Oddness of residually reducible Galois representations

We show that suitable congruences between polarized automorphic forms over a CM field always produce elements in the Selmer group for exactly the +/--Asai (aka tensor induction) representation that is critical in the sense of Deligne. For this we relate the oddness of the associated polarized Galois representations (in the sense of the Bella\"iche-Chenevier sign being +1) to the parity condition for criticality. Under an assumption similar to Vandiver's conjecture this also provides evidence for the Fontaine-Mazur conjecture for polarized Galois representations of any even dimension.


Introduction
Ribet proved in [26] that, if a prime p > 2 divides the numerator of the kth Bernoulli number for k ≥ 4 an even integer, then p divides the order of the part of the class group of Q(µ p ) on which Gal(Q(µ p )/Q) acts by the (1 − k)th power of the mod p cyclotomic character . The argument (in fact a slight variance of Ribet's proof outlined in [19]) is as follows: One considers the weight k Eisenstein series for SL 2 (Z) and proves the existence of a weight k cuspidal eigenform that is congruent to the Eisenstein series. Ribet then shows that one can find a lattice in the irreducible odd p-adic Galois representation ρ f associated to f such that its mod p reduction gives a non-split extension ( 1 * 0 k−1 ). Note that this proves one direction of the Bloch-Kato conjecture for the Riemann zeta function (proven in [6, Theorem 6.1(i)]) in so far as it relates the p-divisibility of the value ζ(k) π k for the even integer k to that of the This is an Open Access article published by World Scientific Publishing Company. It is distributed under the terms of the Creative Commons Attribution 4.0 (CC-BY) License. Further distribution of this work is permitted, provided the original work is properly cited.
order of the Selmer group for the critical motive Q(1 − k). For odd k this motive is not critical and Vandiver's conjecture predicts, in fact, that the corresponding Selmer group is trivial. We note that ρ f is polarized in the sense that and that the characters occurring in the reduction get swapped under this polarization. We generalize Ribet's strategy using polarized (essentially conjugate dual) Galois representations of CM fields to prove part of the Bloch-Kato conjecture for general Asai (or tensor induction) L-functions, our main result in Theorem 4.1. For the Asai L-functions induced from CM fields a similar parity condition to that for the Riemann zeta function is required for criticality. We demonstrate in Corollary 4.3 that this parity condition is linked to the oddness of the polarized Galois representation (where the representation is odd if the sign defined in [3] equals +1, see Definition 2.2).
Before we explain our work we want to discuss a bit more of this parity issue in the situation considered by Ribet. By the Fontaine-Mazur conjecture in [14] all representations σ : G Q → GL 2 (Q p ) that are "geometric" (i.e. unramified at almost all primes and potentially semi-stable at p) are modular and therefore odd (in the sense that det(σ(c)) = −1 for any complex conjugationc), or they must be a Tate twist of an even representation with finite image. Calegari proved the following remarkable result about the non-existence of even geometric representations: Theorem 1.1 ([9]). Let p > 7 be a prime and σ : G Q → GL 2 (Q p ) a continuous representation such that σ is unramified at all but finitely many primes, and at p is potentially semi-stable with distinct Hodge-Tate weights. Further assume that the residual representation σ is absolutely irreducible and not of dihedral type, and that σ| Dp is not a twist of a representation of the form ( * 0 1 ). Then σ is odd.
In the residually reducible case one can use Ribet's argument to prove the following: Assume that σ is unramified outside of p, and either ordinary at p or crystalline at p with Hodge-Tate weights in [0, p − 3]. Then Vandiver's conjecture (p #Cl(Q(µ p ) + )) implies that σ is odd.
Proof. By [26] one can find a lattice such that the mod p reduction is a non-split extension of the form ( 1 * 0 m ) that is unramified away from p. One can show that this extension splits at p. This follows from Fontaine-Laffaille theory if σ is "short crystalline" as in the statement, or from ordinarity, which implies that σ| Dp has an unramified quotient. This implies (see e.g. [19, p. 275]) that p | #Cl(Q(µ p ))( −m ). By Vandiver's conjecture m has to be odd, which implies that det(σ) is odd. Remark 1.3. The assumptions on the ramification of σ can be weakened, e.g. by allowing σ| I ∼ = ( 1 * 0 1 ) for primes = p satisfying m−1 ≡ 1 mod p. At p crystallinity alone is not enough to be able to deduce that the extension is split at p, as the following example (pointed out to us by Kevin Buzzard) shows: Consider p = 3 and σ = ρ ∆,3 the 3-adic Galois representation associated to the weight 12 level 1 cuspform ∆. Then ρ ss ∆,3 = 1 ⊕ , but the class number of Q(µ 3 ) is one.
We will prove in Corollary 5.3 a higher-dimensional generalization of Proposition 1.2 for residually reducible polarized Galois representations of any dimension. This arises as a consequence of the contrapositive of our main result Theorem 4.1, as we will explain in the following: Given a representation ρ : G K → GL n (E) for K/K + a CM field and E a finite extension of Q p one can define two extensions of ρ ⊗ ρ c to G K + , which we denote by As ± (ρ) (for details see Sec. 2.1). Theorem 4.1 concerns the Bloch-Kato conjecture for these representations and generalizes our previous result in [4] for Asai representations of 2-dimensional representations of an imaginary quadratic field to this setting, including extensions of adjoint representations of polarized regular motives as studied in [17]. We explore in this paper the subtle connection between polarizations, signs and criticality that this exposes.
To explain this we need to introduce some more notations: Let Ψ : G K + → E * be a Hecke character and R : R is residually reducible with two summands that get swapped under the polarization). Using Shapiro's lemma one can relate the K + -Selmer group for As ±Ψ(cv ) (ρ) ⊗ Ψ to the part of its K-Selmer group on which the action by a complex conjugation c v for v | ∞ is given by ±Ψ(c v ). In Lemma 4.2, we show that on H 1 (K, As −Ψ(cv) (ρ)⊗Ψ) the action by c v agrees with an involution ⊥ cv corresponding to the polarization condition satisfied by R. A result of [2] implies that the involution ⊥ cv acts on the G K -extension constructed from R by the sign associated to (R, Ψ, c v ) (introduced in Definition 2.2). Combining these, we get in Theorem 4.1 a construction of an element in the Bloch-Kato Selmer group of As ± (ρ) ⊗ Ψ, where the sign is given by −Ψ(c v )sign(R, Ψ, c v ). In Proposition 3.3, we prove that this As ± (ρ) ⊗ Ψ is critical in the sense of Deligne (for characters Ψ where this corresponds to a twist in the left-hand side of the critical strip of As(ρ)) if and only if sign(R, Ψ, c v ) = 1. That automorphic Galois representations R are odd in this sense has been proved in [3] for Galois representations associated to regular algebraic cuspidal polarized representations of GL m over CM fields. We expect it to hold more generally, see the discussion in Remark 2.7. The (non-critical) G K + -representation As Ψ(cv) (ρ) ⊗ Ψ does, in fact, not depend on Ψ(c v ). This allows us in Corollary 5.3 to deduce total oddness of certain residu-ally reducible polarized Galois representations if we assume that the Selmer group H 1 Σ + (K + , As Ψ(c) (ρ) ⊗ Ψ) is trivial for a fixed choicec of complex conjugation.

Notation and Definitions
If L is a number field then we fix algebraic closures L and L v for every place v of L and compatible embeddings L → L v → C and write D Lv and I Lv (or I v if L is understood) for the corresponding decomposition and inertia subgroups of G L := Gal(L/L). We fix a prime p > 2 and denote by : G L → Z * p the p-adic cyclotomic character. If v is a finite place of L, not dividing p, then its value on the arithmetic Frobenius element (Frob v ) = q v , where the latter denotes the size of the residue field of L at v. For a p-adic representation V of G L , we write V (n) := V ⊗ n for n ∈ Z. For the coefficients of our p-adic representations, we will take E to be a (sufficiently large) finite extension of Q p inside Q p with ring of integers O and residue field F. We fix a choice of a uniformizer .
We fix an isomorphism Q p ∼ = C. If χ : A * L /L * → C * is a Hecke character of type (A 0 ) then we write χ gal for the associated Galois character G L → Q * p (for details see e.g. [29], but we use the local Artin reciprocity maps mapping uniformizers to arithmetic Frobenius elements).
Let K/K + be a CM field, and write χ K/K + : G K + → {±1} for the corresponding quadratic character. We fix a complex conjugationc ∈ G K + . For a G K -representation (ρ, V ) we define the conjugate representation (ρ c , V c ) by ρ c (g) = ρ(cgc).

Asai L-function and tensor induction
We describe the definition of the Asai L-function of a cuspidal automorphic representation π for GL m (K) for a CM field K/K + (adapting the treatment for m = 2 in [21], see also [25,Sec. 6]). The Langlands dual of the algebraic group R K/K + GL m is given by given by For each place v we denote by r ± v the local L-group homomorphisms obtained from r ± by restriction.
Let π be a cuspidal automorphic representation for GL m (A K ). Then we may consider π as a representation of R K/K + GL m (A) and factorize it as a restricted under the local Langlands correspondence and put As ± (π) := ⊗ v As ± (π v ). By Langlands functoriality these should be isobaric automorphic representations of GL m 2 (A), the ± Asai transfers of π. For m = 2 this has been proven by Krishnamurty [20] and Ramakrishnan [24].
We will, however, only be referring to the corresponding Langlands L-function, which is defined by For unramified v and π v spherical, the latter is given by For more details and its analytic properties we refer the reader to [16].
We will also consider the twisted Asai L-function L(s, π, r ± ⊗ Ψ) for Ψ : G K + → Q * p . For unramified places its Euler factors are given by The operation for Galois representations corresponding to the Asai transfer is usually called tensor (or multiplicative) induction: Consider a representation ρ : G K → GL(V ) for an n-dimensional vector space V over a field. Then we define the following extensions of V ⊗ V c to G K + : given by We note that As − (ρ) ∼ = As + (ρ) ⊗ χ K/K + .

Bloch-Kato Selmer groups
Let L be either K or K + . Consider a continuous finite-dimensional representation V of G L over E. Let T ⊆ V be a G L -stable O-lattice and put W = V/T and W n = {x ∈ W : n x = 0}. Let Σ be a finite set of places of L. Following Bloch-Kato (see also [12,Sec. 2.1]) we define for M = W, W n the following Selmer groups: where and

Polarized Galois representations and signs
In this section we consider complex conjugations c v ∈ G K + for v | ∞ and conjugate representations R cv defined by R cv (g) = R(c v gc v ) (which are all isomorphic to R c defined using our fixed complex conjugationc). By Schur's Lemma there exists a matrix A v ∈ GL m (Q p ), unique up to scalar, such that One can then apply Schur's Lemma again (see e.g. [8]) to deduce that the matrix A v is in fact symmetric or antisymmetric. Remark 2.4. This lemma shows that our notion of "totally odd polarized" agrees with that in [1,Sec. 2], except that we do not demand that Ψ(c v ) = −1 for all v | ∞, which can, however, easily be achieved by replacing Ψ by Ψχ K/K + .

Example 2.5 (Essentially self-dual G K + -representation).
(1) LetR : This implies thatR is either (generalized) symplectic or orthogonal, i.e. that there exists a sympletic or orthogonal pairing for the vector space underlying R. Lemma 2.6 in [8] shows that (R| GK , Ψ) is odd at v either ifR is symplectic and Ψ(c v ) = −1 or ifR is orthogonal and Ψ(c v ) = 1.
In particular, this shows that the notion of oddness in Definition 2.2 generalizes the notion of oddness for 2-dimensional representations ρ : . This means that sign(R) = 1 is again equivalent to the oddness of the polarization character det(R).
We recall from e.g. [29, Theorem 2.2] that for (π, χ) a RAECSDC automorphic representation of GL m (A K ) there exists a continuous semisimple representation Kv is the local Langlands correspondence for GL m (K v ) using the arithmetic Frobenius normalization.  1(1)]). Let (π, χ) be a RAECSDC representation of GL m (A K ). Then (any irreducible factor of ) ρ π is totally odd with respect to 1−m χ gal . Remark 2.7. In certain cases, Galois representations have been constructed for irregular π (see, in particular, the recent papers of [15,23] in the case that the descent of π to a unitary group has a non-degenerate limit of discrete series as archimedean component).
In Theorem 4.1, we consider representations that are residually reducible and not necessarily Hodge-Tate regular (for an example see [4, Theorem 5.1]). We are therefore interested in the extension of Bellaïche-Chenevier's result to π that are not regular.
If p > 2 and ρ π is residually irreducible then the signs of ρ π and its reduction agree (see e.g. [3,Sec. 2.3]). This implies that in such cases oddness is also known for irregular π if there is a congruence to a RAECSDC representation. Since oddness of ρ π corresponds to the associated G K + -representation being valued in the L-group of (an appropriate covering group of) the unitary group (see [7,Sec. 8.3]) the work of Lafforgue on pseudorepresentations ([22, Proposition 11.7]) should more generally allow to relax the regularity assumption in Theorem 2.6.

Bloch-Kato Conjecture
Let π be a regular L-algebraic cuspidal automorphic representation of GL m (A K ). Harris et al. [18] and Scholze [28] have constructed Galois representations ρ π : G K → GL m (E) associated to π (for E a sufficiently large finite extension of Q p ) such that its Frobenius eigenvalues match the Hecke eigenvalues of π (we again take the arithmetic Frobenius normalization).
In this section, we state the Bloch-Kato conjecture (following the exposition in [13]) for the Asai L-value L(1, π, r ± ⊗ Ψ) for Ψ : G K + → E * a continuous Galois character and analyze the criticality of As ± (ρ π )(−m).
Let F be a number field (including the field of definition of the Satake parameters of π), and p | p a place of F with F p ∼ = E. Let As ± (ρ π ) : G K + → GL m (E) be the Asai plus/minus representation defined in Sec. 2.1. By [10, Conjecture 4.5], (applied to the conjectural Asai transfer As ± (π) to GL m 2 (A)) there should exists a motive M(π, r ± ⊗ Ψ) of rank m 2 with coefficients in F such that if ρ M(π,r ± ⊗Ψ),p : is the Satake parameter in L (R K/K + GL m )(C). Note that this p-adic realization is dual to As ± (ρ π ) ⊗ Ψ due to the arithmetic Frobenius normalization we chose for ρ π .
Put Assume p 0 (see [13,Sec. 4] for details, but in particular, so that (As ± (ρ π ) ⊗ Ψ)| Dv for v | p can be described using Fontaine-Laffaille theory) and π is not ramified at any places dividing p.
where V is the version of the Fontaine-Laffaille functor used in [12].
Let Ω be a Deligne period scaled according to the above choice, i.e. the determinant of the isomorphism calculated with respect to the bases of (T ± B ) + and T ± dR /Fil 1 , so well-defined up to O * (p) .  Conjecture 3.1 ([6; 13, Conjecture 4.1]). If Σ + is a finite set of primes, containing all v where π v or K/K + is ramified, but not containing p then , with the dual action of G K + , and # denotes a Fitting ideal.

Proof. We need to show
(In the following, we will write r for r (−1) m+1 .) We note that the center of the Hodge decomposition of H dR (M ∨ (π, r)(−m)) has dimension h wt(ρ),wt(ρ) = n since the motive corresponding to ρ is assumed to be pure and regular. If we list (a, b) occurring in the Hodge decomposition of H dR (M ∨ (π, r)(−m)) in order of increasing a then under our assumption on m the pair immediately to the left of center has a negative a. This implies that Let t B = 2πit be a rational basis for K + (1) B = (2πi)K + and e χ a basis vector for the Betti realization of the Artin motive K + (χ K/K + ) of rank 1 over K + attached to the character χ K/K + . As a model of H B (M ∨ (π, r)(−m)) we take This implies that the vectors , a < b} form a basis for H B (M ∨ (π, r (−1) m+1 )(−m)) + , so that its dimension is therefore also

Construction of Elements in Critical Selmer Groups for General Asai Representations
This section proves the main result (Theorem 4.1) which establishes the Galois part of the proof of one direction of the Bloch-Kato conjecture for general Asai representations (i.e. we assume the existence of a residually reducible representation R that could be constructed by establishing suitable congruences of polarized automorphic forms). Note that the assumption in the theorem on the reduction of R is that R ss has two summands that get swapped under the involution σ → σ c∨ ⊗ Ψ −1 corresponding to the polarization of (R, Ψ).

Oddness of residually reducible Galois representations 11
Assume there exists an absolutely irreducible representation that is unramified away from a finite set of places Σ and short crystalline at v | p such that If Σ + is the set of places of K + lying below Σ then and this extension is not split.
By using a result of Bellaïche and Chenevier we can now show that the extension (4.1) constructed using [ Under some mild additional assumptions one can check that the Theorem implies that this construction always produces elements in the Selmer group for the Asai motive that is critical in the sense of Deligne. For Ψ| GK = −w (e.g. the polarization character for the Galois representations associated to RACSDC representations of GL w+1 (A K )) and sign(R, Ψ,c) = 1 we have As −Ψ(c)sign(R,Ψ,c) (ρ(i)) ⊗ Ψ = As (−1) w+1 (ρ)(−w + 2i). We can therefore deduce the following corollary to Theorem 4.1 and Proposition 3.3: Corollary 4.3. Let p > 2, K/K + a CM field and ρ : G K → GL n (E) be an residually irreducible continuous representation arising from a regular motive that is pure of some weight. Consider an absolutely irreducible representation with sign(R, Ψ,c) = 1 that is pure of weight w, unramified away from a finite set of places Σ and short crystalline at v | p and satisfies Assume that for some i ∈ Z such that −w + 2i lies in the left-hand side of the critical strip of As(ρ) we have If Σ + is the set of places of K + lying below Σ then p | #H 1 Σ + (K + , As (−1) w+1 (ρ)(−w + 2i) ⊗ (E/O)) and As (−1) w+1 (ρ)(−w + 2i) is critical.
Remark 4.4. By the Bloch-Kato Conjecture (Conjecture 3.1) the Selmer group in the corollary should be governed by the L-value L(1 + w − 2i, π, r (−1) w+1 ) (on the right-hand side of the critical strip). Corollary 4.3 provides evidence for Conjecture 3.1 as we expect the existence of R to be governed by the p-divisibility of the corresponding normalized Asai L-value. For n = 2 and i = 0 we described in [4, Secs. 6 and 7] a strategy to prove suitable congruences between the theta lift of the Bianchi modular form and stable Siegel cuspforms giving rise to R as in the corollary.

Fontaine-Mazur Conjecture
By combining Fontaine-Mazur's conjecture ([14, Conjecture 1]) with Langlands automorphy conjectures one arrives at the following conjecture: Then there exists an algebraic essentially conjugate dual cuspidal automorphic representation π of GL m /K such that ρ π ∼ = R. Remark 5.2. Results toward this conjecture have been proven by Barnet-Lamb-Gee-Geraghty-Taylor [1] and Thorne [29]: Theorem 4.5.1 of [1] proves potential automorphy for odd R that are residually irreducible (and satisfying some further conditions, e.g. potential diagonalizability at v | p).
Theorem 7.1 of [29] proves the conjecture for certain residually reducible R with and R ss ∼ = ρ 1 ⊕ ρ 2 with ρ i | GK (ζp) adequate and R of Schur type, i.e. ρ 1 ∼ = ρ 2 and 1−m ρ c 1 ∼ = ρ ∨ 2 (so residual constituents do not get swapped by involution, different to our set-up). Apart from some other technical conditions [29] assumes R ordinary at v | p and that R ss is automorphic, but not that R is odd. The deformation theory techniques used to study residual representations of Schur type do not apply in our setting.
Similar to Calegari's result (see Theorem 1.1) the oddness of automorphic Galois representations (Theorem 2.6 and Remark 2.7) and Conjecture 5.1 implies the nonexistence of even geometric representations (except for certain exceptional cases like Tate twists of even representations with finite image). In the residually reducible case our result allows the following evidence toward Conjecture 5.1: