L-Invariant for Siegel–Hilbert Forms

Giovanni Rosso

Received: January 23, 2015

Communicated by Otmar Venjakob

Abstract. We prove a formula for the Greenberg–Benois L-invariant of the spin, standard and adjoint Galois representations associated with Siegel–Hilbert modular forms. In order to simplify the calculation, we give a new definition of the L-invariant for a Galois representation V of a number field F ≠ ℚ; we also check that it is compatible with Benois’ definition for Ind_{F}^{ℚ}(V).

2000 Mathematics Subject Classification: 11R23, 11F80, 11F46, 11S25 Keywords and Phrases: Iwasawa Theory, L-invariants, p-adic Lfunctions, p-adic families of automorphic forms

1 Introduction

Since the historical results of Kummer and Kubota–Leopold on congruences for Bernoulli numbers, people have been interested in studying the p-adic variation of special values of L-functions. More precisely, fix a motive M over ℚ. We suppose that M is Deligne critical at s = 0 and that there exists a Deligne’s period Ω(M) such that \frac{L(M,0)}{Ω(M)} is algebraic. Fix a prime p and two embeddings

ℂ_{p} ↩ \overline{ℚ} ↪ ℂ.

Let V be the p-adic realization of M and suppose that V is semistable (à la Fontaine). Thanks to work of Coates and Perrin-Riou, we have precise conjectures on how the special values should behave p-adically; we fix a regular sub-module of V. This corresponds to the choice of a sub-(φ, N)-module of D_{st}(V) which is a section of the exponential map

D_{st}(V) → t(V) ≅ \frac{D_{st}(V)}{Fil^{0}D_{st}(V)}.

Let h be the valuation of the determinant of φ on D. We can state the following conjecture; Conjecture 1.1. There exists a formal series L_{p}^{D}(V, T) ∈ ℂ_{p}[[T]] which grows as log_{p}^{h} such that for all non-trivial, finite-order characters ε : 1 + pℤ_{p} → μ_{p^{∞}} we have

L_{p}^{D}(V, ε(1 + p) − 1) = C_{ε}(D)\frac{L(M ⊗ ε, 0)}{Ω(M)}.

Moreover, for ε = 1 we have

L_{p}^{D}(V, 0) = E(D)\frac{L(M, 0)}{Ω(M)},

where E(D) is an explicit product of Euler-type factors depending on D and (D_{st}(V)/D)^{N=0}.

It may happen that one of the factors of E(D) vanishes and then we say that trivial zeros appear. Since the seminal work of [MTT86], people have been interested in describing the p-adic derivative of L_{p}^{D}(V, (1 + p)^{s} −1) when trivial zeros appear. We suppose for simplicity that L(M, 0) is not vanishing. We have the following conjecture by Greenberg and Benois;

Conjecture 1.2. Let t be the number of vanishing factors of E(D). Then

• ord_{s=0}L_{p}^{D}(V, (1 + p)^{s} − 1) = t,

• L_{p}^{D}(V, 0)^{∗} = L(V^{∗}(1), D^{∗})E^{∗}(D)\frac{L(M,0)}{Ω(M)}.

Here E^{∗}(D) is the product of non-vanishing factors of E(D) and L(V^{∗}(1), D^{∗}) is a number, defined in purely Galois theoretical terms (see Section 3.1), for the dual Galois representation V^{∗}(1).

L-Invariant for Siegel–Hilbert Forms 1229

The error factor L(V, D) is quite mysterious. It has been calculated in only few cases for the symmetric square of a (Hilbert) modular form by Hida, Mok and Benois and for symmetric power of Hilbert modular forms by Hida and Harron– Jorza. Unless V is an elliptic curve over ℚ with multiplicative reduction at p we can not prove the non-vanishing of L(V, D). The aim of this paper is to calculate it in some new cases; let F be a totally real field (we make no assumptions on the ramification at p) and π be an automorphic representation of GSp_{2g}_{/F} of weight \underline{k} = (k_{τ})_{τ}, where τ runs through the real embeddings of F and (k_{τ}) = (k_{1,τ} , …, k_{g,τ}; k_{0}) (note that k_{0} does not depend on τ). We say that π is parallel of weight k, k ∈ ℤ_{≥0} if k_{i,τ} = k for all τ and i = 1, …, g and k_{0} = gk. We suppose that it has Iwahoric level at all 𝔭 | p. We suppose moreover that π_{𝔭} is either Steinberg (see Definition 4.8) or spherical. We partition consequently the prime ideals of F above p in S^{Stb} ∪ S^{Sph}. We have conjecturally two Galois representations associated with π, namely the spinorial one V_{spin} and the standard one V_{sta}. Let V be one of these two representations. We choose for each prime 𝔭 of F dividing p a regular sub module D_{𝔭} of D_{st}(V_{|_{G_{F𝔭}}}). Consider a family of Siegel–Hilbert modular forms as in [Urb11] passing through π. Let us denote by β_{𝔭}(κ) the eigenvalue of the normalized Hecke operators U_{1,𝔭} (see Definition 4.9) on this family. Let S^{Sph,1} = S^{Sph,1}(V, D) be the subset of S^{Sph} for which (D_{st}(V_{𝔭})/D_{𝔭})^{N=0} does contain the eigenvalue 1. Conjecturally, it is empty for the spin representation. The eigenvalues 1 always appears in D_{st}(V_{𝔭}) for V the standard representation but it may appear in D_{𝔭} (this is already the case for the symmetric square of a modular form). Let t_{Stb} be the cardinality of S^{Stb} and t_{Sph} be the cardinality of S^{Sph,1}. We define f_{𝔭} = [F_{𝔭}^{ur} : ℚ_{p}].

Theorem 1.3. Let π be as above, of parallel weight k. Let V = V_{spin} and suppose hypothesis LGp of Section 4.2, then the expected number of trivial zeros for L_{p}^{D}(V (k − 1), T) is t_{Stb} and

L(V (k − 1), D) = ∏_{𝔭∈S^{Stb}} −\frac{1}{f_{𝔭}} \frac{d log_{p} β_{𝔭}(k)}{dk}_{|_{k=\underline{k}}}.

Let V = V_{std}, then the conjectural number of trivial zero for L_{p}^{D}(V, T) is t_{Stb} + t_{Sph} and

L(V, D) = L(V, D)^{Sph} ∏_{𝔭∈S^{Stb}} −\frac{1}{f_{𝔭}} \frac{d log_{p} β_{𝔭}(k)}{dk}_{|_{k=\underline{k}}},

where L(V, D)^{Sph} is a priori global factor. It is 1 if t_{Sph} = 0.

In Section 4.2 we shall provide also a formula for the L-invariant of V_{std}(s) (min(k − g − 1, g − 1) ≥ s ≥ 1).

The proof of the theorem is not different from the one of [Ben10, Theorem 2] which in turn is similar to the original one of [GS93]. Let now g = 2. Let t be the number of primes above p in F. We consider the 2t-dimensional eigenvariety for GSp_{4}_{/F} with variables k = {k_{𝔭,1}, k_{𝔭,2}}_{𝔭}(see Section 5) and let us denote by F_{𝔭,i}(k) (i = 1, 2) the first two graded pieces of D_{rig}^{†}(V_{spin}). The 10-dimensional Galois representation Ad(V_{spin}) has a natural regular sub-(φ, N)-module induced by the p-refinement of D_{rig}^{†}(V_{spin}) and which we shall denote by D_{Ad}. With this choice of regular sub module, Ad(V_{spin}) presents 2t conjectural trivial zeros. In Section 5 we prove the following theorem;

Theorem 1.4. Let π be an automorphic form of weight \underline{k}. Suppose that hypothesis LGp of Section 4.2 is verified for V_{spin} and the point corresponding to π in the eigenvariety X′ (as defined in Section 5) is étale over the weight space. We have then

L(Ad(V_{spin}(π)), D_{Ad}) = ∏_{𝔭} \frac{2}{f_{𝔭}^{2}} det( \begin{matrix}\frac{∂ log_{p} F_{𝔭_{i},1}(k)}{∂k_{𝔭_{j} ,1}} & \frac{∂ log_{p} F_{𝔭_{i},2}(k)}{∂k_{𝔭_{j} ,1}} \\ \frac{∂ log_{p} F_{𝔭_{i},1}(k)}{∂k_{𝔭_{j} ,2}} & \frac{∂ log_{p} F_{𝔭_{i},2}(k)}{∂k_{𝔭_{j} ,2}} \end{matrix} )_{1≤i,j≤t}_{|_{k=\underline{k}}}.

We remark that this theorem is the first to really go beyond GL_{2} and its representations Sym^{n}.

The motivation for Theorem 1.3 lies in a generalization of [Ros15] to Siegel forms. In loc. cit. we use Greenberg–Stevens method to prove a formula for the derivative of the symmetric square p-adic L-function and calculate the analytic L-invariant and the same method of proof could possibly be generalized to finite slope Siegel forms thanks to the overconvergent Maß-Shimura operators and overconvergent projectors of Z. Liu’s thesis. With some work, it could also be generalized to totally real field where p is inert, as already done for the symmetric square [Ros13].

We hope to calculate the L-invariant for V_{std} and Ad(V_{spin}) for more general forms in a future work.

In Section 2 we recall the theory of (φ, Γ)-module over a finite extension of ℚ_{p}. It will be used in Section 3 to generalize the definition of the L-invariant à la Greenberg–Benois to Galois representations V over general number field F (note that we do not suppose p split or unramified). This definition does not require one to pass through Ind_{F}^{ℚ}(V) to calculate the L-invariant which in turn simplifies explicit calculation. We shall check that this definition coincides with Benois’ definition for Ind_{F}^{ℚ}(V). We prove the above-mentioned theorems in Section 4 and 5, inspired mainly by the methods of [Hid07].

Acknowledgement This paper is part of the author’s PhD thesis and we would like to thank J. Tilouine for his constant guidance and attention. We would like to thank A. Jorza for telling us that the study of the L-invariant in the Steinberg case was within reach. We would also like to thank D. Hansen, É. Urban and S. Shah for useful conversations and the anonymous referee for his/her remarks and corrections. The paper has been written while the author was a PhD Fellow of the Fund for Scientific Research - Flanders, at KU Leuven. Part of it has been written during a visit at Columbia University which the author would like to thank for the excellent working condition. During this work, the author has been supported by a FWO travel grant (V4.260.14N) and an ANR grant (ANR-10-BLANC 0114 ArShiFo).

2 Some results on rank one (φ, Γ)-module

Let L be a finite extension of ℚ_{p}. The aim of this section is to recall certain results concerning (φ, Γ)-modules over the Robba ring R_{L}. Let L_{0} be the maximal unramified extension contained in L. Let L′_{0} be the maximal unramified extension contained in L_{∞} := L(μ_{p^{∞}}) and L′ = L · L′_{0}. Let e_{L} := [L(μ_{p^{∞}}) : L_{0}(μ_{p^{∞}})] = [Γ_{ℚ_{p}} : Γ_{L}], where Γ_{L} := Gal(L_{∞}/L). We define

B_{L,rig}^{†,r} = {f = ∑_{n∈ℤ} a_{n}π_{L}^{n}|a_{n} ∈ L′_{0}, such that f(X) = ∑_{n∈ℤ} a_{n}X^{n}
is holomorphic on p^{− \frac{1}{e_{L}r}} ≤ |X|_{p} < 1\} ,
B_{L,rig}^{†} := ⋃_{r} B_{L,rig}^{†,r},

where π_{L} is a certain uniformizer coming from the theory of field of norms. Note that B_{L,rig}^{†} is classically called the Robba ring of L′_{0}. For sake of notation, we shall denote write R_{L} := B_{L,rig}^{†}. We hope that this will cause no confusion in what follows. We have an action of φ on R_{L}. If L = L_{0}, there is no ambiguity and we have:

φ(π_{L}) = (1 + π_{L})^{p} − 1, φ(a_{n}) = φ_{L′_{0}}(a_{n}).

Otherwise the action on π_{L} is more complicated. Similarly, we have a Γ_{L}-action. If L = L_{0} we have

γ(π_{L}) = (1 + π_{L})^{χ_{cycl}(γ)} − 1,

where χ_{cycl} is the cyclotomic character. If L is ramified we also have an action of Γ_{L} on the coefficients given by

γ(a_{n}) = σ_{γ}(a_{n})

where σ_{γ} is the image of γ via

Γ_{L} → Γ_{L}/Γ_{L′} →^{≅} Gal(L′_{0}/L_{0}).

If a_{n} is fixed by φ and Γ_{L}, then is it in ℚ_{p}. We have rk_{R_{ℚp}} R_{L} = [L_{∞} : ℚ_{p,∞}].

Let δ : L^{×} → E^{×} be a continuous character. Let R_{L}(δ) be the rank one (φ, Γ_{L})-module defined as follows; fix a uniformizer ϖ_{L} of L and write δ = δ_{0}δ_{1} with δ_{0}|_{O_{L}^{×}} := δ|_{O_{L}^{×}}, δ_{0}(ϖ_{L}) := 1 and δ_{1} is trivial on O_{L}^{×} and δ_{1}(ϖ_{L}) := δ(ϖ_{L}). As δ_{0} is a unitary character, it defines by class field theory a unique one dimensional Galois representation δ̃_{0}. Fontaine’s theorem on the equivalence of category between (φ, Γ_{L})-modules and Galois representations [Fon90] gives us a one dimensional (φ, Γ_{L})-module D_{rig}^{†}(δ̃_{0}). We define R_{L}(δ_{1}) := R_{L} ⊗_{ℚ_{p}} Ee_{δ_{1}} so that φ^{f_{L}}(e_{δ_{1}}) = δ_{1}(ϖ_{L})e_{δ_{1}} (here f_{L} is the degree of L_{0} over ℚ_{p}), γ(e_{δ_{1}}) = e_{δ} and φ does not act on the E-coefficient. Finally, we define R_{L}(δ) = D_{rig}^{†}(δ̃_{0}) ⊗_{R_{L}} R_{L}(δ_{1}).

We now classify the cohomology of such a (φ, Γ_{L})-modules. It will be useful to calculate it explicitly in terms of C_{φ,γ}-complexes [Ben11, §1.1.5]. We fix then a generator γ_{L} of Γ_{L}; if clear from the context, we shall drop the subscript _{L} and write simply γ.

Proposition 2.1. We have H^{0}(R_{L}(δ)) = 0 unless δ(z) = ∏_{τ} τ(z)^{m_{τ}} with m_{τ} ≤ 0 for all τ; in this case we have H^{0}(R_{L}(δ)) ≅ E. We shall denote its basis by t^{−\underline{m}} ⊗ e_{δ}, where

t^{−\underline{m}} = (t^{−m_{τ}}) ∈ ∏_{τ} B_{dR}^{+} ⊗_{L,τ} E.

If δ(z) = ∏_{τ} τ(z)^{m_{τ}} with m_{τ} ≤ 0, then

dim_{E}H^{1}(R_{L}(δ)) = [L : ℚ_{p}] + 1.

If δ(z) = |N_{L/ℚ_{p}}(z)|_{p} ∏_{τ} τ(z)^{k_{τ}} with k_{τ} ≥ 1, then

dim_{E}H^{1}(R_{L}(δ)) = [L : ℚ_{p}] + 1.

Otherwise

dim_{E}H^{1}(R_{L}(δ)) = [L : ℚ_{p}].

We have H^{2}(R_{L}(δ)) = 0 unless δ(z) = |N_{L/ℚ_{p}}(z)|_{p} ∏_{τ} τ(z)^{k_{τ}} with k_{τ} ≥ 1; in this case we have H^{2}(R_{L}(δ)) ≅ E.

Note that when we choose t^{−\underline{m}} as a basis we are implicitly using the fact that we can embed certain sub-rings of R_{L} into B_{dR}^{+} (see [Ben11, §1.2.1]).

Proof. The same results is stated in [Nak09, Proposition 2.14, 2.15, Lemma 2.16] for E − B-pairs, but the proof for (φ, Γ)-modules is the same. Recall that have a canonical duality [Liu08] given by cup product

H^{i}(D) × H^{2−i}(D^{∗}(χ_{cycl})) → H^{2}(χ_{cycl}).

The last fact is then a direct consequence.

This allows us to define a canonical basis of H^{2}(R_{L}(|N_{L/ℚ_{p}}(z)|_{p} ∏_{τ} τ(z)^{k_{τ}})). We define H_{f}^{1}(D) as the H^{1} of the complex

D_{cris}(D) → t_{D} ⊕ D_{cris}(D)

and we have immediately [Nak09, Proposition 2.7]

dim_{E}H_{f}^{1}(D) = dim_{E}(H^{0}(D)) + dim_{E}t_{D}.  (2.2)

Hence Lemma 2.3. If δ(z) = ∏_{τ} τ(z)^{m_{τ}} with m_{τ} ≤ 0, then

dim_{E}H_{f}^{1}(R_{L}(δ)) = 1.

If δ(z) = |N_{L/ℚ_{p}}(z)|_{p} ∏_{τ} τ(z)^{k_{τ}} with k_{τ} ≥ 1, then

dim_{E}H_{f}^{1}(R_{L}(δ)) = d.

Proposition 2.4. Let D be a semi-stable (φ, Γ)-module over R_{L} with nonnegative Hodge–Tate weight. Suppose that D_{st}(D) = D_{st}(D)^{φ=1}. Then D is crystalline,

D ≅ ⊕R_{L}(δ_{i})

with δ_{i}(z) = ∏_{τ} τ(z)^{m_{i,τ}}, m_{i,τ} ≤ 0 and D_{st}(D) = D_{cris}(D) = H^{0}(D).

Proof. We follow closely the proof [Ben11, Proposition 1.5.8]. As Nφ = pφN we obtain immediately that N = 0, hence D is crystalline. Let r be the rank of D over R_{L}. We write the Hodge–Tate weight as (\underline{m_{i}})_{i=1}^{r} where \underline{m_{i}} = (m_{i,τ})_{τ}. We prove the proposition by induction; the case r = 1 is easy. If D is not split, for r = 2, we can suppose, as D is de Rham, that for each τ we have −m_{1,τ} ≤ −m_{2,τ}, hence \underline{m_{1}} = 0 by twisting. Let δ be defined by ∏_{τ} τ(z)^{m_{τ}}. So we have an extension of R_{L}(δ) by R_{L}. Let d_{2} be a lift to D of a basis of R_{L}. As φ = 1 we have φd_{2} = d_{2}. As the extension is crystalline we know that γ acts trivially too, hence the extension splits. Suppose now r > 2. Take v in the Fil^{−\underline{m_{0}}}D_{st}(D), the smallest filtered piece of D_{st}(D). We can associate to it R_{L}(δ), where δ(z) = ∏_{τ} τ(z)^{m_{0,τ}}. We have

0 → R_{L}(δ) → D → D′ → 0.

By inductive hypothesis D′ ≅ ⊕_{i=1}^{d−1}R_{L}(δ_{i}). We can write

Ext(D′, R_{L}(δ)) = ⊕_{i=1}^{d−1}Ext(R_{L}(δ_{i}), R_{L}(δ))

and we are reduced to the case r = 2 which has already been dealt.

We now want to calculate H_{f}^{1}(R_{L}(δ)) for δ(z) = ∏_{τ} τ(z)^{m_{τ}} with m_{τ} ≤ 0. We recall the following lemma [Ben11, Lemma 1.4.3]

Lemma 2.5. The extension cl(a, b) in H^{1}(R_{L}(δ)) corresponding to the couple (a, b) is crystalline if and only if the equation (1 − γ)x = b has a solution in R_{L}(δ) [\frac{1}{t}]

The following proposition in an immediate consequence of the above lemma [Ben11, Theorem 1.5.7 (i)] (see also the construction of [Nak09] at page 900)

Proposition 2.6. Let e_{δ} be a basis for R_{L}(δ). Then x\underline{m} = cl(t^{−\underline{m}}, 0)e_{δ} is a basis of H_{f}^{1}(R_{L}(δ)).

Remark 2.7. If δ is the trivial character then x_{0} corresponds (via class field theory) to the unramified ℤ_{p}-extension of Hom(G_{L}, E^{×}) ≅ H^{1}(G_{L}, E).

We now have to cut out another “canonical” one-dimensional subspace in H^{1}(R_{L}(δ)) which trivially intersects H_{f}^{1}(R_{L}(δ)) (and reduces to the cyclotomic ℤ_{p}-extension in the sense of the previous remark). We recall that for L = ℚ_{p} Benois has defined in [Ben11, Proposition 1.5.9] a canonical complement H_{c}^{1}(R_{ℚ_{p}}(z^{m})) of H_{f}^{1}(R_{ℚ_{p}}(z^{m})) inside H^{1}(R_{ℚ_{p}}(z^{m})). He has also defined a canonical basis y_{m} of H_{c}^{1}(R_{ℚ_{p}}(z^{m})). We hence define the extension

y\underline{m} := \frac{1}{e_{L}} log_{p}(χ_{cycl}(γ_{L}))cl(0, t^{−\underline{m}})e_{δ}.

When L = ℚ_{p}, this is the same element y_{m} as defined by Benois. We can calculate cohomology of induced (φ, Γ_{ℚ_{p}})-module. Indeed, we now consider two p-adic fields K and L, L a finite extension of K. The main reference for this part is [Liu08, §2.2]. Let D be a (φ, Γ_{L})-module, we define

Ind_{Γ_{L}}^{Γ_{K}} (D) = {f : Γ_{K} → D|f(hg) = hf(g) ∀h ∈ Γ_{L}} .

It has rank [L : K]rk_{R_{L}}(D) over R_{K}; indeed R_{L} is a R_{K}-module of rank [L : K]/|Γ_{K}/Γ_{L}| = [L′_{0} : K′_{0}]. (The unramified part of L/K plus the ramified part which is disjoint by K_{∞}. See after [Liu08, Theorem 2.2].) If D comes from a G_{L}-representation V we have

D_{rig}^{†}(Ind_{G_{L}}^{G_{K}} (V)) = Ind_{Γ_{L}}^{Γ_{K}} (D_{rig}^{†}(V)).

We have then the equivalent of Shapiro’s lemma

H^{i}(D) ≅ H^{i}(Ind_{Γ_{L}}^{Γ_{K}} (D)).

Moreover, the aforementioned duality for (φ, Γ)-modules is compatible with induction [Liu08, Theorem 2.2]. If D ≅ R_{L}(δ) is free of rank one, then we have an explicit description of Ind_{Γ_{L}}^{Γ_{K}} (D). Let e_{∞} = |Γ_{K}/Γ_{L}|, we write {ω^{i}}_{i=0}^{e_{∞}−1} for (Γ_{K}/Γ_{L})^{∧}. The Ind_{Γ_{L}}^{Γ_{K}} (D) is the R_{L}-span of f_{i}, where f_{i}(g) = ω^{i}(g)δ(χ_{cycl}(g))e_{δ}.

We go back to the previous setting where K = ℚ_{p} (hence e_{∞} = e_{L}). Suppose δ(z) = ∏_{τ} τ(z)^{m_{τ}} with m_{τ} ≤ 0 and let D = Ind_{Γ_{L}}^{Γ_{ℚp}} (R_{L}(δ)). Note that in this case D_{st}(D) ≅ E^{f_{L}} is a filtered φ-module where φ acts as a permutation of length f_{L}. To D_{st}(D)^{φ=1} corresponds (by Proposition 2.4 over ℚ_{p}) a rank-one (φ, Γ)-module R_{ℚ_{p}}(z^{m_{0}}), for m_{0} the minimum of the m_{τ} ’s (hence −m_{0} is the greatest Hodge–Tate weight of D). The identifications

H^{0}(R_{ℚ_{p}}(z^{m_{0}})) = D_{st}(R_{ℚ_{p}}(z^{m_{0}}))^{φ=1} = D_{st}(D)^{φ=1} = H^{0}(D) = H^{0}(R_{L}(δ)) induces (via the maps cl(0, ) and cl( , 0)) an injection

H^{1}(R_{ℚ_{p}}(z^{m_{0}})) ↪ H^{1}(Ind_{L}^{ℚ_{p}}(R_{L}(δ))).  (2.8)

which sends x_{m_{0}} to x\underline{m} and y_{m_{0}} to y\underline{m}.

We consider a (φ, Γ)-module M which sits in the non-split exact sequence

0 → M_{0} := ⊕_{i=1}^{r}R_{L}(δ_{i}) → M → M_{1} := ⊕_{i=1}^{r}R_{L}(δ′_{i}) → 0,  (2.9)

where δ_{i}(z) = |N_{L/ℚ_{p}}(z)|_{p} ∏_{τ} τ(z)^{m_{i,τ}} with m_{i,τ} ≥ 1 for all τ and δ′_{i}(z) = ∏_{τ} τ(z)^{k_{i,τ}} with k_{i,τ} ≤ 0 for all τ. We say that M is of type U_{m,k} if the image of M in H^{1}(M_{1}) is crystalline.

Proposition 2.10. Suppose that M as above is not of type U_{m,k}. Then we have dim_{E}(H^{1}(M)) = 2[L : ℚ_{p}]r and H^{2}(M) = H^{0}(M) = 0. Moreover, if we write

0 → H^{0}(M_{1}) →^{Δ_{0}} H^{1}(M_{0}) →^{f_{1}} H^{1}(M) →^{g_{1}} H^{1}(M_{1}) →^{Δ_{1}} H^{2}(M_{0}) → 0

we have H^{1}(M_{0}) = Im(Δ_{1}) ⊕ H_{f}^{1}(M_{0}), Im(f_{1}) = H_{f}^{1}(M) and H^{1}(M_{1}) = Im(g_{1}) ⊕ H_{f}^{1}(M_{1}).

Proof. We have H^{0}(M) = 0 by definition of M. Note that M^{∗}(χ_{cycl}) is a module of the same type, hence H^{2}(M) = H^{0}(M^{∗}(χ_{cycl})) = 0. We can write

0 → H^{0}(M_{1}) → H^{1}(M_{0}) →^{f_{1}} H^{1}(M) →^{g_{1}} H^{1}(M_{1}) → H^{2}(M_{0}) → 0

and conclude by Proposition 2.1. Note that dim_{E}H_{f}^{1}(M) = rd by (2.2). By hypothesis, we have that Im(Δ_{1}) ∩ H_{f}^{1}(M_{0}) = 0 and the first statement follows from dimension counting. The third statement follows from duality. For the second statement H_{f}^{1}(M_{0}) injects into H_{f}^{1}(M). As both have the same dimension, we conclude.

We give the following key lemma for the definition of the L-invariant

Lemma 2.11. The intersection of T := Im(H^{1}(M)) and Im(H^{1}(R_{ℚ_{p}}(z^{m_{0}}))) in Im(H^{1}(M_{1})) is one dimensional.

Proof. The intersection is non-empty as the sum of their dimension is d+2 and Im(H^{1}(M_{1})) has dimension d + 1. We have that H_{f}^{1}(M_{1}) is contained in the image of H^{1}(R_{ℚ_{p}}(z^{m_{0}})) via (2.8) and by the previous proposition the former is not in the image of g_{1} and we are done.

In particular, we deduce that T surjects into the image of H_{c}^{1}(R_{ℚ_{p}}(z^{m_{0}})).

3 L-invariant over number fields

Let F be a number field. We consider a global Galois representation

V : G_{F} → GL_{n}(E)

where E is p-adic field. We suppose that it is unramified outside a finite number of places S containing all the p-adic places. We suppose moreover that it is semistable at all places above p (i.e. D_{st}(V_{|_{F𝔭}}) is of rank n over F_{𝔭}^{ur} ⊗_{ℚ_{p}} E, being F_{𝔭}^{ur} the maximal unramified extension of ℚ_{p} contained in F_{𝔭}^{ur}). In this section we generalize Greenberg–Benois definition of the L-invariant for V whenever it presents trivial zeros. Note that we do not require p split or unramified in F. Let t be the number of trivial zeros. The classical definition by Greenberg [Gre94] describes the L-invariant as the “slope” of a certain t-dimension subspace of H^{1}(G_{ℚ_{p}}, ℚ_{p}^{t}) which is a 2t-dimensional space with a canonical basis given by ord_{p} and log_{p}. In our setting, the main obstacle is that the cohomology of the (φ, Γ)-module R_{F_{𝔭}} is no longer two-dimensional and it is not immediate to find a suitable subspace. Inspired by Hida’s work for symmetric powers of Hilbert forms [Hid07], we consider the image of H^{1}(R_{ℚ_{p}}) inside H^{1}(R_{F_{𝔭}}). If t denotes the number of expected trivial zeros, we show that we can define, similarly to [Ben11], a t-dimensional subspace of H^{1}(G_{F,S}, V) whose image in H^{1}(R_{ℚ_{p}}) has trivial intersection with the crystalline cocycle. This is enough to define the L-invariant; we further check that our definition is compatible with Benois’.

3.1 Definition of the L-invariant

We define local cohomological conditions L_{v} in order to define a Selmer group; we denote by G_{v} a fixed decomposition group at v in G_{F,S} and by I_{v} the inertia. For v ∤ p we define

L_{v} := Ker (H^{1}(G_{v}, V) → H^{1}(I_{v}, V)) .

If v | p we define

L_{v} := H_{f}^{1}(F_{v}, V) = Ker(H^{1}(G_{v}, V) → H^{1}(G_{v}, V ⊗_{E} B_{cris})).

If D_{rig}^{†}(V) denotes the (φ, Γ)-module associated with V we also have L_{p} = H_{f}^{1}(D_{rig}^{†}(V)). We define then the Bloch-Kato Selmer group

H_{f}^{1}(V) := Ker (H^{1}(G_{F,S}, V) → ∏_{v∈S} \frac{H^{1}(G_{v}, V)}{L_{v}}) .

We make the following additional hypotheses:

C1) H_{f}^{1}(V) = H_{f}^{1}(V^{∗}(1)) = 0,

C2) H^{0}(G_{F,S}, V) = H^{0}(G_{F,S}, V^{∗}(1)) = 0,

C3) φ on D_{st}(V_{|_{F𝔭}}) is semisimple at 1 ∈ F_{𝔭}^{ur} ⊗_{ℚ_{p}} E and p^{−1} ∈ F_{𝔭}^{ur} ⊗_{ℚ_{p}} E for all 𝔭 | p,

C4) D_{rig}^{†}(V_{|_{F𝔭}}) has no saturated sub-quotient of type U_{m,k} for all 𝔭 | p.

Note that if V satisfies the previous four conditions, so does V^{∗}(1). The first two conditions tell us that the Poitou–Tate sequence reduces to

H^{1}(G_{F,S}, V) ≅ ⨁_{v∈S} \frac{H^{1}(G_{v}, V)}{H_{f}^{1}(F_{v}, V)}.  (3.1)

For each 𝔭 | p we denote by V_{𝔭} the restriction to G_{F_{𝔭}} of V. We choose a regular sub-module D_{𝔭} ⊂ D_{st}(V_{𝔭}) and define a filtration (D_{𝔭,i}) of D_{st}(V_{𝔭}).

D_{𝔭,i} = \{ \begin{matrix}0 & i = −2, \\ (1 − p^{−1}φ)D_{𝔭} + N(D_{𝔭}^{φ=1}) & i = −1, \\ D_{𝔭} & i = 0, \\ D_{𝔭} + D_{st}(V_{𝔭})^{φ=1} ∩ N^{−1}(D_{𝔭}^{φ=p^{−1}}) & i = 1, \\ D_{st}(V_{𝔭}) & i = 2. \end{matrix}  (3.2)

We have that D_{𝔭,1}/D_{𝔭,−1} coincides with the eigenvectors of φ on D_{st}(V_{𝔭}) of eigenvalue 1 (resp. p^{−1}) and which are in the kernel (resp. in the image) of N. This filtration induces a filtration on D_{rig}^{†}(V_{𝔭}). Namely, we pose

F_{i}D_{rig}^{†}(V_{𝔭}) = D_{rig}^{†}(V_{𝔭}) ∩ (D_{𝔭,i} ⊗ R_{F_{𝔭,log}}[t^{−1}]).

We define

W_{𝔭} := F_{1}D_{rig}^{†}(V_{𝔭})/F_{−1}D_{rig}^{†}(V_{𝔭}).

The same proof as [Ben11, Proposition 2.1.7] tells us that we can find a unique decomposition

W_{𝔭} = W_{𝔭,0} ⨁ W_{𝔭,1} ⨁ M_{𝔭}

such that t_{𝔭,0} = dim_{E}H^{0}(W_{𝔭}^{∗}(1)) = rank_{R_{F𝔭}} W_{𝔭,0}, t_{𝔭,1} = dim_{E}H^{0}(W_{𝔭}) = rank_{R_{F𝔭}} W_{𝔭,1} and M_{𝔭} sits in a sequence

0 → M_{𝔭,0} →^{f} M_{𝔭} →^{g} M_{𝔭,1} → 0

such that gr^{0}(D_{rig}^{†}(V_{𝔭})) = W_{𝔭,0}⊕M_{𝔭,0} and gr^{1}(D_{rig}^{†}(V_{𝔭})) = W_{𝔭,1}⊕M_{𝔭,1}. Moreover M_{𝔭} is non-split; by construction we have H^{0}(M_{𝔭}) = H^{2}(M_{𝔭}) = 0 and if the exact sequence were split we would have H^{0}(M_{𝔭}) ≠ 0 and H^{2}(M_{𝔭}) ≠ 0.

We can prove exactly in the same way as [Ben11, Proposition 2.1.7 (i)] that C4 implies rank_{R_{F𝔭}} M_{𝔭,1} = rank_{R_{F𝔭}} M_{𝔭,0}. In order to define the L-invariant we shall follow verbatim Benois’ construction. For sake of notation, we write D_{𝔭}^{†} for D_{rig}^{†}(V_{𝔭}). We obtain from [Ben11, Proposition 1.4.4 (i)]

H_{f}^{1}(gr^{2}(D_{𝔭}^{†})) = H^{0}(gr^{2}(D_{𝔭}^{†})) = 0.

We deduce the following isomorphism

H_{f}^{1}(F_{1}D_{𝔭}^{†}) = H_{f}^{1}(D_{𝔭}^{†}) = H_{f}^{1}(F_{𝔭}, V).  (3.3)

As the Hodge–Tate weights of F_{−1}D_{𝔭}^{†} are < 0, we obtain from [Ben11, Proposition 1.5.3 (i)] and Poiteau–Tate duality H^{2}(F_{−1}D_{𝔭}^{†}) = 0. Using the long exact sequence associated with

0 → F_{−1}D_{𝔭}^{†} → F_{1}D_{𝔭}^{†} → W_{𝔭} → 0

we see that

\frac{H^{1}(W_{𝔭})}{H_{f}^{1}(W_{𝔭})} = \frac{H^{1}(F_{−1}D_{𝔭}^{†})}{H_{f}^{1}(F_{𝔭}, V)}.

As Greenberg and Benois do, we make the extra assumption that

C5) W_{𝔭,0} = 0 for all 𝔭 | p.

Using Proposition 2.4 we can write gr^{1}(D_{𝔭}^{†}) = ⊕_{i=1}^{t_{𝔭,1}+r_{𝔭}}R_{F_{𝔭}}(∏_{τ_{𝔭}} τ_{𝔭}(z)^{m_{i,τ𝔭}}). We define the 2(t_{𝔭,1} + r_{𝔭})-dimensional subspace obtained as the image of

Ind_{𝔭} := \{ ∑_{i=0}^{t_{𝔭,1}+r_{𝔭}} Ex\underline{m_{i}} + Ey\underline{m_{i}}\} ⊂ H^{1}(gr^{1}(D_{𝔭}^{†})).  (3.4)

We define

T_{𝔭} = (H^{1}(F_{1}D_{𝔭}^{†}) ∩ Ind_{𝔭})/H_{f}^{1}(F_{𝔭}, V).

It has dimension t_{𝔭,1} + r_{𝔭}. Write t = ∑_{𝔭} t_{𝔭,1} + r_{𝔭}. We have a unique t-dimensional subspace H^{1}(D, V) of H^{1}(G_{F,S}, V) projecting via (3.1) to ⊕_{𝔭}T_{𝔭}. We have an isomorphism (cfr. [Ben11, Proposition 1.5.9])

Ind_{𝔭} ≅ D_{cris}(W_{𝔭,1} ⊕ M_{𝔭,1}) ⊕ D_{cris}(W_{𝔭,1} ⊕ M_{𝔭,1}) ≅ E^{t_{𝔭,1}+r_{𝔭}} ⊕ E^{t_{𝔭,1}+r_{𝔭}},

where the first (resp. the second) factor is identified with E^{t_{𝔭,1}+r_{𝔭}} via the basis \{x\underline{m_{i}}\} (resp. \{y\underline{m_{i}}\}). We shall denote the two projections by ι_{f,𝔭} and ι_{c,𝔭}.

We denote by ι_{f} (resp. ι_{c}) the projection of H^{1}(D, V) to E^{t} via ⊕ι_{f,𝔭} (resp. ⊕ι_{c,𝔭}). By the remark after Lemma 2.11 and the definition of T_{𝔭}, we have that ι_{c} is surjective. Summing up, we can give the following definition; Definition 3.5. The L-invariant of the pair (V, D) is

L(V, D) := det(ι_{f} ∘ ι_{c}^{−1}),

where the determinant is calculated w.r.t. the basis (x\underline{m_{i}}, y\underline{m_{j}})_{1≤i,j≤t}.

Remark 3.6. There is no a priori reason for which L(V, D) should be non-zero.

In the case W_{𝔭} = M_{𝔭} we see from the description of H^{1}(F_{1}D_{𝔭}^{†}) that the space T_{𝔭} depends only on V_{|_{F𝔭}} exactly as in the classical case.

3.2 Comparison with Benois’ definition

Fix a global field F and let {𝔭} be the set of primes above p. Let G_{p} denote a fixed decomposition group at p in G_{ℚ} and let 𝔭_{0} be the corresponding place of F. Let G_{𝔭_{0},F} be the decomposition group at 𝔭_{0} in G_{F}. For each other place 𝔭 above p in F, we have G_{𝔭} = τ_{𝔭}G_{p}τ_{𝔭}^{−1}. We shall denote by G_{𝔭,F} the corresponding decomposition group in G_{F}. Consider a p-adic Galois representation

V : G_{F} → GL_{n}(E).

We shall suppose E big enough to contain the Galois closure of F_{𝔭}, for all 𝔭. As before, we suppose V semistable at all primes above p. We have then

Ind_{F}^{ℚ}(V)_{|_{Gp}} ≅ ⨁_{𝔭} τ_{𝔭}^{−1}Ind_{G_{𝔭,F}}^{G_{𝔭}} V_{|_{G_{𝔭,F}}}

where τ_{𝔭} ∈ G_{p} \ Hom(F, \overline{ℚ}). Consider the (φ, Γ)-module

D^{†} := D_{rig}^{†} (Ind_{F}^{ℚ}V ) .

We let D be the regular (φ, N)-module of D_{st}(D^{†}) induced by {D_{𝔭}}_{𝔭}. As before we have a filtration (F_{i}D^{†}) on D^{†} induced by the filtration on D. We denote by W the quotient F_{1}D^{†}/F_{−1}D^{†}. Note that it is semistable. We write W = W_{0} ⊕ M ⊕ W_{1}. We suppose that V satisfies the hypotheses C1-C5 of the previous section.

Lemma 3.7. Let M be as in (2.9). We have

0 → Ind(M_{0}) → Ind(M) → Ind(M_{1}) → 0.

We can now compare our definition of L-invariant with Benois’.

Proposition 3.8. We have a commutative diagram

\xymatrix{H^{1}(G_{ℚ,S}, Ind(V)) \ar[d] & H^{1}(Ind(V), Ind(D)) \ar[l] \ar[r]^{^{Res^{p}}} \ar[d] & \frac{H^{1}(F_{1}D^{†}(Ind(V)))}{H_{f}^{1}(G_{p},Ind(V))} = \frac{H^{1}(F_{−1}D^{†})}{H_{f}^{1}(G_{p},Ind(V))}. \ar[d]^{ι_{p}} \\ H^{1}(G_{F,S}, V) & H^{1}(V, D) \ar[l] \ar[r]^{⊕_{𝔭}Res_{𝔭}} & ∏_{𝔭} T_{𝔭}}

whose vertical arrows are isomorphism.

Proof. We follow [Hid06, §3.4.4]. Recall that we wrote D_{𝔭}^{†} for D_{rig}^{†}(V_{𝔭}). Shapiro’s lemma tells us that

\frac{H^{1}(G_{p}, Ind_{F}^{ℚ}V)}{H_{f}^{1}(G_{p}, Ind_{F}^{ℚ}V)} ≅^{ι_{p}} ⨁_{𝔭} \frac{H^{1}(D_{𝔭}^{†})}{H_{f}^{1}(D_{𝔭}^{†})}.

We are left to show that H^{1}(F_{1}D^{†}(Ind(V))) is sent by ι_{p} into (H^{1}(F_{1}D_{𝔭}^{†})∩Inv_{𝔭}) and we shall conclude by dimension counting. We have then an injection

F_{1}D^{†}(Ind(V)) ↪ ⊕_{𝔭}Ind(F_{1}(D_{rig}^{†}(V_{𝔭}))).

Then clearly the image of ι_{p} lands in H^{1}(F_{1}D_{𝔭}^{†}). But we have also the injection

gr^{1}(D_{rig}^{†}(IndV)) ↪ ⊕_{𝔭}Ind(gr^{1}(D_{rig}^{†}(V_{𝔭})))

which by (2.8) tells us that the image of ι_{p} lands in Inv_{𝔭} and we are done.

Corollary 3.9. We have L(V, D) = L(Ind_{F}^{ℚ}(V), Ind_{F}^{ℚ}(D)).

4 Siegel–Hilbert modular forms, the local case

The calculation of the L-invariant requires to produce explicit cocycles in H^{1}(D, V); when V appears in Ad(V′) for a certain representation V′ we can sometimes use the method of Mazur and Tilouine [MT90] to produce these cocycles. This has been done in many case for the symmetric square [Hid04, Mok12] and generalized to symmetric powers of the Galois representation associated with Hilbert modular forms in [Hid07, HJ13]. The main limit of this approach is that for most representations V it is computationally heavy to obtain V as the quotient of an adjoint representation. In the case D_{rig}^{†}(V) = W = M the situation is way simpler; if t = 1 it has been proved in [Ben10] that to produce the cocycle in H^{1}(V, D) it is enough to find deformations of V |_{ℚ_{p}}. We shall generalized the method of Benois to our situation in the case W_{𝔭} = M_{𝔭} and r_{𝔭} = 1. This will allow us to give a complete formula for the L-invariant of the Galois representations associated with a Siegel–Hilbert modular form which is Steinberg at all primes above p.

4.1 The case t_{𝔭} = r_{𝔭} = 1

We now suppose that W_{𝔭} = M_{𝔭} and r_{𝔭} = 1. For sake of notation, in this section we shall drop the index _{𝔭}. In particular, in this subsection F = F_{𝔭}. All that we have to do is to check that the calculation of [Ben11, Theorem 2] works in our setting. We write as before

0 → M_{0} → M → M_{1} → 0

and, only in this subsection, we shall write δ for the character defining M_{0} and ψ for the character defining M_{1}. We suppose δ = δ′ ∘ N_{F/ℚ_{p}} for δ′(z) = |z|_{p}z^{k} with k ≥ 1 and ψ = ψ′ ∘ N_{F/ℚ_{p}} with ψ′(z) = z^{m} with m ≤ 0. We consider an infinitesimal deformation

0 → M_{0,A} → M_{A} → M_{1,A} → 0,

over A = E[T]/(T^{2}). We suppose that M_{0,A} (resp. M_{1,A}) is an infinitesimal deformation of M_{0} (resp. M_{1}) which still factors through N_{F/ℚ_{p}}. We shall write δ_{A}, δ′_{A}, ψ_{A} and ψ′_{A} for the corresponding one-dimensional character.

Theorem 4.1. Suppose that d log_{p}(δ′_{A}ψ′_{A}^{−1})(χ_{cycl}(γ_{ℚ_{p}}))) ≠ 0; then

L(M, M_{0}) = − log_{p}(χ_{cycl}(γ_{ℚ_{p}}))\frac{f^{−1}d log_{p}(δ_{A}ψ_{A}^{−1})(ϖ)}{d log_{p}(δ′_{A}ψ′_{A}^{−1})(χ_{cycl}(γ_{ℚ_{p}}))}.

Proof. Recall the definition of Ind in (3.4). We have a vector v = ax_{m} + by_{m} in H^{1}(F_{1}D^{†}) ∩ Ind. By definition L(M) = ab^{−1}. The extension M_{j,A} provides us with connecting morphisms B_{j}^{i} : H^{i}(M_{j}) → H^{i+1}(M_{j}). We have by definition

B_{1}^{0}(t^{−m}e_{m}) =cl(dlog(ψ′_{A})(p)t^{−m}e_{m}, dlog(ψ′_{A})(χ_{cycl}(γ_{ℚ_{p}}))t^{−m}e_{m})
=dlog(δ′_{A})(p)x_{m} + dlog(δ′_{A})(χ_{cycl}(γ_{ℚ_{p}}))y_{m}.  (4.2)

As in [Ben10, §3.2] we consider the dual extension

0 → M_{1}^{∗}(χ_{cycl}) → M^{∗}(χ_{cycl}) → M_{0}^{∗}(χ_{cycl}) → 0,

and we shall denote with a ^{∗} the corresponding map in the long exact sequence of cohomology. We have hence ker(Δ_{1})⊥Im(Δ_{0}^{∗}) under duality, and a map

H^{1}(M_{1}^{∗}) → H^{1}(R_{ℚ_{p}}(|z|z^{1−m})).

By duality again, we deduce that the image of Δ_{0}^{∗} inside the target of the above arrow is

aα_{1−m} + bβ_{1−m},

where α_{1−m} (resp. β_{1−m}) is the dual of x_{m} (resp. y_{m}) as in [Ben10, Proposition 1.1.5]. We now consider the map

B_{1}^{1}^{∗} : H^{1}(M_{1}^{∗}(χ_{cycl})) → H^{2}(M_{1}^{∗}(χ_{cycl})) = H^{2}(R_{ℚ_{p}}(|z|z^{m})) ≅ E.

We can use [Ben10, Proposition 2.4] to see that after the above identification of H^{2} with E we have

B_{1}^{1}^{∗}(α_{1−m}) = clog_{p}(χ_{cycl}(γ_{ℚ_{p}}))^{−1}d log_{p}(ψ′_{A}^{−1}(χ_{cycl}(γ_{ℚ_{p}})),  (4.3)
B_{1}^{1}^{∗}(β_{1−m}) = cd log_{p}(ψ′_{A}^{−1}(p)),  (4.4)

where c ∈ E^{×}. We consider the following anti-commutative diagram

\xymatrix{H^{0}(M_{0}^{∗}(χ_{cycl})) \ar[d]^{_{B_{0}^{1}_{∗}}} \ar[r]^{Δ_{0}^{∗}} & H^{1}(M_{1}^{∗}(χ_{cycl})) \ar[d]^{_{B_{1}^{1}_{∗}}} \\ H^{1}(M_{0}^{∗}(χ_{cycl})) \ar[r]^{Δ_{1}^{∗}} & H^{2}(M_{1}^{∗}(χ_{cycl}))}

which means

B_{1}^{1}^{∗}Δ_{0}^{∗} = −Δ_{1}^{∗}B_{0}^{1}^{∗}.

We calculate this identity on t^{1−k}e_{1−k}. Applying (4.3) and (4.4) to ψ′_{A}^{−1}χ_{cycl}, (4.2) to δ′_{A}^{−1}χ_{cycl} and using [Ben10, (3.6)] which says

Δ_{1}^{∗}B_{0}^{1}^{∗}(t^{1−k}) = c (bd log_{p}(δ′_{A})(p) + ad log_{p}(δ′_{A})(χ_{cycl}(γ_{ℚ_{p}})))

we get

b^{−1}a = − log_{p}(χ_{cycl}(γ_{ℚ_{p}}))\frac{d log_{p}(δ′_{A}ψ′_{A}^{−1})(p)}{d log_{p}(δ′_{A}ψ′_{A}^{−1})(χ_{cycl}(γ_{ℚ_{p}}))}.

We conclude as δ′_{A}(p)^{f} = δ_{A}(ϖ).

Remark 4.5. In particular, this theorem proves that this definition of Linvariant is compatible with the Fontaine-Mazur one [Pot14, Zha14].

4.2 Calculation of the L-invariant for Steinberg forms

We fix a totally real field F. Let I be the set of real embeddings. Fix two embeddings

ℂ_{p} ↩ \overline{ℚ} ↪ ℂ

as before. We partition I = ⊔_{𝔭}I_{𝔭} according to the p-adic place which each embedding induces. We shall denote by q_{𝔭} = p^{f_{𝔭}} the residual cardinality for each prime ideal 𝔭. We consider an irreducible representation π of GSp_{2g}_{/F}, algebraic of weight k = (k_{τ})_{τ}, where (k_{τ}) = (k_{τ,1}, …, k_{τ,g}; k_{0}) (k_{0} is a parallel weight for Res_{F}^{ℚ}(𝔾_{m})) with k_{τ,1} ≤ k_{τ,2} … ≤ k_{τ,g}. If k_{τ,1} ≥ g + 1 for all τ, then the weight is cohomological. The cohomological weight of π is then

(μ_{τ})_{τ} = (k_{τ})_{τ} − (g + 1, …, g + 1; 0)_{τ} .

For parallel weights k, we shall choose k_{0} = gk. We now describe the conjectural Galois representation associated with π. We have a spin Galois representation V_{spin} (whose image is contained in GL_{2^{g}}) and a standard Galois representation V_{sta} (whose image is contained in GL_{2g+1}) given respectively by the spinorial and the standard representation of GSpin_{2g+1} = ^{L}GSp_{2g}. Thanks to the work of Scholze [Sch15] we now dispose of the standard Galois representation (see for example [HJ13, Theorem 18]). We also know the existence of the spin representation in many cases [KS14]. We now recall some expected properties of these Galois representations. Our main reference is [HJ13, §3.3]. We will make the following assumption on π at p;

for each 𝔭 | p either π_{𝔭} is spherical or Steinberg.

We explain what we mean by Steinberg. Consider the Satake parameters at 𝔭, normalized as in [BS00, Corollary 3.2], (α_{𝔭,1}, …, α_{𝔭,g}). We have the following theorem on Iwahori-spherical representation of GSp_{2g}(F_{𝔭}) [Tad94, Theorem 7.9].

Theorem 4.6. Let α_{1}, …, α_{g}, α be g + 1 character of F_{𝔭}^{×}. Let B_{GSp_{2g}} be

the Borel subgroup of Sp_{2g}(F_{𝔭}). Then Ind_{B_{GSp_{2g}}}^{GSp_{2g}(F_{𝔭})}(α_{1} × ⋯ × α_{g} ⋊ α) is not irreducible if and only if one of the following conditions is satisfied:

i) There exist at least three indexes i such that α_{i} has exact order two and the α_{i}’s are mutually distinct;

ii) There exists i such that α_{i} = |N( )|_{𝔭}^{±1};

iii) There exist i and j such that α_{i} = |N( )|_{𝔭}^{±1}α_{j}^{±1}.

Remark 4.7. As shown in [HJ13, Lemma 19], such a points are contained in a proper subset of the Hecke eigenvariety for GSp_{2g}.

Definition 4.8. We say that π_{𝔭} is Steinberg if α_{i} = |N( )|_{𝔭}^{i−1}α_{1}.

If π_{𝔭} is Steinberg at p, then α_{𝔭,i}(ϖ_{𝔭}) = q_{𝔭}^{i}α_{𝔭,1}(ϖ_{𝔭}). Trivial zeros appear also for automorphic forms which are only partially Steinberg at 𝔭 and can be dealt exactly at the same way as the parallel one but for the sake of notation we prefer not to deal with them.

To each g + 1 non-zero elements (t_{1}, …, t_{g}; t_{0}) ∈ (A^{×})^{g+1} we associate the diagonal matrix

u(t_{1}, …, t_{g}; t_{0}) := (t_{1}, …, t_{g}, t_{0}t_{g}^{−1}, …, t_{0}t_{1}^{−1})

of GSp_{2g}(A). For 1 ≤ i ≤ g − 1 we denote by u_{𝔭,i} the diagonal matrix associated with (1, …, 1, ϖ_{𝔭}^{−1}, …, ϖ_{𝔭}^{−1}; ϖ_{𝔭}^{−2}), where ϖ_{𝔭} appears i times; we also denote by u_{𝔭,0} the diagonal matrix corresponding to (1, …, 1; ϖ_{𝔭}^{−1}).

Definition 4.9. The Hecke operators U_{𝔭,i}, for 1 ≤ i ≤ g are defined as the double coset operator [Iwu_{𝔭,g−i}Iw].

We have that U_{𝔭,g} is the “classical” U_{p} operator [BS00, §0]. We shall say then that π is of finite slope for U_{𝔭,g} if U_{𝔭,g} has eigenvalue α_{𝔭,0} ≠ 0 on π_{𝔭}.

We are interested to study the possible p-stabilization of π (i.e. Iwahori fixed vectors). If π_{𝔭} is unramified at 𝔭, we have then 2^{g}g! choices (see [HJ13, Lemma 16] or [BS00, Proposition 9.1]). If π_{𝔭} is Steinberg, we have instead only one possible choice, as the monodromy N has maximal rank.

Suppose that we can lift π to an automorphic representation π^{(2^{g})} of GL_{2^{g}}. We suppose also that we can lift π to an automorphic representation π^{(2g+1)} of GL_{2g+1}. Let V = V_{spin} (resp. V_{sta}) be the Galois representation associated with π^{(2^{g})} (resp. π^{(2g+1)}). We make the following assumption

LGp) V is semistable at all 𝔭 | p and strong local-global compatibility at l = p holds.

These hypotheses are conjectured to be always true for f as above. Arthur’s transfer from GSp_{2g} to GL_{2g+1} has been proven in [Xu] (note that it is now unconditional [MW]) and for V = V_{sta} this hypothesis is then verified thanks to [Car14, Theorem 1.1]. These hypotheses are also satisfied in many cases for V = V_{spin} in genus 2 (see [AS06, PSS14]). Roughly speaking, we require that

WD(V_{|F_{𝔭}})^{ss} ≅ ι_{n}^{−1}π_{𝔭}^{(n)},

where WD(V_{|F_{𝔭}}) is the Weil-Deligne representation associated with V_{|F_{𝔭}} à la Berger, π_{𝔭}^{(n)} is the component at 𝔭 of π^{(n)}, and ι_{n} is the local Langlands correspondence for GL_{n}(F_{𝔭}) geometrically normalized (n = 2g + 1 when V is the standard representation and n = 2^{g} when V is the spinorial representation). When π_{𝔭} is an irreducible quotient of Ind_{B}^{GSp_{2g}}(α_{𝔭,1} ⊗ ⋯ ⊗ α_{𝔭,g}) we have that the Frobenius eigenvalues on WD(V_{spin}_{|F_{𝔭}})^{ss} are the 2^{g} numbers

(α_{𝔭,0} ∏ ; \begin{matrix}0 ≤ r ≤ g \\ 1 ≤ i_{1} < … < i_{r} ≤ g \end{matrix} α_{𝔭,i_{1}}(ϖ_{𝔭}) ⋯ α_{𝔭,i_{r}}(ϖ_{𝔭})) .

The ones on WD(V_{sta}_{|F_{𝔭}})^{ss} are

(α_{𝔭,g}^{−1}(ϖ_{𝔭}), …, α_{𝔭,1}^{−1}(ϖ_{𝔭}), 1, α_{𝔭,1}(ϖ_{𝔭}), …, α_{𝔭,g}(ϖ_{𝔭})) .

Moreover, the monodromy operator should have maximal rank (i.e. onedimensional kernel) if we are Steinberg or be trivial otherwise. (This is also a consequence of the weight-monodromy conjecture for V.) Let 𝔭 be a p-adic place of F and let τ be a complex place in I_{𝔭}. The Hodge–Tate weights of V_{spin}_{|_{F𝔭}} at τ are then

(\frac{k_{0}}{2} + \frac{1}{2} ∑_{i=1}^{g} ε(i)(k_{i,τ} − i))_{ε},

where ε ranges among the 2^{g} maps from {1, …, g} to {±1}. The one of V_{sta}_{|_{F𝔭}} are (1 − k_{τ,g}, …, g − k_{τ,1}, 0, k_{τ,1} − g, …, k_{τ,g} − 1). Thanks to work of Tilouine-Urban [TU99], Urban [Urb11], Andreatta-Iovita-Pilloni [AIP15] we have families of Siegel modular forms;

Theorem 4.10. Let W = Hom_{cont} (ℤ_{p}^{×} × ((O_{F} ⊗_{ℤ} ℤ_{p})^{×})^{g}, ℂ_{p}^{×}) be the weight space. There exist an affinoid neighborhood U of κ_{0} = ((z, (z_{i})_{i=1}^{g}) ↦ z^{k_{0}} ∏_{τ∈I} ∏_{i} τ(z_{i})^{k_{i,τ}} ) in W, an equidimensional rigid variety X = X_{π} of dimension dg + 1, a finite surjective map w : X → U, a character Θ : H^{Np} → O(X), and a point x in X above \underline{k} such that x ∘ Θ corresponds to the Hecke eigensystem of π. Moreover, there exists a dense set of points x of X coming from classical cuspidal Siegel–Hilbert automorphic forms of weight (k_{i,τ}; k_{0}) which are regular and spherical at p.

Remark 4.11. Assuming Leopoldt’s conjecture, the multiplicative group appearing in the definition of W is, up to a finite subgroup, ((O_{F} ⊗_{ℤ} ℤ_{p})^{×})^{g+1}/\overline{O_{F}^{×}} (i.e. the ℤ_{p}-points of the torus of Res_{F}^{ℚ}(GSp_{2g}) modulo the ℤ_{p}-points of the center).

This allows us to define two pseudo-representations R_{?} : G_{ℚ} → O(X), for ? = spin, sta, interpolating the trace of the representations associated with classical Siegel forms [BC09, Proposition 7.5.4]. Suppose now that V_{?} is absolutely irreducible (this is conjectured to hold when π is Steinberg at least at one prime); we have then, shrinking U around \underline{k} if necessary, a big Galois representation ρ_{?} with value in GL_{n}(O(X)) such that Tr(ρ_{?}) = R_{?} [BC09, page 214].

For 1 ≤ j < g we define λ_{𝔭}(u_{𝔭,g−j}) = ϖ_{𝔭}^{∑_{τ∈I𝔭} k_{τ,1}+···+k_{τ,j}−k_{0}} and

λ_{𝔭}(u_{𝔭,0}) = ϖ_{𝔭}^{∑_{τ∈I𝔭} (k_{τ,1}+···+k_{τ,g}−k_{0})/2}. We have analytic functions β_{𝔭,j} := Θ(U_{𝔭,j}|λ_{𝔭}(u_{𝔭,g−j})|_{p}) ∈ O(X). We proceed now as in [HJ13]. We recall the following theorem [Liu13, Theorem 0.3.4];

Theorem 4.12. Let ρ : G_{F_{𝔭}} → GL_{n}(O(X)) be a continuous representation. Suppose that there exist κ_{1}(x), …, κ_{n}(x) in F_{𝔭} ⊗_{ℚ_{p}} O(X), F_{1}(x), …, F_{d}(x) in O(X), and a Zariski dense set of points Z ⊂ X such that

• for any x in X, the Hodge–Tate weights of ρ_{x} are κ_{1}(x), …, κ_{n}(x);

• for any z in Z, ρ_{z} is crystalline;

• for any z in Z, κ_{τ,1}(z) < … < κ_{τ,n}(z), for all τ ∈ I_{𝔭};

• for any z in Z, the eigenvalues of φ^{f_{𝔭}} on D_{cris}(V_{z}) are ∏_{τ∈I_{𝔭}} τ(ϖ_{𝔭})^{κ_{τ,1}(z)}F_{1}(z), …, ∏_{τ∈I_{𝔭}} τ(ϖ_{𝔭})^{κ_{τ,n}(z)}F_{n}(z);

• for any C in ℝ, defines Z_{C} ⊂ Z as the set of points z such that for all I, J ⊂ {1, …, n} such that | ∑_{i∈I} κ_{i,τ} (z) − ∑_{j∈J} κ_{τ,j}(z)| > C for all τ ∈ I_{𝔭}. We require that for all z ∈ Z and C ∈ ℝ, Z_{C} accumulates at z.

• for 1 ≤ i ≤ n there exist character χ_{i} : O_{F_{𝔭}}^{×} → O(X)^{×} such that its derivative at 1 is κ_{i} and at each z ∈ Z we have χ_{i}(u) = ∏_{τ} τ(u)^{κ_{τ,i}(z)}.

Then, for all x in X non-critical and regular (κ_{1}(x) < … < κ_{n}(x) and the eigenvalues of φ on ⋀^{i} D_{cris}(V_{x}) are distinct for all i) there exists a Zariski neighbourhood U of x such that ρ_{U} is trianguline and its graded pieces are R_{U}(χ_{i}).

Here the rank one (φ, Γ)-module R_{U}(χ_{i}) over U is defined similarly as in Section 2 following [Liu13, §0.2]. We can apply this theorem and show that the (φ, Γ)-module associated with ρ_{?}_{|G_{ℚp}} is trianguline. We now explicit the triangulation, given in [HJ13, §3.3].

As seen before, a p-stabilization of π_{𝔭} corresponds to a permutation ν and a map ε. The eigenvalues of φ^{f_{𝔭}} are given by

∏_{τ∈I_{𝔭}} τ(ϖ_{𝔭})^{c_{1}+μ_{1,τ}} β_{𝔭,1},

∏_{τ∈I_{𝔭}} τ(ϖ_{𝔭})^{c_{i}+μ_{i,τ}} \frac{β_{𝔭,i−1}}{β_{𝔭,i}} ,

∏_{τ∈I_{𝔭}} τ(ϖ_{𝔭})^{c_{g}+μ_{g,τ}} \frac{β_{𝔭,g−1}}{β_{𝔭,g}^{2}}

where c_{i}’s are a positive integer independent of the weight. We define the following characters of F_{𝔭} with value in O(X):

χ_{𝔭,1}(ϖ_{𝔭}) = β_{𝔭,1},
χ_{𝔭,i}(ϖ_{𝔭}) = \frac{β_{𝔭,i−1}}{β_{𝔭,i}} ,
χ_{𝔭,g}(ϖ_{𝔭}) = \frac{β_{𝔭,g−1}}{β_{𝔭,g}^{2}} ,

and χ_{𝔭,i}(u) = ∏_{τ∈I_{𝔭}} τ(u)^{c_{i}+μ_{i,τ}}.

From [HJ13, Lemma 19] we have that the graded pieces of D_{rig}^{†}(V_{sta}_{|_{𝔭}}) are then given by the characters χ_{𝔭,g}, …, 1, …, χ_{𝔭,g}^{−1}.

Concerning V_{spin}, we number the subsets of {1, …, g} as I_{1}, I_{2}, …, I_{2^{g}}. Each I_{j} corresponds to a map ε_{j} : {1, …, g} → ±1. We have then the graded pieces δ_{𝔭,j} are given by the characters

δ_{𝔭,ε_{j}}(u) = ∏_{τ∈I_{𝔭}} τ(u)^{d_{j}+^{\frac{k_{0}+∑_{i} ε_{j} (i)k_{i,τ}}{2}}} ,
δ_{𝔭,ε_{j}}(ϖ_{𝔭}) =β_{𝔭,g} ∏_{i∈I_{j}} χ_{𝔭,i}(ϖ_{𝔭}).

Let V be either V_{Sta} or V_{spin}. If π_{𝔭} is Steinberg, there is only one choice of a regular (φ, N)-sub-module D_{𝔭} of D_{st}(V_{G_{F𝔭}}), where V is one of the two representations associated with π described above. If the form is not Steinberg at 𝔭 many different regular sub-module can be chosen. In any case, we expect (and we shall assume in the follow) that there is at most one trivial zero for each 𝔭. Consider now the representation π of parallel weight \underline{k} (i.e. associated with N_{F/ℚ}(det\underline{k}), \underline{k} ∈ ℤ) as in the introduction. We give a preliminary proposition on the factorization of the L-invariant. Recall the set S^{Sph,1} and S^{Stb} defined in the introduction, we have the following;

Proposition 4.13. We have the following factorization

L(V, D) = L(V, D)^{Sph} ∏_{𝔭∈S^{Stb}} L(V, D)_{𝔭},

where L(V, D)^{Sph} comes from the prime in S^{sph} and the factors L(V, D)_{𝔭} are local.

Proof. We follow [Hid07, §1.3]. In the notation of Section 3, we write W_{1} = ⊕_{𝔭∈S^{Stb}}W_{𝔭,1} and M_{1} = ⊕_{𝔭∈S^{Sph,1}}M_{𝔭,1}. We are left to show that the endomorphism ι_{f} ∘ ι_{c}^{−1} of D_{cris}(W_{1} ⊕ M_{1}) ≅ E^{t} keeps stable D_{cris}(M_{1}) and on the quotient it respects the direct sum decomposition ⊕_{𝔭∈S^{Stb}}D_{cris}(W_{𝔭,1}).

Consider a prime 𝔭_{0} ∈ S^{Stb} and a cocycle c ∈ H^{1}(D, V) such that res_{𝔭}(c) = 0 for all 𝔭 ≠ 𝔭_{0}. This means that res_{𝔭}(c) ∈ H_{f}^{1}(F_{𝔭}, V) = H_{f}^{1}(F_{𝔭}, M_{𝔭}) (by (3.3)). We have hence ι_{c,𝔭}(c) = 0 for all primes 𝔭 ≠ 𝔭_{0} as H_{c}^{1} is the direct sum complement of H_{f}^{1} (see [Ben11, Proposition 1.5.9]). If 𝔭 in S^{Stb} by Proposition 2.10 we also have ι_{f,𝔭}(c) = 0. The proposition then follows from standard linear algebra as in [Hid07, Corollary 1.9].

Remark 4.14. A key ingredient in the proof of the factorization at Steinberg places is that each prime ideal brings a single trivial zero.

We now consider the case V = V_{sta}. We have a contribution to trivial zeros from the π_{𝔭}’s which are Steinberg and possibly from the π_{𝔭} which are spherical. In particular, if we choose the regular sub-module coming from an ordinary filtration, we always have a trivial zero coming from each place. For all 1 ≤ s ≤ min(k − g − 1, g − 1) we have also e_{Stb} trivial zeros for V (s).

Theorem 4.15. For π_{𝔭} Steinberg we have

L(V, D)_{𝔭} = −\frac{1}{f_{𝔭}} \frac{d log_{p} β_{𝔭,1}(k)}{dk}_{|_{k=\underline{k}}},

where k is the parallel weight variable. For 1 ≤ s ≤ min(k − g − 1, g − 2) we also have

L(V (s), D(s))_{𝔭} = −\frac{1}{f_{𝔭}} \frac{d log_{p}(β_{𝔭,s−1}β_{𝔭,s}^{−1}(k))}{dk}_{|_{k=\underline{k}}}

and if g − 1 ≤ k − g − 1 we have

L(V (g − 1), D(g − 1))_{𝔭} = −\frac{1}{f_{𝔭}} \frac{d log_{p}(β_{𝔭,g−1}β_{𝔭,g}^{−2}(k))}{dk}_{|_{k=\underline{k}}}.

Proof. We note that we can specialize to a parallel family, so that no contribution from the denominator appears. We can apply Theorem 4.1 for δ_{A}ψ_{A}^{−1}(ϖ_{𝔭}) = χ_{𝔭,i}(ϖ_{𝔭}). The factor log_{p}(u) disappears because of the change of variable T ↦ u^{k} − 1 (u any topological generator of ℤ_{p}^{×}).

Remark 4.16. The presence of f_{𝔭} in the denominator can be explained in terms of the p-adic L-function for the induced representation, its missing Euler factors at p and Conjecture 1.2. See [Hid09, pag. 1348].

From now on, V = V_{spin}(k − 1) (s = k − 1 is the only critical integer); if π_{𝔭} is spherical it should not give any trivial zeros (as the corresponding p-adic representation is conjectured to be crystalline and consequently the β_{i}’s are Weil numbers of non-zero weight). So we are left to see what happen at the primes Steinberg at 𝔭. Twisting by β_{𝔭,g} the triangulated (φ, Γ)-module of ρ_{spin} we are in the hypothesis of Theorem 4.1 and we have Theorem 4.17. For π_{𝔭} Steinberg we have

L(V, D)_{𝔭} = −\frac{1}{f_{𝔭}} \frac{d log_{p} β_{𝔭,1}(k)}{dk}_{|_{k=\underline{k}}},

where k is a parallel weight variable.

5 The case of the adjoint representation

We prove Theorem 1.4 of the introduction. We consider only the case g = 2. Fix an automorphic representation π of weight \underline{k} = (\underline{k}_{τ,1}, …, \underline{k}_{τ,g}; \underline{k}_{0})_{τ} and let V = V_{spin} be the spin representation associated with π. Let ρ = ρ_{spin} be the corresponding big Galois representation.

We specializes the eigenvariety X of Theorem 4.10 to the subspace of the weight space given by the equations k_{i,τ} = k_{i,τ′} if τ and τ′ induce the same p-adic place 𝔭 and k_{0} = \underline{k}_{0}. We shall denote the new variable by k_{𝔭,i} and this eigenvariety by X′. For simplicity, we rewrite the graded pieces of V as

δ_{𝔭,1}(ϖ_{𝔭}) = F_{𝔭,1}^{−1}(k), δ_{𝔭,1}(u) = N_{F_{𝔭}/ℚ_{p}}(u)^{\frac{k_{0}+k_{𝔭,1}+k_{𝔭,2}−3}{2}} ,
δ_{𝔭,2}(ϖ_{𝔭}) = F_{𝔭,2}^{−1}(k), δ_{𝔭,2}(u) = N_{F_{𝔭}/ℚ_{p}}(u)^{\frac{k_{0}+k_{𝔭,2}−k_{𝔭,1}+1}{2}} ,
δ_{𝔭,3}(ϖ_{𝔭}) = F_{𝔭,2}(k), δ_{𝔭,3}(u) = N_{F_{𝔭}/ℚ_{p}}(u)^{\frac{k_{0}−k_{𝔭,2}+k_{𝔭,1}−1}{2}} ,
δ_{𝔭,4}(ϖ_{𝔭}) = F_{𝔭,1}(k), δ_{𝔭,4}(u) = N_{F_{𝔭}/ℚ_{p}}(u)^{\frac{k_{0}−k_{𝔭,1}−k_{𝔭,2}+3}{2}}

where k = (k_{𝔭,1}, k_{𝔭_{2}}; k_{0})_{𝔭}. The representation space of Ad(V) is given by the matrices

𝔖𝔭_{4} = {X ∈ 𝔖𝔏_{4}|X^{t}J + JX = 0} .

The p-stabilization on V induces a natural p-stabilization and consequently a regular sub-module D_{Ad} on Ad(V_{spin}). We have

D_{Ad}_{−1} = {nilpotent X} ,
D_{Ad}_{0} = {unipotent X} .

The basis for the space D_{Ad}_{0}/D_{Ad}_{−1} is given by the two diagonal matrices d_{1} = [−1, 0, 0, 1] and d_{2} = [0, −1, 1, 0]. We shall denote by d_{𝔭,i} these matrices when seen as a vector for Ad(V_{𝔭}).

Proposition 5.1. Suppose that C1-C4 holds for V. Suppose that the classical E-point x in the eigenvariety X′ corresponding to π is étale above the weight space. Then, the space L(D_{Ad}, V) is generated by the image of (\frac{d log_{p} δ_{𝔭,i}}{dk_{𝔭′,j}} d_{𝔭,i})_{𝔭′,j=1,2}.

Proof. The proof is standard and goes back to [MT90], so we shall only sketch it. Let A = E[T]/(T^{2}). Consider an infinitesimal deformation of ρ given by

ρ_{A} = V ⊕ ρ′;

note that ρ′ can be written as the first order truncation of \frac{∂ρ}{∂v}, where v is any direction in the weight space. From ρ_{A} we can construct a cocyle c_{x,A} defined by

G_{F} ∋ σ ↦ ρ′(σ)V^{−1}(σ).

It is easy to check that this defines a cocycle with values in V ⊗V^{∗}. Moreover its image lands in Ad(V) ⊂ V ⊗V^{∗} as the determinant is fixed (by our choice of the Hodge–Tate weight on X′). Writing explicitly the matrix for the (φ, Γ)-module associated with ρ_{A} we obtain

( \frac{∂δ_{𝔭,1}}{∂v} ∗  ∗  ∗ ; \frac{∂δ_{𝔭,2}}{∂v} ∗  ∗ ; \frac{∂δ_{𝔭,3}}{∂v} ∗ ; \frac{∂δ𝔭,4}{∂v} )_{|_{k=\underline{k}}} ( \begin{matrix}δ_{𝔭,1}^{−1} & ∗ & ∗ & ∗ \\ & δ_{𝔭,2}^{−2} & ∗ & ∗ \\ & & δ_{𝔭,3}^{−1} & ∗ \\ & & & δ_{𝔭,4}^{−1} \end{matrix} )_{|_{k=\underline{k}}}

In particular, they are upper triangular and their projection via ι_{f} onto the vector d_{𝔭,1} is \frac{d log_{p} F_{𝔭,1}(k)}{dv}_{|_{k=\underline{k}}}. Similarly for d_{𝔭,2}.

We also have that the projection via ι_{c} onto d_{𝔭,1} is −\frac{∂(k_{𝔭,1}+k_{𝔭,2})/2}{∂v}_{|_{k=\underline{k}}}. By hypothesis, the projection to the weight space is étale at x and hence \{\frac{∂}{∂k_{𝔭,i}} \}_{𝔭,i=1,2} is a base of the tangent space at x in X′ and we are done.

We can now prove Theorem 1.4 which we recall;

Theorem 5.2. Let π be an automorphic form of weight k. Suppose that hypothesis LGp is verified for V_{spin} and the point corresponding to π in the eigenvariety X′ is étale over the weight space. We have then

L(Ad(V_{spin}), D_{Ad}) = ∏_{𝔭} \frac{2}{f_{𝔭}^{2}} det( \begin{matrix}\frac{∂ log_{p} F_{𝔭_{i},1}(k)}{∂k_{𝔭_{j} ,1}} & \frac{∂ log_{p} F_{𝔭_{i},2}(k)}{∂k_{𝔭_{j} ,1}} \\ \frac{∂ log_{p} F_{𝔭_{i},1}(k)}{∂k_{𝔭_{j} ,2}} & \frac{∂ log_{p} F_{𝔭_{i},2}(k)}{∂k_{𝔭_{j} ,2}} \end{matrix} )_{1≤i,j≤t}_{|_{k=\underline{k}}}.

Proof. By hypothesis we can use Proposition 5.1, so we just have to follow the proof of [Hid06, Theorem 3.73]. The matrix of ι_{f} is exactly what appears in the Theorem, while the matrix of ι_{c} can be directly calculated using the

formula \frac{d log_{p}(u^{±k_{𝔭,i}})}{dk_{𝔭′,j}} = ±δ_{𝔭,𝔭′}δ_{i,j} (where δ_{a,b} here is Kronecker delta) and gives a contribution of 2^{−1} for each prime ideal 𝔭.

References

[AIP15] Fabrizio Andreatta, Adrian Iovita, and Vincent Pilloni. p-adic families of Siegel modular cuspforms. Ann. of Math. (2), 181(2):623–697, 2015.

[AS06] Mahdi Asgari and Freydoon Shahidi. Generic transfer from GSp(4) to GL(4). Compos. Math., 142(3):541–550, 2006.

[BC09] Joël Bellaïche and Gaëtan Chenevier. Families of Galois representations and Selmer groups. Astérisque, (324):xii+314, 2009.

[Ben10] Denis Benois. Infinitesimal deformations and the ℓ-invariant. Doc. Math., (Extra volume: Andrei A. Suslin sixtieth birthday):5–31, 2010.

[Ben11] Denis Benois. A generalization of Greenberg’s L-invariant. Amer. J. Math., 133(6):1573–1632, 2011.

[BS00] S. Böcherer and C.-G. Schmidt. p-adic measures attached to Siegel modular forms. Ann. Inst. Fourier (Grenoble), 50(5):1375–1443, 2000.

[Car14] Ana Caraiani. Monodromy and local-global compatibility for l = p. Algebra Number Theory, 8(7):1597–1646, 2014.

[Fon90] Jean-Marc Fontaine. Représentations p-adiques des corps locaux. I. In The Grothendieck Festschrift, Vol. II, volume 87 of Progr. Math., pages 249–309. Birkhäuser Boston, Boston, MA, 1990.

[Gre94] Ralph Greenberg. Trivial zeros of p-adic L-functions. In p-adic monodromy and the Birch and Swinnerton-Dyer conjecture (Boston, MA, 1991), volume 165 of Contemp. Math., pages 149–174. Amer. Math. Soc., Providence, RI, 1994.

[GS93] Ralph Greenberg and Glenn Stevens. p-adic L-functions and p-adic periods of modular forms. Invent. Math., 111(2):407–447, 1993.

[Hid04] Haruzo Hida. Greenberg’s L-invariants of adjoint square Galois representations. Int. Math. Res. Not., (59):3177–3189, 2004.

[Hid06] Haruzo Hida. Hilbert modular forms and Iwasawa theory. Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, Oxford, 2006.

[Hid07] Haruzo Hida. On a generalization of the conjecture of Mazur-Tate-Teitelbaum. Int. Math. Res. Not. IMRN, (23):Art. ID rnm102, 49, 2007.

[Hid09] Haruzo Hida. L-invariants of Tate curves. Pure Appl. Math. Q., 5(4, Special Issue: In honor of John Tate. Part 1):1343–1384, 2009.

[HJ13] Robert Harron and Andrei Jorza. On symmetric power L-invariants of Iwahori level Hilbert modular forms. preprint available at http://arxiv.org/abs/1310.6244, 2013.

[KS14] Arno Kret and Sug Woo Shin. Galois representations for some automorphic representations of the general symplectic group over a totally real field. preprint, 2014.

[Liu08] Ruochuan Liu. Cohomology and duality for (ϕ, Γ)-modules over the Robba ring. Int. Math. Res. Not. IMRN, (3):Art. ID rnm150, 32, 2008.

[Liu13] Ruochuan Liu. Triangulation of refined families. preprint available at http://arxiv.org/abs/1202.2188, 2013.

[Mok12] Chung Pang Mok. L-invariant of the adjoint Galois representation of modular forms of finite slope. J. Lond. Math. Soc. (2), 86(2):626–640, 2012.

[MT90] B. Mazur and J. Tilouine. Représentations galoisiennes, différentielles de Kähler et “conjectures principales”. Inst. Hautes Études Sci. Publ. Math., (71):65–103, 1990.

[MTT86] B. Mazur, J. Tate, and J. Teitelbaum. On p-adic analogues of the conjectures of Birch and Swinnerton-Dyer. Invent. Math., 84(1):1– 48, 1986.

[MW] Colette Moeglin and Jean-Loup Walspurger. Stabilisation de la formule des traces tordue X: stabilisation spectrale. preprint. Available

at http://arxiv.org/abs/1412.2981.

[Nak09] Kentaro Nakamura. Classification of two-dimensional split trianguline representations of p-adic fields. Compos. Math., 145(4):865–914, 2009.

[Pot14] Jonathan Pottharst. The L-invariant, the dual L-invariant, and families. preprint available at http://vbrt.org/writings/Linv.pdf, 2014.

[PSS14] A Pitale, A. Saha, and R. Schmidt. Transfer of Siegel cusp forms of degree 2. Mem. Amer. Math. Soc., 232(1090):107, 2014.

[Ros13] Giovanni Rosso. Derivative at s = 1 of the p-adic L-function of the symmetric square of a Hilbert modular form. accepted Israel Journal of Mathematics, 2013.

[Ros15] Giovanni Rosso. Derivative of symmetric square p-adic L-functions via pull-back formula. In Arithmetic and Geometry, volume 420 of London Math. Soc. Lecture Note Ser., pages 373–400. Cambridge Univ. Press, Cambridge, 2015.

[Sch15] Peter Scholze. On torsion in the cohomology of locally symmetric varieties. Ann. of Math. (2), 182(3):945–1066, 2015.

[Tad94] Marko Tadić. Representations of p-adic symplectic groups. Compositio Math., 90(2):123–181, 1994.

[TU99] J. Tilouine and E. Urban. Several-variable p-adic families of Siegel-Hilbert cusp eigensystems and their Galois representations. Ann. Sci. École Norm. Sup. (4), 32(4):499–574, 1999.

[Urb11] Eric Urban. Eigenvarieties for reductive groups. Ann. of Math. (2), 174(3):1685–1784, 2011.

[Xu] Bin Xu. Endoscopic classification of representations of GSp(2n) and GSO(2n). PhD thesis, University of Toronto.

[Zha14] YuanCao Zhang. L-invariants and logarithm derivatives of eigenvalues of frobenius. Science China Mathematics, 57(8):1587–1604, 2014.

Giovanni Rosso DPMMS, Centre for Mathematical Sciences Wilberforce Road Cambridge CB3 0WB United Kingdom gr385@cam.ac.uk 1254

Documenta Mathematica 20 (2015)