family of number fields over F. In the second, one asks how \(\text {Sel}_{F}\, M^{\chi }\) varies as \(\chi \) moves through a family of twists. In the particular examples considered above, M is taken either to be \({\mathbb {Q}}/{\mathbb {Z}}\) with the unramified local conditions, or the divisible Tate module of E with the standard local conditions.
We can often translate between these two kinds of problems. Suppose K/F is a quadratic extension of number fields, and take \(\chi \) to be its corresponding quadratic character. If M is a \(G_F\)-module for which multiplication by 2 is invertible, the module \(\text {Ind}_{K/F} M\) splits as a direct sum
Specifically, taking \(\sigma \) to be a generator for \(\text {Gal}(K/F)\), the subspace M consists of the elements of the form \([\sigma ] \otimes x + [1] \otimes \sigma x\) with x in M, while \(M^{\chi }\) consists of the elements of the form \([\sigma ] \otimes x - [1] \otimes \sigma x\) with x in M.
Applying this decomposition to class groups using (7.3) gives
for any odd prime \(\ell \), where the local conditions are set as in Sect. 7. Here the action of \(G_F\) on \(\text {Cl}\,^*K\) is given by the formula
for \(\sigma \in G_F\), \(\phi \in \text {Cl}\,^*K\) and \(\tau \in \text {Gal}(H_K/K)\).
If multiplication by 2 on M is not invertible, the relationship between \(\text {Sel}_K \,M\) and \(\text {Sel}_F\, M^{\chi }\) can be more complicated. In this case, the Cassels–Tate pairing proves to be a convenient tool for moving from Selmer groups over extensions to Selmer groups of twists. As an example, we will prove the following as the \(\ell =2\) case of Proposition 7.4.
Proposition 1.5
Take K/F to be a quadratic extension of number fields with associated quadratic character \(\chi \), and take S to be the set of places of F not dividing 2 where K/F is ramified. Suppose there is no element of \(F^{\times }\setminus (F^{\times })^2\) which lies in \((F_v^{\times })^{2}\) at every place v in S, and which has even valuation at all remaining finite primes.
Then we have an exact sequence
of finite \(\text {Gal}(K/F)\)-modules.
The first term \( \left( \text {Cl}\,^*K\right) ^{\text {Gal}(K/F)} [2^{\infty }]\) is essentially determined from genus theory, making it an analogue of the subgroup \((\text {Cl}\,\, K)[2]\) in Gerth’s conjecture. Meanwhile, the final term \(\text {Sel}({\mathbb {Q}}_2/{\mathbb {Z}}_2)^{\chi }\), which agrees with \(2(\text {Cl}\,\, K) [2^\infty ]\) in the situation considered by Gerth, is more in line with what the Cohen–Lenstra heuristics can handle.
This work plays a role in the second author’s work on the distribution of Selmer groups in twist families [28, 29]. In particular, in [29, Example 3.11], Proposition 7.4 is combined with [29, Theorem 2.14] to give distributional results about class groups in families of cyclic extensions.
The idea of studying the \(\ell ^{\infty }\) part of the class group of K by studying the module \(\text {Ind}_{K/F} {\mathbb {Q}}_{\ell }/{\mathbb {Z}}_{\ell }\) and its irreducible components is not new; see e.g [23, 1.6.4], [6], or [30]. The novelty of our work lies in the ability to handle “bad” choices of \(\ell \).
1.3 Layout of the paper
In Sect. 2 we summarize the notation used throughout the paper and review the material we need from [19].
Section 3 addresses a technical point that we have so far sidestepped, showing that the Cassels–Tate pairing is, in a suitable sense, independent of the choice of embeddings \(F^s\hookrightarrow F^s_v\) used to define restriction maps on cohomology. In Sect. 4, we define the functor \(\text {Ind}_{K/F} :\text {SMod}_K \rightarrow \text {SMod}_F\) for a finite separable extension K/F and prove Theorem 1.1. These sections rely on some basic properties of group change homomorphisms for which we have not found a suitable reference. We give this necessary background material in Appendix A.
In Sect. 5, we prove our symmetry result for class groups of number fields with many roots of unity. As mentioned above, this recovers a result of Lipnowski, Sawin, and Tsimerman [15], who encode the symmetry result using the language of bilinearly enhanced groups. In Sect. 6, we show how one can use the Cassels–Tate pairing to define such a bilinearly enhanced structure, giving an alternative to the approach of [15]. Unlike in that work, we are able to allow \(\ell =2\) by drawing on the framework of theta groups from [19].
Section 7 contains the material introduced in Sect. 1.2.2, and Sect. 8 gives two miscellaneous applications of the theory of the Cassels–Tate pairing. The first of these, to calculating involution spins, is needed for calculations of fixed point Selmer groups in [29, Sect. 5].
2 Notation and review of [19]
2.1 Conventions
We will keep the same notation as in [19, Sect. 2]. In an effort to be self-contained, we recall some of our conventions now.
Throughout, F will denote a global field with fixed separable closure \(F^s/F\) and absolute Galois group \(G_F=\text {Gal}(F^s/F)\). For each place v of F, \(F_v\) denotes the completion of F at v, \(F_v^s\) denotes a fixed separable closure of \(F_v\), and \(G_v\) denotes the Galois group of \(F^s_v/F_v\). We write \(I_v\) for the inertia subgroup of \(G_v\). For each place v we fix an embedding \(F^s\hookrightarrow F_v^s\), via which we view \(G_v\) as a closed subgroup of \(G_F\).
Given a topological group G, a discrete G-module M, and \(i\ge 0\), \(H^i(G, M)\) will denote the continuous group cohomology of G acting on M. We often work with these groups using the language of inhomogeneous i-cochains [21, Sect. 2], the group of which we denote \(C^i(G, M)\). We denote by \(Z^i(G,M)\) the subgroup of \(C^i(G, M)\) consisting of i-cocycles. Given \(\phi \) in \(C^i(G_F, M)\) or \(H^i(G_F, M)\), we write \(\phi _v\) for the restriction of \(\phi \) to \(G_v\).
Finally, given a \(G_F\)-module M, we write \(M^\vee \) for the dual \(G_F\)-module \(\text {Hom}(M,\, (F^s)^{\times })\), and denote by \(\cup _M\) the cup product constructed from the standard evaluation pairing \(M \times M^{\vee }\rightarrow (F^s)^{\times }\) (cf. [19, Notation 2.2]).
2.2 A review of [19]
We briefly recap some of the relevant material from [19]. The reader familiar with that work may safely skip this section.
2.2.1 The category \(\text {SMod}_F\)
In [19, Definition 1.1] the category \(\text {SMod}_F\) was introduced. Its objects are pairs \((M,{\mathscr {W}})\) consisting of a finite \(G_F\)-module M of order indivisible by the characteristic of F along with a compact open subgroup \({\mathscr {W}}\) of the restricted product 
. Here the restricted product is taken with respect to the subgroups \(H^1_{\text {ur}}(G_v,M)\) of unramified classes. It is shown in [19, Sect. 4] that \(\text {SMod}_F\) is a quasi-abelian category. We recall also that:
-
The dual of an object \((M,{\mathscr {W}})\) is the pair \((M^\vee , {\mathscr {W}}^\perp )\), where \({\mathscr {W}}^\perp \) is the orthogonal complement of \({\mathscr {W}}\) with respect to the pairing induced by Tate local duality; we denote by \(\vee \) the contravariant functor on \(\text {SMod}_F\) sending an object to its dual.
-
A sequence
$$\begin{aligned} E = \left[ 0 \rightarrow (M_1,\,{\mathscr {W}}_1) \xrightarrow {\,\,\iota \,\,} (M,\, {\mathscr {W}}) \xrightarrow {\,\,\pi \,\,} (M_2,\, {\mathscr {W}}_2) \rightarrow 0\right] \end{aligned}$$(2.1)in \(\text {SMod}_F\) is exact if the underlying sequence of \(G_F\)-modules is exact and if
$$\begin{aligned} \iota ^{-1}({\mathscr {W}}) = {\mathscr {W}}_1\quad \text {and}\quad \pi ({\mathscr {W}}) = {\mathscr {W}}_2. \end{aligned}$$
With these definitions, the duality functor \(\vee \) preserves exact sequences.
Associated to any object \((M,{\mathscr {W}})\) in \(\text {SMod}_F\) is the Selmer group
The definition of the morphisms in \(\text {SMod}_F\) ensure that the map sending \((M,{\mathscr {W}})\) to its Selmer group defines a functor \(\text {Sel}\) from \(\text {SMod}_F\) to the category of finite abelian groups.
2.2.2 The Cassels–Tate pairing
The main object of interest in [19] is a version of the Cassels–Tate pairing associated to each short exact sequence in \(\text {SMod}_F\).
Theorem 2.1
([19], Theorem 1.3) Given a short exact sequence E in \(\text {SMod}_F\), as depicted in (2.1), there is a bilinear pairing
with left kernel \(\pi (\text {Sel}M)\) and right kernel \(\iota ^\vee (\text {Sel}M^\vee )\).
Two basic properties of the pairing that are established in [19] are the following:
-
Given E as above, and identifying \(M_2\) with \((M_2^{\vee })^{\vee }\) in the standard way, we have the duality identity
$$\begin{aligned} \text {CTP}_{E}(\phi , \psi ) = \text {CTP}_{E^\vee }(\psi , \phi ) \end{aligned}$$(2.3)for all \(\phi \) in \(\text {Sel}\,M_2\) and \(\psi \) in \(\text {Sel}\, M_1^{\vee }\).
-
Given a morphism of short exact sequences
in \(\text {SMod}_F\), we have the naturality property
$$\begin{aligned} \text {CTP}_{E } (\phi , \,{f_1^{\vee }}(\psi ) ) \,=\, \text {CTP}_{E'}(f_{2} (\phi ), \psi ), \end{aligned}$$(2.5)for all \(\phi \) in \(\text {Sel}\, M_{2 }\) and \(\psi \) in \(\text {Sel}\, (M_{1}')^{\vee }\).
One source of diagrams of the form (2.4) comes from the pullback and pushout operations for short exact sequences, which we defined in [19, Definition 4.1]. Briefly, given an exact sequence E as above and a morphism \(f :N_2 \rightarrow M_2\) in \(\text {SMod}_F\), there is a natural short exact sequence in \(\text {SMod}_F\) forming the top row of a commutative diagram
in which the right square is cartesian. We refer to it as the pullback of E along f. Dually, the pushout of E along a morphism \(g :M_1 \rightarrow N_1\) is an exact sequence in \(\text {SMod}_F\) fitting into a commutative diagram
Pullback and pushout in \(\text {SMod}_F\) induce the corresponding operations on underlying \(G_F\)-modules.
We will occasionally need to work with the explicit definition of the Cassels–Tate pairing in this paper, so we include this next. This pairing is modeled on the Weil pairing definition of the Cassels–Tate pairing for abelian varieties [17, I.6.9]. That the various choices involved can be made, and that the result is independent of these choices, is justified in [19, Sect. 3].
Definition 2.2
([19], Definition 3.2) Take E to be a short exact sequence in \(\text {SMod}_F\), as depicted in (2.1). Take \(\phi \in \text {Sel}\, M_2\) and \(\psi \in \text {Sel}\, M_1^{\vee }\). To define \(\text {CTP}_E(\phi , \psi )\) we make ‘global’ choices consisting of:
-
cocycles \({\overline{\phi }}\) and \({\overline{\psi }}\) representing \(\phi \) and \(\psi \) respectively,
-
a cochain \(f\in C^1( G_{F}, M)\) satisfying \({\overline{\phi }} = \pi \circ f,\) and
-
a cochain \(\epsilon \) in \(C^2(G_F, (F^s)^{\times })\) so
$$\begin{aligned} d\epsilon \,=\, \iota ^{-1}(df) \,\cup _{M_1}\, {\overline{\psi }}. \end{aligned}$$
We further make ‘local’ choices consisting of:
-
cocycles \({\overline{\phi }}_{v, M}\) in \(Z^1(G_v, M)\) for each place v of F, so that \(\pi ({\overline{\phi }}_{v, M}) = {\overline{\phi }}_v\), and so that \(({\overline{\phi }}_{v, M})_v\) represents a class in \({\mathscr {W}}\).
Then
lies in \(Z^2\left( G_v, (F^s)^{\times }\right) \) for each v, and we have
3 Independence of the choice of embeddings
In this section we address a technical point concerning the category \(\text {SMod}_F\) that is needed in the proof of Theorem 4.5. In what follows, we refer to Appendix A for the relevant background for various operations on cohomology.
Notation 3.1
Recall that we have fixed a field embedding \( F^s\hookrightarrow F^s_v\) for each place v of F. Call this embedding \(i_v\). This embedding determines the embedding of \(G_v\) in \(G_F\), and these embeddings are used in the construction of \(\text {SMod}_F\).
Suppose we have a second set of embeddings \(i'_v :F^s\hookrightarrow F^s_v\). Then there is \(\tau _v\) in \(G_F\) so \(i'_v\) equals \(i_v \circ \tau _v\), and the associated decomposition group is \(\tau _v^{-1} G_v \tau _v\). Given a discrete \(G_F\)-module M, we have a natural isomorphism
given by conjugation at each place.
Taking \(\text {SMod}'_F\) to be the variant of \(\text {SMod}_F\) defined using the embeddings \(i'_v\) instead of \(i_v\), we can define an equivalence of categories \(\text {SMod}_F \rightarrow \text {SMod}'_F\) by
This respects duality and Selmer groups. The following proposition shows it also respects the Cassels–Tate paring.
Proposition 3.2
Given \(\text {SMod}_F\) and \(\text {SMod}_F'\) as above, take E to be a short exact sequence in \(\text {SMod}_F\), and take \(E'\) to be the corresponding sequence in \(\text {SMod}_F'\). Then
The proof relies on the following. Given a profinite group G, a closed subgroup H of G, a discrete H-module M, and \(\tau \in G\), we can consider the \(\tau H \tau ^{-1}\)-module \(\tau M\), as detailed in Definition A.2. As in Definition A.6, for each \(n\ge 0\) we have a conjugation operation
on cochains. In the case \(G = H\), we have a natural isomorphism \(M \cong \tau M\) given by the operation \(c_\tau \) of Fig. 2, and composing this with conjugation gives a chain map
This chain map is homotopic to the identity. Explicitly, we may follow [3, Appendix 2] or [20, 4.5.5] to define a map
by the formula
On \(C^0\) we take \(h_{\tau }\) to be the zero map.
Given any f in \(C^n(G, M)\), this construction satisfies
We may also verify that, given G-modules \(M_1\) and \(M_2\), integers \(k,j\ge 0\), and
we have the identity
where \(\tau g\) is shorthand for \(\text {conj}_\tau ^{G} g\). These identities are most easily checked by considering the corresponding operations on homogeneous cochains (cf. Sect. A.1).
Proof of Proposition 3.2
Take
to be a short exact sequence in \(\text {SMod}_F\), and choose \(\phi \in \text {Sel}M_2\) and \(\psi \in \text {Sel}M_1^{\vee }\). Take
to be a tuple for computing \(\text {CTP}_E(\phi , \psi )\) as in Definition 2.2. With \(h_\tau \) as in (3.1) above, the proposition reduces to the claim that, for each place v of F,
represents the same class in \(H^2(G_v, \,(F^s_v)^{\times })\) as
To prove this claim, we start by using (3.2) to calculate that their difference equals
From (3.3) we have
so the difference of the expressions is equal to
which is the coboundary of
The two expressions thus represent the same cohomology class, giving the claim. \(\square \)
4 Changing the global field
In this section we consider the behavior of Selmer groups under change of field and prove Theorem 1.1. As in the previous section, we will frequently refer to Appendix A for the relevant background on cohomological operations related to field change.
Take K/F to be an extension of global fields with K contained in \(F^s\). Given a \(G_{K}\)-module M of order indivisible by the characteristic of F, we consider the corresponding induced \(G_{F}\)-module
We wish to extend the assignment \(M\mapsto \text {Ind}_{K/F}M\) to a functor from \(\text {SMod}_{K}\) to \(\text {SMod}_{F}\). For this we need to understand how local conditions subgroups behave under induction. This uses a local version of the Shapiro isomorphism, which we turn to now.
4.1 Local and global Shapiro isomorphisms
For M as above, Shapiro’s lemma gives a natural isomorphism
that we call the global Shapiro isomorphism.
An equivalent definition to the following local version of (4.1) can be found in [25, 3.1.2], with (4.4) appearing as [25, (3.3)].
Definition 4.1
Given a place v of F and a place w of K over v, the fields \(F^s_v\) and \(K_w^s\) are isomorphic. We have embeddings
corresponding to these places. Fixing some isomorphism between \(F^s_v\) and \(K^s_w\), there is some choice of \(\tau _w \in G_F\) so that \(i_w\) is identified with \(i_v \circ \tau _w\). We then have
The set \(\{\tau _w\}_{w\mid v}\) gives a set of representatives for the double coset \(G_v\backslash G_F/G_K\), and for any discrete \(G_K\)-module M we can consider the isomorphism
of Definition A.5. We abbreviate 
as \(\text {sh}_{\text {loc}, \,v}\). In the case \(k = 1\), we can apply (A.8) with \(C_1 = I_v\) to show that this restricts to an isomorphism
Combining the maps \(\text {sh}_{\text {loc}, \,v}\) for every v gives a topological isomorphism of restricted products
4.2 The functor \(\text {Ind}_{K/F}\)
We now define the sought functor \(\text {Ind}_{K/F}\).
Definition 4.2
The isomorphism (4.3) sends compact open subgroups to compact open subgroups, so we may define a faithful additive functor
by setting
A special case of (A.7) gives the commutative square
from which we see that the global Shapiro map gives an isomorphism
Here the subscript indicates whether a functor is being considered on \(\text {SMod}_F\) or \(\text {SMod}_K\). Consequently, the Shapiro isomorphism defines a natural isomorphism
We now consider how the functor \(\text {Ind}_{K/F}\) combines with the duality functors on \(\text {Sel}_F\) and \(\text {Sel}_K\). For this we require the following definition.
Definition 4.3
Given \((M, {\mathscr {W}})\) in \(\text {SMod}_{K}\), we consider the perfect \(G_{F}\)-equivariant bilinear pairing
defined by
where \(\langle ,\,\rangle \) is the evaluation pairing from \(M \otimes M^{\vee }\) to \((F^s)^{\times }\). By tensor-hom adjunction, this pairing corresponds to a \(G_{F}\)-equivariant isomorphism
Per the following proposition, these maps define a natural isomorphism
on \(\text {SMod}_K\).
Proposition 4.4
Let \((M,{\mathscr {W}})\) be an object in \(\text {SMod}_{K}\). We then have
In particular, t gives an isomorphism
We will prove this proposition in Sect. 4.5. From this proposition, the tuple \((\text {Ind}_{K/F}, \, t)\) forms a morphism
of additive categories with duality in the sense of [1, Definition 1.1.8].
The functor \(\text {Ind}_{K/F}\) is exact, and we may compare the Cassels–Tate pairing of a short exact sequence E in \(\text {SMod}_K\) with the pairing for \(\text {Ind}_{K/F} E\). The following theorem, which formalizes Theorem 1.1, shows that these pairings are essentially equivalent.
Theorem 4.5
Choose any short exact sequence
in \(\text {SMod}_K\). Then, for any \(\phi \) in \(\text {Sel}_K\, M_2\) and \(\psi \) in \(\text {Sel}_K\, M_1^{\vee }\), we have
As with Proposition 4.4, we will prove this in Sect. 4.5.
Example 4.6
Given K/F as above and an abelian variety A/K, we may consider the restriction of scalars \(R_{K/F}\,A\), which is an abelian variety over F defined as in e.g. [16]. We have a canonical isomorphism of discrete \(G_F\)-modules
Given an integer n coprime to the characteristic of F, we have an isomorphism
in \(\text {SMod}_F\), where \({\mathscr {W}}_n^F\) and \({\mathscr {W}}_n^K\) denote the local conditions defining the n-Selmer groups of \(R_{K/F}\,A/F\) and A/K respectively. The Shapiro isomorphism gives an identification
In this language, Theorem 4.5 says that the Cassels–Tate pairing of A over K is identified with the Cassels–Tate pairing of \(R_{K/F}\,A\) over F.
Remark 4.7
If we instead define \(\text {SMod}_F\) and \(\text {SMod}_K\) using the weaker notion of suitable local conditions given in [19, Definition 4.16], then Definition 4.2 still defines a functor
and Proposition 4.4 and Theorem 4.5 still apply.
In addition, we may use the explicit constructions of Sect. A.3 to extend \(\text {Ind}_{K/F}\) to a functor from \(\text {SMod}_{K, \ell }^{\infty }\) to \(\text {SMod}_{F, \ell }^{\infty }\) for each \(\ell \) not equal to the characteristic of F, and to extend the natural isomorphism \(\text {sh}_{K/F}\) in this context. Here \(\text {SMod}_{K, \ell }^{\infty }\) and \(\text {SMod}_{F, \ell }^{\infty }\) are the categories of potentially-infinite Galois \({\mathbb {Z}}_{\ell }\)-modules decorated with local conditions introduced in [19, Sect. 4.2]. We still have Proposition 4.4 and Theorem 4.5 in this context, as can be proved from [19, Lemma 4.9] and the definition of the Cassels–Tate pairing in these categories.
Before turning to the proofs of Proposition 4.4 and Theorem 4.5, we give some basic consequences of these results.
4.3 The effect of corestriction and restriction
By (A.2), restriction and corestriction can be phrased in terms of the Shapiro isomorphism, so Theorem 4.5 can be used to understand their effect on the Cassels–Tate pairing. We begin with a definition. Here and below, we refer to Fig. 2 for the definition of the maps i, \(\nu \), \(i_1\) and \(\nu _1\).
Definition 4.8
Let M be a \(G_F\)-module. For each place w of K dividing a place v of F, we define natural homomorphisms
by setting
Here, by an abuse of notation, we write \(\tau _w\) as shorthand for the map \(\text {conj}_{\tau _w}^{G_w}\) of Notation A.4, where \(\tau _w\in G_F\) is fixed as in Definition 4.1. By (A.6), the w component of \(\text {sh}_{\text {loc},\, v}\) is the composition of the homomorphisms
and the w component of the inverse is the composition
If M is a \(G_F\)-module we also have a commutative diagram
as can be seen by verifying the result for \(k = 0\).
In what follows we abbreviate \(\text {cor}_{G_{F}}^{G_{K}}\) as \(\text {cor}\) and \(\text {res}_{G_{K}}^{G_{F}}\) as \(\text {res}\).
Corollary 4.9
Given exact sequences
associated to the same exact sequence of \(G_{F}\)-modules, we have the following:
-
(1)
Suppose
$$\begin{aligned} \sum _{v\text { of }F}\sum _{w\mid v}\text {cor}_{w\mid v}(\phi _w)\in {\mathscr {W}}_F\quad \text {for all}\quad (\phi _w)_w \in {\mathscr {W}}_K. \end{aligned}$$Then, for any \(\phi \in \text {Sel}_K\, M_2\) and \(\psi \in \text {Sel}_F\, M_1^{\vee }\), we have
$$\begin{aligned} \text {CTP}_{E_F}\left( \text {cor}\,\phi , \,\psi \right) \, =\, \text {CTP}_{E_K}\left( \phi , \,\text {res}\, \psi \right) . \end{aligned}$$ -
(2)
Suppose
$$\begin{aligned} \sum _{v\text { of } F}\sum _{w\mid v}\text {res}_{w\mid v}(\phi _v)\in {\mathscr {W}}_K \quad \text {for all}\quad (\phi _v)_v \in {\mathscr {W}}_F. \end{aligned}$$Then, for any \(\phi \in \text {Sel}_F\, M_2\) and \(\psi \in \text {Sel}_K\, M_1^{\vee }\), we have
$$\begin{aligned} \text {CTP}_{E_F}\left( \phi ,\, \text {cor}\, \psi \right) \, =\, \text {CTP}_{E_K}\left( \text {res}\,\phi , \,\psi \right) . \end{aligned}$$ -
(3)
Suppose the conditions of the previous two parts both hold. Then, for any \(\phi \) in \(\text {Sel}_F\, M_2\) and \(\psi \) in \(\text {Sel}_F\, M_1^{\vee }\), we have
$$\begin{aligned} \text {CTP}_{E_K}\left( \text {res}\,\phi ,\, \text {res}\,\psi \right) = [K: F] \cdot \text {CTP}_{E_F}(\phi , \,\psi ). \end{aligned}$$
Proof
From (A.2), we have
For the first part, we start with the commutative diagram of \(G_F\)-modules
with exact rows. From (4.9) and our assumptions, this induces a commutative diagram in \(\text {SMod}_{F}\) with top row \(\text {Ind}_{K/F} E_K\) and bottom row \(E_F\). We also note that the diagram
is commutative. The first part then follows from Theorem 4.5, (A.2), and naturality applied to (4.10). The proof for the second part is similar.
Under the conditions of the third part, we find that \(\text {res}\, \phi \) is in \(\text {Sel}_{K} \,M_2\). The conditions of the first part also hold, so
giving the claim. \(\square \)
Remark 4.10
The first two parts of this corollary were proved by Fisher for the Cassels–Tate pairing of an elliptic curve over a number field [7]. For more general abelian varieties, a sketched proof for these identities appears in [31, Theorem 8]. We mention that the latter work relies on a cochain identity that does not hold for the standard resolutions used to define the Cassels–Tate pairing [31, (6)]. The third part of the corollary extends a result for abelian varieties due to Česnavičius [2, Lemma 3.4].
If \([K :F]\) is coprime to the order of M, Corollary 4.9 (3) shows that the Cassels–Tate pairing for E over F can be recovered from the Cassels–Tate pairing over K. In particular, if M has order a power of \(\ell \), we can compute the Cassels–Tate pairing for E by computing it over any field K for which the image of \(G_{K}\) in \(\text {Aut}(M)\) is an \(\ell \)-group.
4.4 The effect of conjugation
Given K/F as above, and given \(\tau \) in \(G_F\), we can define a functor
by taking
Conjugation by \(\tau \) then defines a natural isomorphism
Given M in \(\text {SMod}_K\), the map \(\rho _{\tau }\) defined in Fig. 2 gives a natural isomorphism
in the category \(\text {SMod}_F\). By (A.3), we have a commutative square
Naturality of the Cassels–Tate pairing and Theorem 4.5 gives the following, which can also be proven directly from the definition of the Cassels–Tate pairing.
Proposition 4.11
Given a short exact sequence E in \(\text {SMod}_K\), and given \(\tau \) in \(G_F\), we have
We consider one special case of this proposition. Suppose that K/F is Galois, and suppose we have an exact sequence of \(G_F\)-modules
Suppose also that we augment this to an exact sequence
in \(\text {SMod}_K\) so that
Under this assumption, the Selmer group for M, \(M_1\), \(M_2\), and their duals are all \(\text {Gal}(K/F)\)-modules under the conjugation action. The above proposition gives
for all \(\tau \) in \(\text {Gal}(K/F)\), \(\phi \) in \(\text {Sel}_{K} M_2\), and \(\psi \) in \(\text {Sel}_{K} M_1^{\vee }\).
4.5 Proofs of Proposition 4.4 and Theorem 4.5
In Sect. A.2, we defined corestriction using injective resolutions. This contrasts with our definition of the Cassels–Tate pairing, which is defined explicitly by manipulating inhomogeneous cochains.
To prove Theorem 4.5, we need these definitions to be more compatible. For the work below, we do this by redefining corestriction in terms of inhomogeneous cochains. A potentially more elegant approach would be to redefine the Cassels–Tate pairing using injective resolutions. We hope to return to this alternative in future work.
In the proofs of Proposition 4.4 and Theorem 4.5 we will use the following class field theoretic lemma.
Lemma 4.12
Given v a place of F and w a place of K dividing v, we have a commutative diagram
Proof
For v nonarchimedean this is [21, (7.1.4)]. If v is archimedean, it can be verified by separate consideration of the two cases \(K_w = {\mathbb {C}}\) and \(K_w = {\mathbb {R}}\). \(\square \)
Proof of Proposition 4.4
Given a place v of F, and given \(\phi _w\) in \(H^1(G_w, M)\) and \(\psi _w\) in \(H^1(G_w, M^{\vee })\) for each place w of K dividing v, it suffices to show that
For each w, we define \(\tau _w\) in \(G_F\) as in Definition 4.1. We refer to Fig. 2 for the definition of the maps \(\nu \) and \(\nu _1\). The pairing \(t_{M}\) of (4.5) may be expressed as a composition
where B is the evaluation pairing on \(M \otimes M^{\vee }\) and where P is the pairing defined in Lemma A.8 (2). Taking \(B'\) to be the pairing \(\text {Ind}_{K/F}B \circ P\), we have the identity
for any a in \(C^1(G_v, \text {Ind}_{K/F} M)\) and b in \(C^1(G_v, \text {Ind}_{K/F} M^{\vee })\), and for any w dividing v. This is readily verified at the level of cochains.
To ease notation, we take
Note that this is the w-component of \(\text {sh}_{\text {loc},\,v}^{-1}\).
From (4.9) and Lemma 4.12, we have a commutative triangle
The left hand side of (4.15) thus equals
This establishes (4.15), giving the proposition. \(\square \)
Proof of Theorem 4.5
Take \(G = G_F\) and \(H = G_K\), and choose a transversal T as in Definition A.7. For each place w of K, define \(\tau _w\) as in Definition 4.1. Applying Proposition 3.2 as necessary, we may assume the \(\tau _w\) all lie in the image of T.
Choose cocycles \({\overline{\phi }}\) and \({\overline{\psi }}\) representing \(\phi \) and \(\psi \), and choose f in \(C^1(H, M)\) projecting to \({\overline{\phi }}\). Choose \(\epsilon \) in \(C^2(G, (F^s)^\times )\) satisfying \(d\epsilon = \iota ^{-1}(df) \cup _{M_1} {\overline{\psi }}\).
With the Shapiro map \({}^T\text {sh}^H_G\) on cochains as given in Definition A.7, we take
Lemma A.8 then implies that \(\epsilon ' \in C^2\left( G_F,\, \text {Ind}_{K/F} (F^s)^{\times }\right) \) satisfies
where the pairing \(B'\) is defined as in the proof of Proposition 4.4 above.
Choose a tuple of cocycles \(({\overline{\phi }}_{v,\, \text {Ind } M})_v\) representing a class in \(\text {sh}_{\text {loc}}({\mathscr {W}})\) so \({\overline{\phi }}_{v,\, \text {Ind } M}\) projects to the cocycle \({\overline{\phi }}'_v\) for each v of F. Defining \(\Upsilon _w\) as in (4.17), we then take
By part three of Lemma A.8 we have \(\pi \left( {\overline{\phi }}_{w, M}\right) = {\overline{\phi }}_w\) for each v. In the notation of Definition 2.2, we may compute \(\text {CTP}_E(\phi , \psi )\) from the tuple
and we may compute \(\text {CTP}_{\text {Ind}_{K/F}\, E}(\text {sh}_{K/F} \,\phi , \,\text {sh}_{K/F}\, \psi )\) from the tuple
Here \(\nu \) is defined as in Fig. 2. With the pairing \(t_{M_1}\) as in Definition 4.3, for each place v of F we have
where we have suppressed the appearances of \((\text {Ind}_{K/F}\, \iota )^{-1}\) and \(\iota ^{-1}\). Summing over all places v gives the result. \(\square \)
5 Class groups and roots of unity
We now shift our focus to applications of our theory of the Cassels–Tate pairing. Our first goal is to prove Propositions 1.3 and 1.4 on the class groups of global fields with many roots of unity. To start, we follow Flach [8] to translate the reciprocity pairing into a Cassels–Tate pairing.
5.1 The reciprocity pairing as a Cassels–Tate pairing
Notation 5.1
Take F to be a global field, and write \({\mathcal {O}}_F\) for the ring of integers of F. For each positive integer m indivisible by the characteristic of F, take
with the product being over all places of F. We have
where \(H_F\) is the Hilbert class field of F.
To ease notation in what follows, set
Similarly, we will denote by \(\underline{\mu _m}\) the dual object \((\mu _m, {\mathscr {W}}_m^{\text {ur}, \perp })\).
The Selmer group of \(\underline{\mu _m}\) turns out to be isomorphic to the m-Selmer group of F [5, Definition 5.2.4], whose definition we now recall.
Definition 5.2
The m-Selmer group of F is defined to be the group
Here we are thinking of a nonarchimedean place v in terms of the corresponding normalized valuation. Note that given \(\alpha \in F^\times \) representing an element in \( \text {Sel}^m F\), we have \((\alpha )=I^m\) for some fractional ideal I of F. This defines a homomorphism
taking \(\alpha \) to the class of I. The m-Selmer group of F then sits in the short exact sequence
the first map induced by the natural inclusion of \({\mathcal {O}}_{F}^{\times }\) into \(F^\times \).
Notation 5.3
Write \(\delta _m\) for the connecting map
associated to the sequence
For a place v, we write \(\delta _{m,v}\) for the corresponding local isomorphism
For a nonarchimedean place v of F the local Tate pairing
is given by
where \(\text {rec}_{F_v}:F_v^\times \rightarrow G_v^\text {ab}\) is the local Artin map. This follows from [21, Theorem 7.2.13] and antisymmetry of the cup-product.
Lemma 5.4
The map \(\delta _m\) gives an isomorphism
Proof
For each nonarchimedean place v, it follows from (5.2) and [24, Proposition XIII.13] that the orthogonal complement of \(H^1 (G_v/I_v,\tfrac{1}{m}{\mathbb {Z}}/{\mathbb {Z}})\) under the local Tate pairing is equal to
Thus the Selmer group \(\text {Sel}~\underline{\mu _m}\) corresponds under \(\delta _m^{-1}\) to the group
This kernel is precisely the m-Selmer group of F. \(\square \)
Given positive integers a, b indivisible by the characteristic of F, we now consider the exact sequence
in the category \(\text {SMod}_F\). The associated Cassels–Tate pairing gives a pairing
This pairing is related to the reciprocity pairing \(\text {RP}\) considered in (1.5) as follows.
Proposition 5.5
Choose \(\phi \in \text {Cl}^* F[b]\) and \(\psi \in \text {Sel}~\underline{\mu _a} \). Write \(\alpha =\delta _a^{-1}(\psi )\) for the element of \(\text {Sel}^a F\) corresponding to \(\psi \) via Lemma 5.4. Then we have
Proof
View \(\alpha \) as an element in \(F^\times \) by picking a representative for its class in \(\text {Sel}^a F\). For each nonarchimedean place v, choose \(\alpha _v, \beta _v\) in \(F_v^{\times }\) such that \(\alpha _v\) has trivial valuation and \(\alpha = \alpha _v \beta _v^a.\) From [19, Definition 3.8 and Lemma 3.9], we find
As in (5.2), for each nonarchimedean place v we have
Summing these terms gives (5.4). \(\square \)
5.2 Self-duality in the presence of roots of unity
Let p be the characteristic of F and let \({\mathbb {Z}}_{(p)}\) denote the localization of \({\mathbb {Z}}\) at the ideal generated by p (thus \({\mathbb {Z}}_{(p)}={\mathbb {Q}}\) if F has characteristic 0). Fix, once and for all, an isomorphism
between \({\mathbb {Z}}_{(p)}/{\mathbb {Z}}\) and the set of roots of unity in \(F^s\).
Suppose F contains \(\mu _m\) for some integer \(m\ge 1\) indivisible by the characteristic of F. Then the restriction of g to \(\tfrac{1}{m}{\mathbb {Z}}/{\mathbb {Z}}\) is \(G_F\)-equivariant. Furthermore, we have
since \(H^1_{\text {ur}}(G_v, \tfrac{1}{m}{\mathbb {Z}}/{\mathbb {Z}})\) is orthogonal to itself under local Tate duality (this can also be checked by noting that, if the extension \(F_v(\alpha ^{1/m})/F_v\) is unramified for some choice of prime v and \(\alpha \in F_v^{\times }\), then \(\alpha \) must have valuation divisible by m). In particular, we obtain a morphism
which is readily checked to be self-dual. Using Proposition 5.5, we can restate Proposition 1.3 in the following form.
Proposition 5.6
Choose positive integers a, b such that F contains \(\mu _{ab}\), and take E(a, b) to be the exact sequence defined in (5.3). Then we have
In particular, in the case that \(a = b\), the pairing \(\text {CTP}_{E(a, a)}(\text {---}, g_{a}(\text {---}))\) is symmetric.
Proof
From the above discussion we have a morphism of exact sequences
Given \(\phi \) and \(\psi \) as in the proposition statement, we can combine the duality identity and naturality (Eqs. (2.3) and (2.5)) to give
Since \(g_b\) is self-dual, we have the result. \(\square \)
Proof of Proposition 1.3
Take \((\alpha ) = I^{a}\) and \((\beta ) = J^{b}\) as in Proposition 1.3. Noting that J and \(\pi _{\text {Cl},b}(\beta )\) represent the same class in \(\text {Cl}\,\, F\), and noting that \(\chi _{b, \beta }=\delta _b(\beta )\) by definition, we have
Repeating this argument for \(\chi _{b, \beta }(\text {rec}_F(I))\), the claimed identity
is equivalent to
which is a consequence of Proposition 5.6. \(\square \)
5.3 Proof of Proposition 1.4
We now turn to proving Proposition 1.4. Fix F, \(\kappa \), m, d, \(\alpha \), and I as in that proposition. By (1.7), it suffices to consider the case that m is a power of 2, so we assume that we are in this case.
Given Proposition 5.5, we wish to prove the identity
where g is the fixed isomorphism (5.5).
Lemma 5.7
To prove Proposition 1.4, it suffices to establish (5.7) in the case where \(d = 1\) and F contains \(\mu _{m^2}\).
Proof
Take \(\zeta \) to be a fixed generator of \(\mu _{2m^2/d}\). By splitting into cases depending on whether \(\zeta \) lies in F, we can check that the map
defines a field automorphism of \(F(\mu _{2m^2/d})\). Repeating as necessary, we find that \(\kappa \) extends to a field involution on \(F(\mu _{m^2})\) that acts as \(x \mapsto x^{-1}\) on \(\mu _{m^2}\).
Write \(K=F(\mu _{m^2})\). Using a subscript to record the field we are working over, we use the identity \(\chi _{K,\, m, \,\alpha } = \text {res}_{G_K} \chi _{F, \,m,\, \alpha }\) and Corollary 4.9 to give
We thus can establish the proposition for F by applying (5.7) for K with \(d = 1\). This proves the lemma. \(\square \)
Without loss of generality, we now restrict to the case \(d = 1\).
Notation 5.8
For every positive integer n, take \({\mathscr {W}}^0_n\) to be the subset of \({\mathscr {W}}_n^{\text {ur}}\) consisting of tuples \((\phi _v)_v\) that are trivial at all places dividing 2.
Take E to be the exact sequence
in \(\text {SMod}_F\).
Take \(F_0\) to be the subfield of F fixed by the involution \(\kappa \). Then \(F/F_0\) is a quadratic extension. Recall that we defined a functor \(\text {Ind}_{F/F_0}\) from \(\text {SMod}_F\) to \(\text {SMod}_{F_0}\) in Definition 4.2. Writing this functor simply as \(\text {Ind}\), we have an exact sequence
in \(\text {SMod}_{F_0}\), where we have omitted the local conditions decorating these objects.
Our next construction uses the language of theta groups detailed in [19, Sect. 5].
Notation 5.9
Choose any positive divisor n of \(m^2\), and take \(\zeta = g(1/n)\). Any element in \(\text {Ind}\, \tfrac{1}{n}{\mathbb {Z}}/{\mathbb {Z}}\) can be written as a linear combination of the elements
We will denote these elements by \(x_0\) and \(y_0 = \kappa x_0\), respectively. In the case where we need to distinguish between different n, we will write them as \(x_{0,n}\) and \(y_{0,n}\).
Define \({\mathcal {H}}_-(n)\) to be the group
This is a quotient of the discrete Heisenberg group. It fits in the exact sequence
where the fixed primitive root of unity \(\zeta \) in \(\mu _n\) is sent to z, where x is sent to \(x_0\), and where y is sent to \(y_0\). We can check that the \(\text {Gal}(F/F_0)\) action on \({\mathcal {H}}_-(n)\) given by
is well-defined, and that it makes the two maps in (5.8) equivariant. The pushout of (5.8) along \(\mu _n \hookrightarrow (F^s)^{\times }\) defines a theta group
and the associated map \(f_{{\mathcal {H}}(n)}\) (see [19, Definition 5.1]) is the isomorphism
defined by
One sees from this that \(f_{{\mathcal {H}}(n)}\) is equal to the composition of the maps
where t is defined as in (4.6) and \(\rho _{\kappa }\) is defined as in Fig. 2.
For the statements of the following results, we refer to [19, Definition 5.3] for the notion of isotropic local conditions, and to [19, Definition 5.7] for the definition of the Poonen–Stoll class.
Lemma 5.10
Given the setup of Notation 5.9, the local conditions \(\text {sh}_{\text {loc}}({\mathscr {W}}^0_n)\) are isotropic with respect to the theta group \({\mathcal {H}}(n)\).
Proof
Choose \((\phi _v)_v\) in \(\text {sh}_{\text {loc}} ({\mathscr {W}}^0_n)\). Given any place v of \(F_0\), we wish to show that \(q_{G_v}(\phi _v)\) is zero, where
is the connecting map on cohomology associated to (5.9) (cf. [19, Notation 5.2]).
If \(F/F_0\) is not ramified at v, then \(q_{G_v}(\phi _v)\) lies in the image of \(H^2_{\text {ur}}(G_v, \mu _n)\), and this group is trivial. So we can assume \(F/F_0\) is ramified at v.
Now \(F_0(\mu _4)\) is a quadratic extension of \(F_0\) contained in F, so it equals F. Consequently, \(F/F_0\) is only ramified at infinite places and places dividing 2. By the definition of \({\mathscr {W}}^0_n\), \(\phi _v\) is trivial at all such places, so \(q_{G_v}(\phi _v)\) is zero. This gives the lemma. \(\square \)
Lemma 5.11
The Poonen–Stoll class corresponding to the exact sequence \(\text {Ind}\, E\) and the theta group \({\mathcal {H}}(m^2)\) is trivial.
Proof
Taking \({\mathcal {H}}_1\) to be the preimage of \(\text {Ind}\, \tfrac{1}{m}{\mathbb {Z}}/{\mathbb {Z}}\) in \({\mathcal {H}}(m^2)\), we define a homomorphism
by mapping \(x_{0, m}\) to \(x^m\) and \(y_{0, m}\) to \(y^m\), where x and y are taken from the generators of \({\mathcal {H}}(m^2)\). The map s is equivariant, and is a section of the natural map from \({\mathcal {H}}_1\) to \(\tfrac{1}{m}{\mathbb {Z}}/{\mathbb {Z}}\). The cocycle \({\overline{\psi }}_{\text {PS}}\) defined in [19, Remark 5.8] from s is thus trivial, from which the result follows. \(\square \)
Completion of the proof of Proposition 1.4
The map \(f_{{\mathcal {H}}(m^2)}\) above fits in the diagram
in \(\text {SMod}_{F_0}\). Since the local conditions \(\text {sh}_{\text {loc}}({\mathscr {W}}_{m^2}^0)\) are isotropic for \({\mathcal {H}}(m^2)\), and since the associated Poonen–Stoll class is trivial, [19, Theorem 5.10] gives
Taking \(\phi = \text {sh}_{F/F_0}(\psi )\) for some \(\psi \) in \(\text {Sel}\left( \tfrac{1}{m}{\mathbb {Z}}/{\mathbb {Z}},\, {\mathscr {W}}^0_m\right) \), we have
Taking \(\psi =g^{-1} \circ \chi _{m, \,\alpha } \) establishes the \(d=1\) case of (5.7), proving the proposition. \(\square \)
6 Bilinearly enhanced class groups
In [15], Lipnowski, Sawin, and Tsimerman generalized the Cohen–Lenstra heuristics to certain families of fields containing many roots of unity. As part of this work, they introduced the notion of a bilinearly enhanced group, which we give now. The equivalence between this definition and the one appearing in [15, Definition 2.1] is established by [15, Lemma 4.2].
Notation 6.1
Take \(\ell ^n\) to be a positive power of a rational prime and choose a finite abelian \(\ell \)-group G. Writing \(G^*\) for \(\text {Hom}(G, {\mathbb {Q}}_{\ell }/{\mathbb {Z}}_{\ell })\), choose a homomorphism
and an infinite sequence \(\omega _1, \omega _2, \ldots \) of alternating pairings
Taking
to be the evaluation pairing, we call the tuple \((G, \psi , (\omega _i)_{i\ge 1})\) an \(\ell ^n\)-bilinearly enhanced group if
for all \(r \ge 0\) and \(\alpha , \beta \in G^*[\ell ^{n+r}]\), and if
for all \(i \ge 1\), \(\alpha \in G^*[\ell ^{i+1}]\), and \(\beta \in G^*[\ell ^i]\).
Given a number field F containing \(\mu _{\ell ^n}\) with \(\ell \) an odd prime, Lipnowski et al. give an associated \(\ell ^n\)-bilinearly enhanced group
Here, the map \(\psi _F:\text {Cl}\,^*F[\ell ^n] \rightarrow \text {Cl}\,\,F[\ell ^n]\) is straightforward to define, though it does require a choice of a primitive \((\ell ^n)^{th}\) root of unity. In our notation, it is the composition of the maps
The condition (6.1) determines the pairings \(\omega _i\) for \(i \ge n\). For \(i < n\), the definition of \(\omega _i\) is somewhat subtler. In [15], it is defined using Artin–Verdier duality between the étale cohomology groups
Having defined these objects, Lipnowski et al. then give a heuristic for how often a given \(\ell ^n\)-bilinearly enhanced group appears as the tuple (6.3) as F varies through certain families of number fields containing \(\mu _{\ell ^n}\).
The existence of the bilinearly enhanced group (6.3) recovers Proposition 1.3 in the case that \(\ell \) is odd. For suppose we have chosen nonnegative integers c and d satisfying \(c+ d =n\). Then, for \(\alpha \) in \(\text {Cl}\,^* F[\ell ^c]\) and \(\beta \) in \(\text {Cl}\,^*F[\ell ^d]\), we have
This is the full extent of what we can conclude about the class group of F and its reciprocity pairing from the mere existence of the bilinearly enhanced group (6.3). Specifically, we have the following proposition, whose proof we omit.
Proposition 6.2
Take \(\ell ^n\) to be a positive power of an odd prime. Choose a finite abelian \(\ell \)-group G and a homomorphism
Then the following are equivalent:
-
(1)
There is some sequence of alternating pairings \(\omega _1, \omega _2, \ldots \) so that \((G, \psi , (\omega _i)_{i \ge 1})\) is an \(\ell ^n\)-bilinearly enhanced group.
-
(2)
For every pair of positive integers c, d with sum n, and for every \(\alpha \) in \(G^*[\ell ^c]\) and \(\beta \) in \(G^*[\ell ^d]\), we have
$$\begin{aligned}\langle \psi (\alpha ),\beta \rangle = \langle \psi (\beta ), \alpha \rangle .\end{aligned}$$
There is generally more than one way to complete \((\text {Cl}\,\, F[\ell ^{\infty }], \psi _F)\) to a bilinearly enhanced group. If we want to choose one canonically, we may do so using the Cassels–Tate pairing. This gives an alternative to the approach considered in [15, 4.2.2]. It seems likely that our pairings agree with those constructed in [15].
Notation 6.3
Choose n a positive power of \(\ell \) so F contains \(\mu _n\), and fix g as in (5.5).
Take \(E_n\) to be the pushout of E(n, n) along the map \(g_n :\underline{\tfrac{1}{n}{\mathbb {Z}}/{\mathbb {Z}}} \rightarrow \underline{\mu _n}\) (cf. Sect. 2.2). This exact sequence takes the form
The dual sequence takes the form
These are both extensions of \(\underline{\tfrac{1}{n}{\mathbb {Z}}/{\mathbb {Z}}} \) by \(\underline{\mu _n}\). Furthermore, \(E_n\) and \(E_n^{\vee }\) are isomorphic as sequences of abelian groups.
We then define the object \(M_n\) in \(\text {SMod}_F\) as the central term of the Baer difference of these sequences, so we have an exact sequence
Considered in the category of abelian groups, this exact sequence splits. Furthermore, since Baer sum commutes naturally with taking duals, we can find a commutative diagram
in \(\text {SMod}_F\), with the central column \(\gamma _n\) an isomorphism in \(\text {SMod}_F\). (We will describe the underlying \(G_F\)-module of \(M_n\) and detail an explicit choice for \(\gamma _n\) in Remark 6.5).
Take m to be a positive divisor of n. Because \( E_n - E^{\vee }_n\) splits in the category of abelian groups, we can apply the functor \(\text {Hom}({\mathbb {Z}}/m{\mathbb {Z}}, \text {---})\) to produce a new exact sequence of \(G_F\)-modules, which we denote \((E^{\vee }_n - E_n)[m]\). We have a commutative diagram
Here, we have assigned the local conditions \({\mathscr {W}}_{n, m}\) in the top row so that the top three vertical maps are all strictly monic in \(\text {SMod}_F\). The local conditions \({\mathscr {W}}_{n, m}^-\) in the bottom row are assigned so the bottom three vertical maps are all strictly epic.
We immediately see that \({\mathscr {W}}_{n, m}^-\) is contained in \({\mathscr {W}}_{n, m}\). By easy nonsense, the final map in the bottom row is strictly epic. We then have that the final map in the top row is strictly epic. Similarly, the first map in the first row is strictly monic, so the first map in the final row is also strictly monic. So all rows are exact, and \({\mathscr {W}}_{n,m}\) equals \({\mathscr {W}}_{n, m}^-\).
The morphism \((M_n[m], {\mathscr {W}}_{n, m}) \hookrightarrow M_n\), and the composition of morphisms
are both kernels of the multiplication by m morphism. So there is a unique isomorphism
in \(\text {SMod}_F\) that commutes with the two kernel morphisms. This map fits in the commutative diagram (in \(\text {SMod}_F\)) with exact rows
Proposition 6.4
Choose a global field F and a prime \(\ell \) not equal to the characteristic of F. Take n to be a power of \(\ell \) such that F contains \(\mu _{n}\). Given a positive divisor m of n, take D(m) to be the exact sequence \((E_n - E_n^{\vee })[m]\).
Then, for all divisors m of n, the pairing
is antisymmetric.
Further, if a and b are positive integers satisfying \(m = ab\), we have
for all \(\phi \) in \(\text {Cl}\,^*F[m]\) and \(\psi \) in \(\text {Cl}\,^*F[a]\).
Finally, we have
for all \(\phi \) and \(\psi \) in \(\text {Cl}\,^*F[n]\).
Proof
The antisymmetry of \(\text {CTP}_{D(m)}\) follows from the duality identity (2.3) and from applying naturality (2.5) to the morphism \(D(m) \rightarrow D(m)^{\vee }\) given in (6.5).
The relation (6.6) is the statement of naturality for the morphism \(D(a) \rightarrow D(m)\) of exact sequences defined by the natural inclusion at each entry.
The final claim follows from the relation \(E_n - E_n^{\vee } = D(n)\) and the duality identity applied to the sequence \(E_n^{\vee }\). \(\square \)
If \(\ell \) is odd, this proposition shows that the pairings \(\text {CTP}_{D(m)}\) may be used to complete the tuple \((\text {Cl}\,^*F[\ell ^{\infty }], \psi _F)\) to a bilinearly enhanced group.
Remark 6.5
From the definition of the Baer sum, the underlying \(G_F\)-module of the central term \(M_n\) of \(E_n - E_n^{\vee }\) is the subquotient
of \(\mu _{n^2}\oplus \frac{1}{n^2}{\mathbb {Z}}/{\mathbb {Z}}\). The injection \(\mu _n \rightarrow M_n\) sends x to the class of (x, 0), and the surjection \(M_n \rightarrow \tfrac{1}{n}{\mathbb {Z}}/{\mathbb {Z}}\) sends a class represented by (x, y) to ny.
At the level of \(G_F\)-modules, one can take \(\gamma _n :M_n \rightarrow (M_n)^{\vee }\) to be induced by the map
sending (x, y) to \((-y, x )\). This map corresponds to an alternating pairing on \(M_n\).
In the case \(\ell = 2\), we can complement Proposition 6.4 as follows.
Proposition 6.6
Suppose F contains \(\mu _n\) for some n a power of 2. Then, for any positive divisor m of n, the pairing \(\text {CTP}_{D(m)}\) is alternating.
Proof
For \(m=n\) this follows from the final part of Proposition 6.4. Thus we suppose \(m<n\). Since the case \(m = 1\) is trivial, we may also assume \(m > 1\).
Take \(P_1: M_n[2m] \times M_n[2m] \rightarrow \mu _{2m}\) to be the alternating pairing corresponding to the isomorphism \(\gamma _{n, 2\,m}\). Following [19, Definition 5.17] with this pairing and \(e =1\) gives a theta group \({\mathcal {H}}\) over \(M_n[m]\) constructed as a quotient U/N. In the terminology of [19, Definition 5.1], the map \(f_{\mathcal {H}}\) associated to this theta group equals \(\gamma _{n, m}\). The composition of the morphisms
is strictly epic, so the second morphism in this sequence is also strictly epic. By [19, Example 5.18], we see that \({\mathscr {W}}_{n, m}\) is an isotropic subgroup of local conditions for the theta group \({\mathcal {H}}\).
The group U is isomorphic as a \(G_F\) set to \((F^s)^{\times } \times M_n[2\,m]\), and we can check that the map \(\zeta \mapsto (1, \zeta )\) defines a \(G_F\)-equivariant homomorphism from \(\mu _{2\,m}\) to U fitting into the triangle
This map descends to a \(G_F\)-equivariant lift \(\mu _m \rightarrow {\mathcal {H}}\) of the map \(\mu _m \rightarrow M_n[m]\), so we find the Poonen–Stoll class associated to D(m) and \({\mathcal {H}}\) by [19, Definition 5.7] is 0. The proposition now follows from [19, Theorem 5.10]. \(\square \)
7 Decomposing class groups
The Cohen–Lenstra–Martinet heuristics [4, 6] give a conjectural description for how the class group of an extension of global fields K/F varies as K changes within some family. According to these conjectures, the structure of the semisimple decomposition of the \(G_F\)-representation \({\mathbb {Q}}[G_F/G_K]\) has an impact on how the class group of K behaves. Using Construction 7.1 below, the Cassels–Tate pairing seems to be a helpful tool for breaking apart the class group with respect to such a semisimple decomposition.
In what follows, we will work in the categories \(\text {SMod}_{F, \ell }^{\infty }\) and \(\text {SMod}_{K, \ell }^{\infty }\) introduced in [19, Sect. 4.2], so as to allow the underlying modules to be infinite. As in Remark 4.7, the results on base change proved in Sect. 4 remain valid in this setting.
Construction 7.1
Let K/F be a separable extension of global fields, and take \(\ell \) to be a rational prime other than the characteristic of F. Taking \(A = {\mathbb {Q}}_{\ell }/{\mathbb {Z}}_{\ell }\), we have an identification
where the first isomorphism is as defined in Sect. 5.1, and the second isomorphism is the Shapiro isomorphism of Sect. 4. Here, \({\mathscr {W}}^{\text {ur},\, K}\) and \({\mathscr {W}}^{\text {ur},\, F}\) are the subgroups of everywhere unramified local conditions for A and \(\text {Ind}_{K/F}A\). They are identified with each other by the local Shapiro isomorphism as in (4.2).
Suppose we have an isogeny of \(G_F\)-modules
where \(T_1, \ldots , T_r\) is a sequence of compact finitely generated free \({\mathbb {Z}}_{\ell }\)-modules with a continuous action of \(G_F\). Here by the term isogeny we mean that the map is injective and has finite cokernel. Equivalently, taking \(V_i ={\mathbb {Q}}_{\ell } \otimes T_i \) and \(A_i = A \otimes T_i\), the corresponding map
is an isomorphism.
Take M to be the cokernel of (7.2). Applying the snake lemma to the commutative diagram of \(G_F\)-modules
gives, upon noting that the left vertical map is an injection and the central vertical map is an isomorphism, an exact sequence of \(G_F\)-modules
Here the map \(\iota \) is the inverse of the connecting map provided by the snake lemma. We give \(\text {Ind}_{K/F} A\) the unramified local conditions and give M and \(\bigoplus _{i \le r} A_i\) the local conditions so that this sequence is exact.
From (7.1) and [19, Proposition 4.11] (a version of Theorem 2.1 which does not assume finiteness of the relevant \(G_F\)-modules), we have an exact sequence
with the last map given by the Cassels–Tate pairing.
Remark 7.2
In the above, to calculate the local conditions on \(\bigoplus _{i \le r} A_i\) more precisely, we start with the commutative square
which we have for every place v of F. The first vertical map is surjective since \(H^1_{\text {ur}}(G_w, {\mathbb {Q}}_{\ell })\) surjects onto \(H^1_{\text {ur}}(G_w, A)\) for each place w of K. So the local conditions for \(\bigoplus _{i \le r} A_i\) are given by
We will consider Construction 7.1 in detail in the case that K/F is cyclic of degree \(\ell \). We expect that it would give similarly–detailed information in the case that K/F is any abelian extension.
Example 7.3
In Construction 7.1, take K/F to be a cyclic extension of degree \(\ell \), and
to be a nontrivial character associated to this extension. Take \(T ={\mathbb {Z}}_{\ell }\) and \(T^{\chi } = {\mathbb {Z}}_{\ell }[\mu _{\ell }]\). We give T the trivial \(G_F\)-action and define an action on \(T^{\chi }\) by
Take \(A = {\mathbb {Q}}_{\ell }/{\mathbb {Z}}_{\ell }\) as above, and define \(A^{\chi }\) to be \(A \otimes T^{\chi }\). We have an exact sequence
of \(G_F\)-modules, where \(\iota \) is given by
where \(\nu \) is given as in Fig. 2, and where p is given by
Giving \(\text {Ind}_{K/F}A\) the subgroup of unramified local conditions, the resulting local conditions on \(A^{\chi }\) are given by
and the local conditions for A are the unramified ones. The map \(\iota \) is the composition
with \(i_1\) taken as in Fig. 2. From (A.2), the local conditions for \(\tfrac{1}{\ell }{\mathbb {Z}}/{\mathbb {Z}}\) are seen to be
The exact sequence (7.3) then takes the form
the \(*\) denoting a Pontryagin dual. We note that, for \(\sigma \) in \(G_F\), \(\sigma :A^{\chi } \rightarrow A^{\chi }\) is a morphism in \(\text {SMod}_{F, \ell }^{\infty }\), so \(\text {Sel}\, A^{\chi }\) has the structure of a \(\text {Gal}(K/F)\)-module.
From this exact sequence, we can prove the following proposition.
Proposition 7.4
In the above situation, we have an exact sequence
of \(\text {Gal}(K/F)\)-modules, where the leftmost map is the natural inclusion, and where \(\text {Gal}(K/F)\) acts trivially on the final term.
In particular, denote by S the set of places of F not dividing \(\ell \) where K/F is ramified, and suppose that the following assumption is satisfied:
- \(\left( \varvec{\star }\right) \):
-
There is no element of \(F^{\times }\) outside \((F^{\times })^{\ell }\) that has valuation divisible by \(\ell \) at all finite primes and which lies in \((F_v^{\times })^{\ell }\) at every prime v in S.
Then we have an exact sequence of \(\text {Gal}(K/F)\)-modules
Proof
To show (7.4) is exact, it suffices to show that the kernel of p in \(\text {Cl}\,^*K[\ell ^{\infty }]\) is the portion invariant under \(G_F\). So take \(\sigma \) to be a generator of \(\text {Gal}(K/F)\), and take \(\zeta = \chi (\sigma )\). We have an exact sequence
in which the homomorphism f is given by
From the long exact sequence, we see that the map
is injective. In addition, the map \((\rho _{\sigma } - 1):\text {Ind}_{K/F}A \rightarrow \text {Ind}_{K/F}A \) factors as the composition
From these two facts, we see that the kernel of p is precisely the portion of \(\text {Sel}\left( \text {Ind}_{K/F}A\right) \) invariant under \(\rho _{\sigma }\). The exactness of (7.4) follows by (A.3), and the equivariance of this sequence under \(\text {Gal}(K/F)\) follows from the commutativity of the diagram
Finally, suppose that the assumption (\(\varvec{\star }\)) is satisfied. To conclude the proof, we wish to show that \(\text {Sel}\, (\mu _{\ell }, {\mathscr {W}}^{\perp })\) vanishes. For a nonarchimedean place v of F, let
denote the component of \({\mathscr {W}}\) at v. Note that, for each v, \({\mathscr {W}}_v\) contains both the restriction of \(\chi \) to \(G_v\) and the group \(H^1\left( G_v,\,\tfrac{1}{\ell }{\mathbb {Z}}/{\mathbb {Z}}\right) \) of unramified classes. For the places v in S, where \(\chi \) generates \(H^1\left( I_v,\,\tfrac{1}{\ell }{\mathbb {Z}}/{\mathbb {Z}}\right) \), this implies that \({\mathscr {W}}_v\) is equal to \(H^1(G_v,\tfrac{1}{\ell }{\mathbb {Z}}/{\mathbb {Z}})\). In particular, \({\mathscr {W}}_v^\perp \) is equal to 0 at all places \(v\in S\), and it is contained in the orthogonal complement of \(H^1_{\text {ur}}\left( G_v,\,\tfrac{1}{\ell }{\mathbb {Z}}/{\mathbb {Z}}\right) \) at all other finite places v of F. Identifying \(H^1(G_F,\mu _\ell )\) with \(F^\times /(F^{\times })^\ell \) as in Notation 5.3, we see from the discussion in Sect. 5.1 that \(\text {Sel}\, (\mu _{\ell }, {\mathscr {W}}^{\perp })\) is contained in the subgroup of \(\text {Sel}^\ell F\) consisting of elements which lie in \((F_v^\times )^\ell \) for each place v in S. The assumption (\(\varvec{\star }\)) is precisely the condition that this subgroup is trivial. \(\square \)
8 Two miscellaneous examples
8.1 Involution spin
The notion of the spin of a prime ideal was introduced in [9]. Given a number field K, a nontrivial field automorphism \(\kappa \) of K, and a principal ideal \((\alpha ) = {\mathfrak {a}}\) of K coprime to \((\kappa \alpha )\), the spin of \({\mathfrak {a}}\) can be defined as the quadratic residue symbol
For fixed \(\kappa \), these spin symbols seem to be equidistributed as \({\mathfrak {a}}\) varies through the principal primes of K. In most cases, such equidistribution is conditional on other hard results.
One exception occurs when \(\kappa \) is an involution of K. In this case, the symbol \(\left( \frac{\alpha }{\kappa {\mathfrak {a}}}\right) \) is found to be determined by the splitting of the prime factors of \({\mathfrak {a}}\) in a certain extension of K. The distribution of these spin symbols is then determined by the Chebotarev density theorem.
We can explain this phenomenon with our work on theta groups, giving an alternative to the trace-based approach appearing in the proof of [9, Proposition 12.1]. Our version of this proposition is the following.
Proposition 8.1
Take K/F to be a quadratic extension of number fields, and take \(\kappa \) to be the generator of \(\text {Gal}(K/F)\).
Choose a nonzero integer \(\alpha \) in K so that the ideal \({\mathfrak {a}}=(\alpha )\) is coprime to \((\kappa \alpha )\). Further, assume that the extension \(K(\sqrt{\alpha })/K\) splits at all primes of K where K/F is ramified, and is unramified at all primes dividing 2 where K/F is inert.
Then we have the identity
Proof
We will use the exact sequence
of \(G_F\)-modules, where i and \(\nu \) are defined as in Fig. 2. We take \(x_0, y_0\) to be a basis for \(\text {Ind}_{K/F}{\mathbb {F}}_2\) chosen so \(\kappa x_0 = y_0\) and \(\kappa y_0 = x_0\).
Take S to be the set of places of F below some place where \(K(\sqrt{\alpha })/K\) is ramified. Note that the coprimality condition above implies that S only contains places where K/F splits. We define a local conditions subgroup 
for \(\text {Ind}_{K/F} {\mathbb {F}}_2\), with each \({\mathscr {W}}_v\) defined as follows:
-
If K/F ramifies at v, \({\mathscr {W}}_v\) is taken to be zero.
-
If v is in S, \({\mathscr {W}}_v\) is taken to be the image of \(H^1(G_v, \langle x_0 \rangle )\).
-
Otherwise, \({\mathscr {W}}_v\) is taken to be \(H^1_{\text {ur}}\left( G_v, \,\text {Ind}_{K/F} {\mathbb {F}}_2\right) \).
Consider the finite group
which is isomorphic to the dihedral group of order 8. Define an action of \(\text {Gal}(K/F)\) on this group by
Then \({\mathcal {H}}_{-}\) fits in an exact sequence of \(G_F\)-groups
Taking the pushout of this sequence along \(\mu _2\hookrightarrow (F^s)^{\times }\) defines a theta group \({\mathcal {H}}\) for \(\text {Ind}_{K/F} {\mathbb {F}}_2\) in the sense of [19, Definition 5.1].
We now need the following lemma. As in Sect. 5.3, we refer to [19, Definition 5.3] for the definition of isotropic local conditions, and to [19, Definition 5.7] for the definition of the Poonen–Stoll class. \(\square \)
Lemma 8.2
Given the above setup, choose \(\delta \in F^{\times }\) so \(K = F(\sqrt{\delta })\). Then:
-
(1)
The local conditions \({\mathscr {W}}\) are isotropic with respect to the theta group \({\mathcal {H}}\),
-
(2)
The Poonen–Stoll class \(\psi _{\text {PS}}\) associated to the exact sequence E and the theta group \({\mathcal {H}}\) is the quadratic character \(\chi _{F,\, -\delta } \in H^1(G_F, \mu _2)\) corresponding to the extension \(F(\sqrt{-\delta })/F\).
Proof
We start by establishing isotropy. For a place v, let \(q_{{\mathcal {H}},G_v}\) be as defined in [19, Notation 5.2]. At v in S, \({\mathscr {W}}_v\) is the image of
in \(H^1(G_v, \text {Ind}_{K/F} {\mathbb {F}}_2)\), so \(q_{{\mathcal {H}}, G_v}({\mathscr {W}}_v) = 0\). At other places v where K/F is not ramified, \(q_{{\mathcal {H}}, G_v}({\mathscr {W}}_v)\) lies in the image of \(H^2(G_v/I_v, \mu _2)\), which is a trivial group. At places v where K/F is ramified, \({\mathscr {W}}_v\) is zero, so \(q_{{\mathcal {H}}, G_v}({\mathscr {W}}_v) = 0\). This proves part (1).
For the second part, we note that the map \(\iota :{\mathbb {F}}_2 \rightarrow \text {Ind}_{K/F} {\mathbb {F}}_2\) does not lift to a homomorphism from \({\mathbb {F}}_2\) to \({\mathcal {H}}_{-}\). However, we can define a lift of \(\iota \) from \({\mathbb {F}}_2\) to \({\mathcal {H}}\) by the formula
We may use this lift to compute the Poonen–Stoll class as in [19, Remark 5.8], giving the part. \(\square \)
We return to the proof of Proposition 8.1. With the lemma proved, [19, Theorem 5.10] gives
for all \(\phi _0\) in \(\text {Sel}({\mathbb {F}}_2, \,\pi ({\mathscr {W}}))\), where we have identified the groups \({\mathbb {F}}_2\) and \(\mu _2\) via the unique isomorphism between them.
For \(\beta \) in \(K^{\times }\), we will use the notation \(\chi _{\beta }\) for the quadratic character associated to \(K(\sqrt{\beta })/K\). Given \(\alpha \) as in the proposition statement, take \(\phi \in H^1(G_F, \text {Ind}_{K/F}{\mathbb {F}}_2)\) to be the image of \(\chi _{\alpha }\) under the Shapiro isomorphism. From the conditions on \(\alpha \) we see that \(\phi \) maps into \({\mathscr {W}}_v\) for all places v outside S. Because of this, \(\nu (\phi )\) lies in \(\text {Sel}({\mathbb {F}}_2, \pi ({\mathscr {W}}))\).
For \(v \in S\), the restriction of \(\phi \) to \(G_v\) is given by the restriction of \(x_0 \cdot \chi _{\alpha } + y_0 \cdot \chi _{\kappa \alpha }\) to \(G_v\). Modulo \({\mathscr {W}}_v\), this equals \(\text {res}_{G_v}\left( i(1) \cdot \chi _{\kappa \alpha }\right) \). We calculate
The result follows from (8.1). \(\square \)
8.2 Kramer’s everywhere-local/global norm group
Combining our results on field change with the results on Baer sums detailed in [19, Sect. 4.1], we can reprove a result of Kramer [14, Theorem 2]. We go through this now.
We assume that F has characteristic other than 2, take a quadratic extension K/F, and take \(\chi \) to be the quadratic character \(\text {Gal}(K/F)\xrightarrow {\,\,\,\sim \,\,\,}\{\pm 1\}\) associated to this extension.
Notation 8.3
Choose an abelian variety A/F. We write \(A^{\chi }\) for the quadratic twist of A by \(\chi \). By definition this is another abelian variety over F equipped with a K-isomorphism
such that for any \(\sigma \) in \(G_F\), the composition \(\psi ^{-1}{}\psi ^\sigma \) is multiplication by \(\chi (\sigma )\) on A.
Note that \(\psi \) induces an isomorphism
of group schemes over F. Via this isomorphism, we can view both \(\text {Sel}^{2}\, A\) and \(\text {Sel}^{2}\, A^\chi \) as subgroups of \(H^1(G_{F},A[2])\).
Given a positive integer n, we view A[n] as an element of \(\text {SMod}_F\) by equipping it with the local conditions used to define the n-Selmer group of A over F, and view \(A^\chi [n]\) as an element of \(\text {SMod}_F\) similarly. We caution that although A[2] and \(A^\chi [2]\) are isomorphic as \(G_F\)-modules, they need not be isomorphic as objects in \(\text {SMod}_F\). With this set, the exact sequences
and
in \(\text {SMod}_F\) give rise to the specialization of the classical Cassels–Tate pairing to the 2-Selmer groups of A and \(A^{\chi }\), respectively.
We are then interested in the pairing
We will make use of the exact sequence
of group schemes over F, where \(R_{K/F}A_K\) is the restriction of scalars of \(A_K\) to F. As in Example 4.6, the \(G_F\)-module of \(F^s\)-points of \(R_{K/F}A_K\) is \(\text {Ind}_{K/F} A(F^s)\). The final map in the above sequence is given on \(F^s\)-points by
where \(\sigma \) is any element of \(G_F\) outside \(G_K\). We will manipulate this sequence using the pushout and pullback operations considered in Sect. 2.2.
Take M to be the subgroup of \(R_{K/F}A_K\) in the image of \((A \times A^{\chi })[4]\), and take
to be the associated isogeny with kernel M. Equipping \(R_{K/F}A_K[\varphi ]\) with the usual local conditions defining the \(\varphi \)-Selmer group, the exact sequence
is then the pushout of \(E \oplus E^{\chi }\) along the map
Here, i is defined as in Fig. 2, and \({\mathscr {W}}_2\) and \({\mathscr {W}}_2^\chi \) are the standard local conditions for A[2] and \(A^\chi [2]\) respectively. Taking a pullback, we finally obtain an exact sequence
in \(\text {SMod}_F\), where \(R_{K/F}A_K[2]\) carries its usual local conditions, and where \(\nu \) is defined as in Fig. 2.
The Cassels–Tate pairing of (8.4) equals the pairing (8.2) by naturality. Applying the functor \(\text {Sel}_F\) to the sequence (8.4) gives the exact sequence
by Shapiro’s lemma and (A.2).
We are thus left with the following:
Theorem 8.4
([14], Theorem 2) The pairing (8.2) has left and right kernels
respectively.
Remark 8.5
In the case where A is an elliptic curve, A is canonically isomorphic to its dual. Thus Theorem 8.4 yields a non-degenerate pairing on the group
This pairing is alternating since the Cassels–Tate pairings for A and \(A^{\chi }\) are both alternating. It follows that the group (8.5) has even \({\mathbb {F}}_2\)-dimension. This is used by Kramer to give a purely local formula for the parity of the 2-infinity Selmer rank of A over K [14, Theorem 1].
If instead A is taken to be an arbitrary principally polarized abelian variety, the corresponding pairing on the group (8.5) is only antisymmetric in general. Nevertheless, it is possible to use the work of Poonen–Stoll [22] to analyze the failure of this pairing to be alternating, allowing one to once again decompose the parity of the 2-infinity Selmer rank of A over K as a sum of local terms. This is carried out in work of the first author [18, Theorem 10.20], where the relative simplicity of (8.4) compared to either E or \(E^{\chi }\) plays an essential role.
This simplification of the Cassels–Tate pairing under quadratic twist was also observed independently by the second author, and plays a key role in the work [26] concerning the distribution of 4-Selmer groups in quadratic twist families. In hindsight, the underlying reason behind this simplification is that, due to (8.4), the sum of the Cassels–Tate pairings \(\text {CTP}_E + \text {CTP}_{E^{\chi }}\) can be simpler to control than either \(\text {CTP}_E\) or \(\text {CTP}_{E^{\chi }}\).
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Acknowledgements
We would like to thank Peter Koymans, Wanlin Li, Melanie Matchett Wood, Carlo Pagano, Bjorn Poonen, Karl Rubin, and Richard Taylor for helpful discussion about this work. We would also like to thank the reviewers for their careful reading of the paper and for their helpful comments and corrections.
Funding
This work was partially conducted during the period when Alexander Smith served as a Clay Research Fellow. Previously, the author was supported in part by the National Science Foundation under Award No. 2002011. Adam Morgan is supported by the Engineering and Physical Sciences Research Council Grant EP/V006541/1. During part of the period in which this work was carried out, they were supported by the Max-Planck-Institut für Mathematik in Bonn and the Leibniz fellow programme at the Mathematisches Forschungsinstitut Oberwolfach, and would like to express their thanks for the excellent conditions provided.
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Appendix A: Group change operations
Appendix A: Group change operations
This appendix reviews the construction and basic properties of group change operations for the cohomology of profinite groups. Much of the material is standard but we include it here for lack of a suitable reference (though see [11, 21]). We refer to [19, Notation 2.1] for the notation concerning inhomogeneous cochains and cup products used below.
1.1 Setup
Take G to be a profinite group, and take \(\text {Mod}_G\) to be the category of discrete G-modules. This is an abelian category with enough injectives [21, (2.6.5)]. Given M in this category, the groups \(H^n(G, M)\) may be defined as the right derived functors of the G-invariants functor \(H^0(G, \text {--})\). The conversion between this definition and the inhomogeneous cochain definition proceeds as follows.
Given a nonnegative integer n, we take \({\mathscr {C}}^n(G, M)\) to be the set of continuous maps from \(G^{n+1}\) to M. These form the homogeneous cochain resolution
in \(\text {Mod}_G\) detailed in [21, I.2]. The map e from \(C^n(G, M)\) to \({\mathscr {C}}^n(G, M)\) defined by
gives an isomorphism from \(C^n(G, M)\) to \(H^0(G,{\mathscr {C}}^n(G, M))\) and the resulting diagram
commutes. The modules \({\mathscr {C}}^n(G, M) \) are \(H^0(G, \text {--})\)-acyclic, so taking cohomology of this diagram yields the sought identification between \(H^n(G, M)\) and \(R^nH^0(G, M)\).
1.2 The basic operations
The following gives a convenient framework for understanding certain cohomological operators on \(\text {Mod}_G\).
Notation A.1
Take \(G_0\), \(G_1\), and \(G_2\) to be closed subgroups of G. Suppose we have chosen exact functors
that send injective modules to injective modules. Given a natural homomorphism
we define the map
for each M in \(\text {Mod}_{G_0}\), and each \(n \ge 0\), by choosing an injective resolution \(0 \rightarrow M \rightarrow I^{\bullet }\) of M and taking the cohomology of the diagram
Definition A.2
There are three basic exact functors we will consider. First, given an open subgroup H of G we have the induction functor
taking M to \({\mathbb {Z}}[G] \otimes _{{\mathbb {Z}}[H]} M\).
For H a closed subgroup of G, we have the forgetful functor
Given a discrete G-module M, we will almost always write \(\text {Res}_H^G(M)\) as M.
Finally, for H a closed subgroup and \(\tau \) in G, we have the conjugation functor
For each M in \(\text {Mod}_H\), \(\text {Conj}_{\tau }^H(M)\) is defined to be M as an abelian group; given m in M we write \(\tau \cdot m\) for the corresponding element of \(\text {Conj}_{\tau }^H(M)\). The \(\tau H \tau ^{-1}\)-action is given by
We will usually use the notation \(\tau M\) to refer to \(\text {Conj}_{\tau }^H (M)\).
Proposition A.3
The three functors of Definition A.2 send injectives to injectives.
Proof
Conjugation is an equivalence of categories, so preserves injectives. For H open in G, \(\text {Ind}_G^H\) is an exact left and right adjoint for the exact functor \(\text {Res}_H^G\). So both functors preserve injectives. For \(\text {Res}_H^G\) when H is not open, see [11, Proposition 4.25]. \(\square \)
In Fig. 1 we give four cohomological group change operations that can be defined using the setup of Notation A.1. They give, respectively, the operations of restriction, conjugation, corestriction, and the isomorphism appearing in Shapiro’s lemma. We will also need some additional natural homomorphisms, which we give in Fig. 2. By functoriality of cohomology (or, equivalently, via the setup of Notation A.1) these morphisms induce operations on cohomology.
Notation A.4
Throughout Figs. 1 and 2, H denotes a subgroup of G, and \(\tau \) denotes an element of G. Natural isomorphisms are marked with the notation 
. When using these objects and functors we often suppress either G or H or both from the notation. By an abuse of notation we will use the lone symbol \(\tau \) to refer to the functor \(\text {Conj}_{\tau }^H\), the natural isomorphism \(\text {conj}_{\tau }^H\), and the natural isomorphism
Four group change homomorphisms
Some useful natural homomorphisms
Identities between the natural morphisms in Figs. 1 and 2 automatically hold on cohomology. Thus, for an open subgroup H of G, and discrete H-module M, the diagrams
commute. If M is a discrete G-module, we have a commutative diagram
If M is instead a discrete H-module, we have a commutative square
for every \(\tau \) in G.
We now define a final group change operation that we will refer to as the restricted Shapiro isomorphism. This is related to the double coset formula relating restriction and corestriction (see e.g. [21, (1.5.6)]).
Definition A.5
Given a profinite group G, take H to be an open subgroup of G, and take C to be a closed subgroup of G. For any \(\tau \) in G, we take the notation
We can then define a natural morphism on the category of discrete H-modules
by the formula
If we take B to be a set of representatives for the double coset \(C\backslash G/H\), the map
with \(\tau \)-component \(r^{C\backslash G/H}_{\tau }\) is a natural isomorphism of C-modules.
We define the restricted Shapiro isomorphism
by composing (A.4) with the isomorphisms
for each \(\tau \) in B.
When \(n = 0\), this isomorphism has \(\tau \) component given by
From this description, we find that the \(\tau \)-component of (A.5) is given by the composition
and the \(\tau \) component of its inverse is the composition
Now suppose that \(C_1\) is a closed subgroup of C, and for \(\epsilon \) in G write
Take \(B_1\) to be a set of representatives for the double cosets in \(C_1\backslash G/H\), chosen so each \(\epsilon \) is of the form \(c \tau \) for some c in C and \(\tau \) in B. Given \(\tau \) in B, we consider the map
where \(\epsilon \) varies through the subset of \(B_1\) in \(C \tau H\). Bundling these together, we are left with a map
fitting into the commutative diagram
This is particularly useful in the case that \(C = G\), where 
equals \(\text {sh}^H_G\).
If \(C_1\) is a normal subgroup of C, and if we choose \(\epsilon \) in \(B_1\) corresponding to \(\tau \) in B, we see that \(D_{1\tau } = D_{1 \epsilon }\). In this case, the \(\epsilon \) component of \(\oplus _{\tau , \epsilon }\, \text {res}_{D_{1\epsilon }}^{D_{\tau }}\) equals \( \text {res}_{D_{1\tau }}^{D_{\tau }}\). Removing the redundant components of (A.7) then leaves a square
1.3 Explicit cochain formulae
We detail here certain maps on inhomogeneous cochains which induce previously considered operations on cohomology.
1.3.1 Conjugation
Definition A.6
Take H to be a closed subgroup of G, and take M to be a discrete H-module. Given \(\tau \) in G and \(n \ge 0\), we have the conjugation operation
defined by
This gives a chain map on the complex of inhomogeneous cochains, and taking cohomology recovers the conjugation operation defined in Fig. 1.
1.3.2 Corestriction and Shapiro maps
Definition A.7
Take H to be an open subgroup of G, and choose a map (giving a transversal for H in G)
whose composition with the projection \(G \rightarrow G/H\) is the identity, and so that \(T(H)=1\).
For \(n \ge 0\), and any discrete G-module M, we define a map
by setting
where we have taken
We have a commutative square
where the bottom row is the corestriction for homogeneous cochains given in [21, I.5.4], defined from the given transversal T. This definition is natural, in the sense that a morphism \(\alpha :M \rightarrow N\) of discrete G-modules gives a commutative square
We then define
where \(i_1\) is the natural homomorphism of Fig. 2. Per (A.1), this yields the Shapiro isomorphism when evaluated on cocycle classes.
The key properties of this construction are as follows.
Lemma A.8
-
(1)
For any f in \(C^n(H, M)\) we have
$$\begin{aligned} d\left( {}^T\text {sh}^H_G f \right) = {}^T\text {sh}^H_G (df).\end{aligned}$$ -
(2)
Given discrete H-modules \(M_1, M_2\), we can define a G-equivariant homomorphism
$$\begin{aligned}P :\text {Ind}^H_G M_1 \otimes \text {Ind}^H_G M_2 \rightarrow \text {Ind}^H_G (M_1 \otimes M_2)\end{aligned}$$by setting, for \(m_1\) in \(M_1\), \(m_2\) in \(M_2\), and \(\sigma _1, \sigma _2\) in G,
$$\begin{aligned}P\big ([\sigma _1] \otimes m_1,\,[\sigma _2] \otimes m_2\big ) = {\left\{ \begin{array}{ll} [\sigma _1] \otimes ( m_1 \otimes \sigma _1^{-1} \sigma _2m_2)&{}\text { if } \sigma _1 H = \sigma _2 H \\ 0 &{}\text { otherwise.}\end{array}\right. }\end{aligned}$$Then, given \(k, j \ge 0\), we have
$$\begin{aligned} {}^T\text {sh}^H_G (f \cup g) \,=\, {}^T\text {sh}^H_G f \,\cup _{P}\, {}^T\text {sh}^H_G g \end{aligned}$$for all f in \(C^k(H, M_1)\) and g in \(C^j(H, M_2)\).
-
(3)
Choose \(\tau \) in G satisfying \(\tau = T(\tau H)\). Then we have the identity
$$\begin{aligned}\nu _1 \circ \tau ^{-1} \circ \text {res}^G_{\tau H \tau ^{-1}} \circ {}^T\text {sh}^H_G (f) \,=\, f\end{aligned}$$for all f in \(C^n(H, M)\).
Proof
Given the commutative square (A.10), the first part is in [21, I.5.4] for the explicit corestriction of homogeneous cochains. The identity for cup products is also immediate from the form of corestriction and cup product for homogeneous cochains, with the latter defined as in [21, I.4].
For the third part, define functions \(g_{\tau '}: G^n \rightarrow M\) so
For \(\sigma _1, \ldots , \sigma _n\) in H, we then have
Applying \(\nu _1\) just leaves
and the definition of the explicit Shapiro map shows this equals \(f(\sigma _1, \ldots , \sigma _n)\). \(\square \)
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Morgan, A., Smith, A. Field change for the Cassels–Tate pairing and applications to class groups. Res. number theory 10, 61 (2024). https://doi.org/10.1007/s40993-024-00545-2
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DOI: https://doi.org/10.1007/s40993-024-00545-2