Invertible analytic functions on Drinfeld symmetric spaces and universal extensions of Steinberg representations

Recently, Gekeler proved that the group of invertible analytic functions modulo constant functions on Drinfeld’s upper half space is isomorphic to the dual of an integral generalized Steinberg representation. In this note, we show that the group of invertible functions is the dual of a universal extension of that Steinberg representation. As an application, we show that lifting obstructions of rigid analytic theta cocycles of Hilbert modular forms in the sense of Darmon–Vonk can be computed in terms of L\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {L}$$\end{document}-invariants of the associated Galois representation. The same argument applies to theta cocycles for definite unitary groups.


Introduction
Let F be a local field and C F the completion of an algebraic closure of F . In [12] Drinfeld introduced the p-adic period space of dimension n − 1: it is a rigid analytic variety over F whose set of C F -points is given by the complement of all F -rational hyperplanes in P n−1 (C F ). It carries a natural PGL n (F )-action.
Let A C F be the ring of analytic functions on that are defined over C F . In [25] (see also [13], Theorem 2.7.11) van der Put constructed for n = 2 a PGL 2 (F )-equivariant isomorphism where Dist 0 (P 1 (F ), Z) denotes the space of Z-valued distributions on P 1 (F ) with total mass 0. Recently, Gekeler (see [18]) generalized van der Put's result to arbitrary dimension and constructed a PGL n (F )-equivariant isomorphism where Gr n−1,n denotes the Grassmannian of hyperplanes in F n or, by duality, the projective space of the dual of F n . A similar result holds if one replaces A C F by the ring of analytic functions defined over some extension E of F (see Theorem 12 for a precise formulation.) Even more recently, Junger gave a completely different proof of this result (cf. [20], Theorem B). By definition, the space of Z-valued distributions of total mass 0 on the Grassmannian Gr n−1,n (F ) is the Z-dual of the integral generalized Steinberg representation attached to a maximal parabolic subgroup of GL n of type (n − 1, 1). The main aim of this note is to determine the class of the extension in representation-theoretic terms. We show that this extension is as non-split as possible. More precisely, in Sect. 1 we study extensions of generalized Steinberg representations attached to maximal parabolic subgroups of arbitrary type. Let r be an integer with 0 < r < n and denote by Gr r,n (F ) the Grassmannian of r-dimensional subspaces of F n . If N is a topological group we denote by v r (N ) the space of continuous functions from Gr r,n (F ) to N modulo constant functions. To each continuous homomorphism λ : F × → N we construct an extension These can be viewed as multiplicative refinements of the extensions studied in [15], Section 2.5, which were inspired by the constructions in Section 2.2 of [11]. Results of Dat (cf. [10]), Orlik (cf. [21]) and Colmez-Dospinescu-Hauseux-Nizioł (cf. [3]) imply that the extension attached to the identity id : F × → F × can be regarded as a universal extension (see Proposition 9).
In the second section we relate the PGL n (F )-module A × C F to a dual of the universal extension. Let E(ι) be the extension associated with the embedding ι : F × → C × F . Taking Hom Z (·, C × F ) induces an exact sequence Basic integration theory (see Sect. 1.2) provides a map Let E(ι) ∨ be the pullback of Hom Z (E(ι), C × F ) along (2). By (1), it sits inside an exact sequence of the form The main result of this article is that there exists a PGL n (F )-equivariant isomorphism E(ι) ∨ ∼ = − −→ A × C F that is compatible with the inverse of Gekeler's isomorphism. Again, one may replace A C F by the ring of analytic functions defined over some extension E of F (see Theorem 13). The case E = F involves the universal extension.
This project started as an attempt to understand the relation between lifting obstructions of rigid analytic theta cocycles in the sense of Darmon-Vonk (cf. [8], [7]) and automorphic L-invariants as introduced by Spieß in [22]. As a first application of our main theorem, we show how lifting obstructions of cuspidal theta cocycles for Hilbert modular groups can be computed in terms of L-invariants of the associated Galois representations. We also give an alternative construction of the Dedekind-Rademacher cocycle that was first constructed by Darmon-Pozzi-Vonk using Siegel units (cf. [6], Theorem A). We also discuss the example of theta cocycles for definite unitary groups of arbitrary rank and their connection with automorphic, respectively, Fontaine-Mazur L-invariants for higher rank groups as introduced in [15]. In all of the above cases, we first compute the lifting obstruction of theta cocycles in terms of automorphic L-invariants. Second, we use the equality of automorphic and Fontaine-Mazur L-invariants as proven in [23], respectively [17].

Notation
The space of continuous maps from a topological space X to a topological space Y is denoted by C(X, Y ). We always endow it with the compact-open topology. If A and B are topological groups, we write Hom(A, B) for the space of continuous homomorphism from A to B. All rings will be commutative and unital. If R is a ring, we write R × for the group of invertible elements of R.

Extensions of generalized Steinberg representations
We construct multiplicative refinements of the extensions of generalized Steinberg representations studied in [15], Sect. 2.5. These refinements were previously constructed in the case n = 2 in Section 6.1 of [2].
Throughout Sects. 1 and 2 we fix a non-Archimedean local field F of residue characteristic p and an F -vector space V of dimension n ≥ 2. If W is any finite-dimensional F -vector space, we denote by GL W (respectively PGL W ) the general (respectively projective) linear group of W viewed as an algebraic group over F . We endow GL W (F ) and PGL W (F ) with the natural topology induced by the one on F . These are locally profinite groups. We often abbreviate G = PGL V (F ).

Generalized Steinberg representations
We fix an integer r with 0 < r < n and write Gr r,V for the Grassmannian variety that parametrizes all r-dimensional subspaces of V . We endow Gr r,V (F ) with the natural topology inherited from the one on F . It is a compact, totally disconnected space. Given an abelian topological group N we define the (continuous) generalized Steinberg representations v r,V (N ) as the space of continuous functions from Gr r,V (F ) to N modulo constant functions, i.e., is an isomorphism for every compact, totally disconnected space X and, therefore, the natural map is an isomorphism.
We fix an F -rational point W 0 ∈ Gr r,V (F ). Its stabilizer P in GL V is a maximal proper parabolic subgroup. The map Thus, we get an isomorphism of G-modules. Proof Let be an element of v r (N ) PGL V (F ) and ∈ C(Gr r,V (F ), N ) a representative of . By (3) we may view as a function on GL V (F ). Invariance of implies that for all g ∈ GL V there exists a constant c(g) ∈ N such that (gg ) = c(g) + (g ) holds for all g ∈ GL V (F ). We may assume that (1) = 0. Thus, c(g) = (g) and, therefore, : GL N (V ) → N is a group homomorphism, which is trivial on P(F ). Since SL V (F ) is the commutator subgroup of GL V , we see that factors over the determinant. Therefore, is trivial since it is trivial on P(F ).

Integration
Let X be a compact, totally disconnected space. We write Dist(X, Z) for the space of Z-valued distributions on X, i.e., Dist(X, Z) = Hom Z (C(X, Z), Z).
The canonical map is an isomorphism and hence, we constructed the desired pairing. In particular, for every abelian prodiscrete group N the integration pairing induces a map int : By similar arguments, we can define a canonical map for every abelian pro-p group N .

Preliminaries on continuous extensions
We say that M 2 is a continuous extension of M 1 by M 3 in this case. Two exact sequences

Continuous induction
Given a topological R[P(F )]-module M we define its continuous induction as

Lemma 2 For every exact sequence
Proof The only non-trivial step is to show that i P (g) is surjective and that it admits a topological section.
By the Bruhat decomposition the map is a locally trivial fibration. Thus, since GL V (F )/P(F ) is totally disconnected and compact, we can find finitely many closed subsets U i ⊆ GL V (F ) such that the maps are homeomorphisms onto their images, their images are disjoint and form an open cover of GL V (F ). Let ϕ : GL V (F ) → M 3 be an element of i P (M 3 ). We construct a preimage : GL V (F ) → M 2 of ϕ as follows: every g in G can be uniquely written as a product u · p with u in a unique U i and p ∈ P(F ). We put (g) = p.s(ϕ(u)), where s : M 3 → M 2 is a fixed section of g. Thus, the homomorphism i P (g) is surjective. Moreover, the map sending ϕ to defines a topological section of i P (g).

Extensions
Let R be a topological ring and N a topological R-module, i.e., N is an abelian topological group and an R-module such that the multiplication map R × N → N is continuous. We consider both R and N as P(F )-modules via the trivial action.
To a continuous group homomorphism λ : for p ∈ P(F ), n ∈ N and r ∈ R. By definition M λ is a continuous extension of R by N . By Lemma 2 the sequence for all p ∈ P(F ) and g ∈ GL V (F ). The group GL V (F ) acts via left multiplication on the first factor. The subspace E(λ) 0 of tuples of the form

Lemma 3 The map
is a group homomorphism that is functorial in R and N .
Proof Functoriality in R and N follows directly from the construction. Thus, we only have to show that the map is a group homomorphism: let λ, λ : P(F ) → A be continuous homomorphisms. The Baer sum of the two extensions E(λ) and E(λ ) is the space of triples ( 1 , 2 , r) with ( 1 , r) ∈ E(λ) and ( 2 , r) ∈ E(λ )) modulo triples of the form ( , − , 0) with ∈ v r (A). Sending a triple ( 1 , 2 , r) to the tuple ( 1 + 2 , r) defines a map from the Baer sum to E(λ + λ ) and thus, they define the same extension class.

Lemma 4 We have b λ,R = 0 if and only if λ can be extended to a continuous homomor-
for all g ∈ GL V (F ). We may assume that (1) = 0. But then c(g) = (g) and, hence, is a homomorphism that extends λ.

Remark 5 Note that the underlying R[G]-module of E(λ) does not depend on the topology on R.
One should view the representations i P (M λ ) as infinitesimal deformations of i P (R): consider R ⊕ N as an R-algebra by putting n 1 · n 2 = 0 for all n 1 , n 2 ∈ N. For example, if N is a free R-module of rank s, there exists an R-algebra isomorphism The map

Homomorphisms
We keep the notations from last section. There is a canonical isomorphism which is functorial in N : every homomorphism from P(F ) to N has to be trivial on the unipotent radical of P(F ). Thus, it factors through the canonical map for unique homomorphisms λ i : F × → N. We will identify λ with the pair (λ 1 , λ 2 ). By the same argument every continuous homomorphism λ : GL V (F ) → N is of the form λ = λ • det for a unique continuous homomorphism λ : F × → N . Therefore, Lemma 4 implies the following: For a continuous group homomorphism λ : ,ct (R, v r (N )) and E(λ) = E(λ, 0).
If λ 1 , λ 2 : F × −→ N are two group homomorphism, the corollary above implies that The next claim follows immediately.

Corollary 7 The map
is an injective group homomorphism that is functorial in R and N .

Universality
It is a natural question to ask whether there exists a class of topological R-modules such that the map in Corollary 7 is an isomorphism. In case R = Z one could reformulate the question as follows: Let c un = c id,Z ∈ H 1 ct (G, v r (F × )) be the extension associated with the identity id : F × → F × . Functoriality implies that for every continuous homomorphism λ : F × → N the equality λ * (c un ) = c λ,Z holds in H 1 ct (G, v r (F × )). Thus, the question is whether c un is a universal extension. We give a partial answer to this question. Let F × (resp. Z) be the profinite completion of F × (resp. Z). We define c un = c i, Z , where i : F × → F × is the natural inclusion. Let N be an abelian profinite group. Continuous homomorphisms from F × to N can be identified with continuous homomorphisms from F × to N and, by functoriality, we have λ * ( c un ) = c λ, Z for every continuous homomorphism λ : F × → N .

Definition 8
Let N be a profinite group. We say that N is pretty good if N is topologically finitely generated and every prime divisor l of the pro-order of N that is prime to p is bon and banal for GL V (F ) in the sense of [10], Section 2.1.5.

Proposition 9 Let F be a p-adic field. For every abelian profinite group N that is pretty good the homomorphism
is an isomorphism.
Proof Let 0 ≤ t ≤ s be integers. It is enough to check that the map surjects onto the space of smooth extensions for l = p and every prime l = p that is bon and banal. The exact sequence We get the following commutative diagram with exact columns and injective horizontal maps: 0 Hom(F × , Z/l s Z) The exactness of the second column follows from Lemma 1. By a simple diagram chase, we see that it is enough to prove the claim above in the case s = t.
The case l = p: By assumption l does not divide the order of the torsion subgroup of F × . In particular, Hom(F × , Z/l s Z) is a free Z/l s Z-module of rank 1. By [10], Theorem 1.3, respectively [21], Theorem 1, the space of smooth extension is also free of rank 1. Thus, the claim follows from the injectivity of (6).
The case l = p: By [3], Theorem 1.10 (2), there exists an isomorphism between Hom(F × , Z/p s Z) and the space of continuous extensions. The claim now follows by injectivity of (6) and the finiteness of Hom(F × , Z/p s Z).

Remark 10
We expect that Proposition 9 also holds in the case that F is a local function field. More precisely, we expect that the map (6) for l = p is inverse to the one constructed in [3].

Invertible analytic functions on Drinfeld symmetric spaces
Using the main theorem of [18] we will prove that the group of invertible analytic functions on Drinfeld's upper half space is isomorphic to a dual of the universal extension (for r = n-1) defined in Sect. 1.7.

Zero cycles
Let us recall that Drinfeld's upper half space = V of dimension n−1 is the complement of all F -rational hyperplanes in P(V ), i.e., It is a rigid analytic variety over F on which the group G = PGL V (F ) acts naturally. Let C F be the completion of an algebraic closure of F with respect to the unique extension of the norm. Let Z 0 ( C F ) = Z[ (C F )] be the free abelian group on the C F -valued points of . We define Z 0 0 ( C F ) as the kernel of the degree map deg : In the following, by an extension E/F we always mean a closed subextension of C F /F and we write ι E : F → E for the inclusion. For such an extension E we put The support of a zero cycle z ∈ Z 0 ( E ) is always defined over a finite extension of E. Let c geo (E) ∈ Ext 1 Z[G] (Z 0 0 ( E ), Z) be the class of the exact sequence

Zero cycles of degree 0 and Steinberg representations
We put V * = Hom F (V, F ). Let z = x a x [x] ∈ Z 0 0 ( E ) be a zero cycle of degree 0. There exist lifts v x ∈ V ⊗ F C F \ {0} of the elements x such that the formal sum x a x [v x ] is invariant under Aut(C F , E). By definition of we have (v x ) = 0 for all nonzero elements ∈ V * . Since the cycle z is of degree 0, the function Choosing different lifts v x changes 0 (z) only up to a constant and, therefore, the induced element (z) ∈ v 1,V * (E × ) does not depend on the chosen lifts. The resulting map is G-equivariant. Here and in the following we always identify GL V and GL V * via the GL V -action on V * given by (g. )(v) = (g −1 (v)). Given a topological abelian group we also abbreviate v(N ) = v 1,V * (N ) ∼ = v n−1,V (N ).

Zero cycles and the universal extension
We fix an element y 0 ∈ P V * (F ) and denote by P ⊆ GL V * ∼ = GL V its stabilizer. Let c un be the universal extension of Z by v(F × ) associated with the identity id : F × → F × and the parabolic subgroup P as in Sect. 1.7.
The following is a generalization of [2], Lemma 6.8, from the case n = 2 to arbitrary dimension.

Proposition 11 For every extension E/F the equality
Proof We fix a lift 0 ∈ V * of y 0 . Under identification (5) we have for all p ∈ P(F ). For x ∈ (C F ) we choose a lift v x ∈ V C F and define the function .
The function is independent of the choices of lifts of y 0 and x and, by (7), fulfils for all g ∈ GL V * (F ) and p ∈ P(F ). We thus get a well-defined map x a x ).
Note that a priori takes values in E(ι C F ) but one can argue as before that it factors through E(ι E ).
For all g, g ∈ GL V * (F ) and all x ∈ (E) we have where k x,g is a constant that does not depend on g. Thus, the homomorphism is Similarly, for x, x ∈ (E) we have where k x,x is a constant independent of g. Thus, the restriction of to cycles of degree 0 agrees with 0 , i.e., the diagram -modules is commutative and, therefore, the claim follows.

Gekeler's theorem
For an extension E/F we write A E = O E ( E ) for the ring of rigid analytic functions on that are defined over E. If N is any abelian group, we identify Hom Z (Z 0 ( C F ), N ) with the space F( (C F ), N ) of all (set theoretic) functions from (C F ) to N . Choosing a base point x ∈ (C F ) we also get a GL V (F )-equivariant isomorphism ) that is independent of the choice of x . Thus, by taking Hom Z (·, C × F ) the map 0 induces the homomorphism

By an easy argument with Riemann sums (or rather Riemann products) we see that the map takes values in
A rationality argument as before shows that the map 0

Theorem 12 For every extension E/F the map
Proof For every extension E the canonical map is an isomorphism. But this follows directly from the proof of [18], Theorem 3.11.

The main theorem
Let E/F be an extension. Applying Hom Z (·, E × ) to the exact sequence Let E ∨ un,E be the pullback of this extension along int, i.e., we have an exact sequence

Theorem 13 For every extension E/F there is a unique PGL
such that the following diagram commutes: As before, we only treat the case E = C F . Applying Hom Z (·, C × F ) to the commutative diagram in the proof of Lemma 11 (with E = C F ) yields the following commutative diagram with exact rows: We know that modulo constants the map takes values in analytic functions. Therefore, C F itself takes values in analytic functions, and the claim follows from Theorem 12.
Let us end this section by giving an explicit description of the "p-adic completion" of E ∨ un,F in the case that F is a p-adic field: let F × be the torsion-free part of the pro-p completion of F × . It is a free Z p -module of rank [F : Q p ] + 1 = d + 1. Write i : F × → F × for the natural map and E(i) for the associated extension of Z p by v( F × ). Let E ∨,p un be the pullback of Hom Z p (E(i), F × ) along Thus, we have an exact sequence Choosing a basis λ 1 , . . . λ d+1 of the free Z p -module Hom(F × , Z p ) we get an isomorphism where E(λ i ) is the extension of Z p by v(Z p ) associated with λ i .

Lifting obstructions of theta cocycles
The aim of this section is to apply our main result to the study of lifting obstructions of theta cocycles. In [8] Darmon and Vonk initiated the theory of rigid meromorphic cocycles, i.e., elements in the cohomology group H 1 (SL 2 (Z[1/p]), M × ), where M × is the group of invertible meromorphic functions on Drinfeld's p-adic upper half plane . They provide a large supply of classes in the space of theta cocycles, i.e., elements of H 1 (SL 2 (Z[1/p]), M × /C × p ) (see also [16] and [19] for a generalization of the theory to other number fields and congruence subgroups). It is thus a natural question to ask whether these classes can be lifted to genuine meromorphic cocycles.
In the following, we want to show that for rigid analytic theta cocycles, i.e., classes in H 1 (PGL 2 the answer is often negative. But one may still lift these classes to elements in H 1 (SL 2 p is a discrete subgroup that can be computed in terms of Galois representations. In fact, we give general results for cuspidal analytic theta cocycles for Hilbert modular groups. In addition, we explain that our methods also yield a new proof of a recent result of Darmon-Pozzi-Vonk on the Dedekind-Rademacher cocycle (cf. [6], Theorem A). We end this note by a generalization of the whole story to higher rank unitary groups.
We will use the following notation throughout this section: If M is an abelian group, we denote its Z-dual by M * = Hom Z (M, Z).

L-invariants of Galois representations
Let F be a p-adic field with absolute Galois group G F . Let ρ : G F → GL 2 (Q p ) be a Galois representation that is an extension of Q p by Q p (1), i.e., it defines a class [ρ] in H 1 (G F , Q p (1)). Local class field theory gives an isomorphism We define the L-invariant of ρ as the orthogonal complement of [ρ] under the local Tate pairing Since the pairing is non-degenerate, L(ρ) is a subspace of codimension at most one. Its codimension is one if and only if ρ is non-split.

Cuspidal theta cocycles for Hilbert modular groups
Let F be a totally real field of degree d and with ring of integers O F . We assume that F has narrow class number one. (One can drop this assumption by formulating everything in the adelic language as in [22] or [14]. For ease of exposition we stick to the case of narrow class number one.) We fix a prime p of F lying above the rational prime p. We write F p for the completion of F at p, O p for the valuation ring of F p with local uniformizer and q for the cardinality of the residue field of O F . We put and The matrix w = 0 1 0 normalizes the Iwahori subgroup I p . We define I p to be the subgroup generated by I p and w and let be the unique non-trivial homomorphism that is trivial on I p . Let f be a cuspidal Hilbert newform with trivial nebentypus and parallel weight 2. We assume that f is Steinberg at p, i.e., p divides the level of f exactly once and U p f = f. For simplicity, we assume that all Hecke eigenvalues of f are rational. In that case one can attach to f an elliptic curve E f /F that has split multiplicative reduction at p. Let ρ f : G F → GL 2 (Q p ) be the associated Galois representation. By Tate's p-adic uniformization theorem, the restriction ρ f,p of ρ f to a decomposition group at p is a non-crystalline extension of Q p by Q p (1). Thus, we can define the discrete subgroup ρ f,p ⊆ F × p . We put St(Z) = C(P 1 (F p ), Z)/Z and for any nonzero ideal n of O F we define 0 (n) = γ ∈ PGL 2 (O F ) + | γ is upper triangular modulo n , where the superscript + denotes matrices with totally positive determinant.
Suppose that p divides n exactly once. Then, by [22], Proposition 5.8 (b), the map given by evaluation at a nonzero Iwahori-fixed vector induces an isomorphism on fisotypic components for the Hecke algebra away from p. In particular, the f -isotypic component vanishes for i = d and is nonzero for i = d and large enough level. Let us give a sketch of the proof: first let us remind ourselves of the definition of a (compactly) induced module. Given a group V , a subgroup U ⊆ V and an The group V acts on c-ind V U A via left translation. The I p -invariants of St p (Z) are a free Z-module of rank 1 that generate St p as a G p -module. The matrix w acts on it by χ p .
Hence, Frobenius reciprocity induces a surjective G p -equivariant homomorphism c-ind In fact, this surjection fits into a short exact sequence of G p -modules that identifies the Steinberg representation with the first cohomology with compact support of the Bruhat-Tits tree (see for example [22], equation (18)). Utilizing Shapiro's Lemma the short exact sequence above induces a long exact sequence in cohomology of the form where W p denotes the Atkin-Lehner operator at p. Since p divides the conductor of f , we see that where the superscript f denotes taking the f -isotypic component. Thus, the claim follows. Let us mention that the above argument also yields the following: if g is a cuspidal Hecke eigenform that is not Steinberg at p, then we have Let A denote the ring of analytic functions on Drinfeld's p-adic upper half plane over F p . By Theorem 12, which in this case is due to van der Put (see [25], Proposition 1.1), we have Moreover, we have if the level n is large enough. By our main theorem the following diagram is commutative: where the horizontal maps are the boundary maps induced by the short exact sequences

Remark 16
Note that only the cohomology of arithmetic groups in degree 0-or in other words the trivial representation-contributes to the cohomology of E(ord p ) * . If the Eisenstein series of weight 2 and level 1 would exist, then it would also contribute to the cohomology of E(ord p ) * . One can enlarge H 1 ( p , E(ord p ) * ) by raising the level away from p. This process is often called smoothing (see for example the constructions of Darmon-Dasgupta in [5]).
It is well-known that Thus, the long exact sequence associated with (10) induces the short exact sequence By the discussion in Section 3.4 of [22] the exact sequences (8), (10) and (11) all fit together nicely into the following big commutative diagram with exact rows and columns: Remember that van der Put's theorem gives a canonical isomorphism H 1 ( p , St p (Z) * ) = H 1 ( p , A × /Q × p ). We denote the image of J triv under this isomorphism also by J triv . The Eisenstein series E 2 (p) defines a class in H 1 ( 0 (p), Z) and, thus, also a class J DR ∈ H 1 ( p , A × /Q × p ) that is unique up to powers of J triv . Arguing as in the previous section we see that this class cannot be lifted to a class in H 1 ( p , A × ). But there exists a discrete subgroup E 2 (p),p ⊆ Q × p of rank one such that J DR can be lifted to a class in H 1 ( p , A × / E 2 (p),p ). Thus, we have to show that this discrete subgroup is homothetic to the one generated by p.
Although Theorem 3.16 of [17] is stated only for cusp forms it is enough to assume that the system of eigenvalues for the full Hecke algebra (including the Hecke operator at p) shows up only in a single degree in cohomology. This is certainly the case for E 2 (p). Thus, one can argue as in loc.cit. to show that E 2 (p),p is homothetic to the lattice generated by any element q ∈ Q × p such that • ord p (q) = 0 and • log p (q) = −2 dα dk | k=2 , where α is the U p -eigenvalue of the Hida family passing through E 2 (p). As the U peigenvalue in the Eisenstein family is constant (and equal to 1) we see that q = p fulfils the two properties above.

Theta cocycles for definite unitary groups
We end this note with some remarks about theta cocycles for definite unitary groups. The author hopes to return to the study of these cocycles and their arithmetic applications in the future. Let GU be the group of unitary similitudes of a definite Hermitian form over a totally real number field F and let G be GU modulo its center. Assume that G is split at a prime p of F , i.e., G(F p ) ∼ = PGL n (F p ). Let π be an automorphic representation which is cohomological with respect to the trivial coefficient system and such that its local component at π p is the Steinberg representation of PGL n (F p ). We assume for simplicity that all Hecke eigenvalues are rational. By the result of many authors (see Theorem 2.1.1 of [1] for an overview), one can attach a Galois representation ρ π to π and the restriction ρ π ,p of ρ π to a decomposition group at p is of the following form: it is upper triangular and the i-th diagonal entry is the n − i-th cyclotomic character.