THE TEITELBAUM CONJECTURE IN THE INDEFINITE SETTING MARCO ADAMO SEVESO
Abstract. Let f be a new cusp form on Γ0 (N ) of even weight k + 2 ≥ 2. Suppose that there is a prime p k N and that we may write N = pN + N − , where N − is the squarefree product of an even number of − primes. There is a Darmon style L-invariant LN (f ) attached to this factorization, which is the Orton L− invariant when N − = 1. We prove that LN (f ) does not depend on the chosen factorization of N and it is equal to the other known L-invariants. We also give a formula for the computation of the logarithmic p-adic Abel-Jacobi image of the Darmon cycles. This formula is crucial for the computations of the derivatives of the p-adic L-functions of the weight variable attached to a real quadratic field K/Q such that the primes dividing N + are split and the primes dividing pN − are inert. Classification: 11F67 (primary), 11F75, 11F80 (secondary). Keywords: L-invariants of modular forms, monodromy modules, Darmon cycles.
Contents 1. Introduction 2. Weights, distributions and families of distributions 3. Hecke operators in cohomology 3.1. Hecke operators 3.2. Lifting the operators to coefficients 4. Slope decompositions and families of modular forms on indefinite quaternion algebras 5. p-adic integration pairings 5.1. Review of the pairings Φlog and Φord from [RS] 5.2. The indefinite p-adic integration pairing 6. Proof of the main results 6.1. Proof of the key proposition 6.2. Proof of the main result 6.3. Application to the p-adic Abel-Jacobi map 7. Appendix: base point free Shapiro’s Lemma 7.1. Application References
1 4 7 7 9 10 15 15 16 17 17 19 20 21 22 22
1. Introduction Let Sk+2 (Γ0 (N )) be the C-vector space of even weight k + 2 ≥ 2 cusp forms on Γ0 (N ) and let p be a prime. Let f ∈ Sk+2 (Γ0 (N )) be a newform for the Hecke Q-algebra H0 (N ) acting on Sk+2 (Γ0 (N )) and let Kf be the number field generated by the eigenvalues of f , that we regard as a subfield of both C and Cp via σ ∞ : Q ,→ C, σ p : Q ,→ Cp . If χ is a Dirichlet character, let Kf (χ) be the field generated over Kf by the values of χ. Let L (f, χ, s) be the complex L-function attached to the twist of f by a Dirichlet character χ. By a theorem of Shimura ∞ there exist complex periods Ωw ∈ C× , with w∞ ∈ {±1}, such that f L∗ (f, χ, j + 1) :=
j!c (χ)
j+1
j
∞ (−2πi) τ (χ) Ωw f
Date: April 11, 2012. 1
L (f, χ, j + 1) ∈ Kf (χ) , 0 ≤ j ≤ k,
k−j
where the sign of the period is choosen so that χ (−1) = (−1) w∞ and c (χ) ∈ N is the conductor of χ. As χ × runs over all Dirichler characters of conductor c (χ) = pr M with r ∈ N, M | ϕ (N ) and ϕ (N ) = # (Z/N Z) , × these special values can be p-adically interpolated. Let ω : Zp,N/pordp (N ) → Cp be the continuous character which is the composition of the projection onto Z× with the Teichmuller character [−]. As explained in p [MTT], there exist p-adic L-functions Lp f, ω j χ, s on Zp such that Lp f, ω j χ, j = ep (j, χ) L∗ (f, χ, j + 1) , j k−j 1 − χ(p)p and α is a root of X 2 − ap (f ) X + pk−1 = 0 such that where ep (j, χ) = αordp1 c(ψ) 1 − χ(p)p α α ordp (α) < k + 1. As remarked in [MTT, §15], the p-adic multiplier ep (j, χ) vanishes for some (j, χ) if and only if p k N and j = k/2. Then it also vanishes for every χ0 such that χ0 (p) = χ (p). In particular, suppose from now on that f is a newform and p k N . It is known that f | Up = −wp pk/2 f , with wp ∈ {±1}. Then α = −wp pk/2 and ep (k/2, χ) vanishes for all Dirichlet characters such that χ (p) = −wp . The following conjecture was formulated in [MTT, §15]: Conjecture 1. There exists a constant Lp (f ) ∈ Cp such that, for every Dirichlet character such that χ (p) = −wp , i d h Lp f, ω k/2 χ, s = Lp (f ) L∗ (f, χ, k/2 + 1) . ds s=k/2 The constant Lp (f ) only depends on the restriction of the Galois representation attached to f to a decomposition group at p. The constant Lp (f ) was only defined when k = 0 in [MTT]. Since then, several definitions have been proposed by many authors. Between the others that we will discuss below, we quote the Coleman L-invariant LC (f ) of [C], the so called Fontaine-Mazur L-invariant LF M (f ) of [M] and the Breuil L-invariant LB (f ) of [Br]. More in the spirit of the present paper are the Teitelbaum L-invariant [T1] and, most of all, the Darmon-style L-invariants defined in [Dar], [O], [G] and [RS], which we are now going to discuss. The above conjecture was first proved when k = 0 in [GS]. It is now a theorem, thanks to Kato-Kurihara-Tsuji and, independently, Stevens (both unpublished), using LF M (f ) and, respectively, LC (f ). Other proofs have been offered in [O], using the Darmon-style L-invariant of [O], and [E], using LB (f ). Let W := Homcont Z× p , Gm be the weight space, that we view as a rigid analytic Qp -variety. The integers Z are embedded in W by n 7→ (t 7→ tn ). Since the modular form f is new, there exists an affinoid neighborhood U ⊂ W of k and a formal q-expansion P f U = i≥1 ai q i ∈ OW (U ) [[q]] such that: • for every integer n ≥ k which belongs to U , P f n := i≥1 ai (n) q i ∈ Sn+2 (Γ0 (N )) • fk = f. Recall we assume p k N and choose a decomposition N = pN + N − into factors prime each other such that N − is squarefree. When N − is divisible by an odd number of primes, called the definite setting, the − Teitelbaum L-invariant LN (f ) of [T1] is known to be equal to the other known L-invariants thanks to [BDI], when f is split, and [BD1], when k = 0 (see also [Se2, Proposition 5.7] for the higher weight non-split case). Suppose from now on the N − is divisible by an even number of primes, called the indefinite setting. − Let us briefly recall how the L-invariant LN (f ) can be defined in this setting. − Let B N be the unique (up to isomorphism) indefinite quaternion Q-algebra of discriminant N − and, for − − − an integer M prime to N − , let RN (M ) be an Eichler Z-order in B N of level M and set ΓN (M ) := 0 0 − − − − × × N N + N + RN (M ) , the norm one elements in R (M ) . We set Γ := Γ (pN ), R := R (N ) [1/p] and 0 0 0 0 0 1 × N− Γ := R1 . Fix an isomorphism ιp : B ⊗ Qp = M2 (Qp ), by means of which Γ acts on the p-adic upper 2
halfplane Hp and the Bruhat-Tits tree Tp attached to GL2 (Qp ). For every field F , let us denote by Pk (F ) the space of one variable polynomials of degree ≤ k, endowed with the right GL2 (F )-action given by at + b a b k (1) (P g) (t) = (ct + d) P ,g= . c d ct + d We write Vk (F ) := HomF (Pk (F ) , F ), which is naturally a left GL2 (F )-module by the rule (gΛ) (P ) = Λ (P g). Let E be the set of oriented edges of Tp . Fix e∞ ∈ E whose stabilizer in Γ is Γ0 and write −
−
Char E, VkN (F ) to denote the space of VkN (F )-valued harmonic cocyles. As explained in [RS, Lemma 2.8], the evaluation at e∞ -morphism induces in cohomology: p−new − − , ' H 1 Γ0 , VkN (F ) ρe∞ : H 1 Γ, Char E, VkN (F ) c
c
where (−)c denotes the cuspidal part. There is an involution W∞ acting on these cohomology groups, which commutes with the action of the Hecke algebras. If H is a Q [W∞ ]-module, we may write H = H + ⊕ H − , where H w∞ for w∞ ∈ {±1} is the subspace where W∞ = w∞ . When F = C, by the JacquetLanglands correspondence and the Eichler-Shimura isomorphism there is a unique (up to non-zero scalar p−new,w∞ − − factor) c = cN ,w∞ ∈ H 1 Γ0 , VkN (C) having the same eigenvalues of f . Let K/Qp be a finite c
extension containing σ p (Kf ). By the universal coefficient theorem there is a corresponding eigenvector in p−new,w∞ − , that we denote again by c. H 1 Γ0 , VkN (K) c k 1 Let us write A P (Qp ) to denote the space of K-valued locally analytic functions on Qp with a pole of order at most k at ∞, endowed with the right GL2 (Qp )-action defined by (1). We also denote by Dk P1 (Qp ) k is continuous K-dual, which is a left GL2 (Qp )-module as above. Write D0,b P1 (Qp ) to denote the subspace of bounded distributions that are zero on Pk (K), as defined for example in [RS], [T1] or [T2]. As explained in [RS] (and recalled before Theorem 13), there is an identification − k R : HN (K) = H (K) := H iN − Γ, D0,b P1 (Qp ) ' H iN − (Γ, Char (E, Vk (K))) . As recalled in §5.1 there are natural integration pairings Φlog and Φord that induce, by cap product, ∨ Φlog , Φord : HiN − Γ, Div0 Hpur ⊗ Pk (K) → H (K) . ur Here Hpur Qur := Qur (resp. Div0 Hpur ) denotes the free K-module of divisors (resp. p p − Qp and Div Hp , where K0 := K ∩ Qur that are fixed by the action of GQur degree zero divisors) supported on Hpur Qur p p p /K0 is the maximal unramified subextension of K. Let i ∂ (2) H2 (Γ, Pk (K)) → H1 Γ, Div0 Hpur ⊗ Pk (K) → H1 Γ, Div Hpur ⊗ Pk (K) be the exact sequence induced by the exact sequence 0 → Div0 Hpur ⊗ Pk (K) → Div Hpur ⊗ Pk (K) → Pk (K) → 0. −
,w∞ Let char = cN be such that c = ρe∞ (char ). The following theorem is proved in [RS]. har
Theorem 2. Let p be the prime of Kf induced by σ p . There exists a unique Darmon-Teitelbaum L-invariant − LN (f ) ∈ Kf,p such that − (3) Φlog (∂p) R−1 (char ) = LN (f ) · Φord (∂p) R−1 (char ) for every p ∈ H2 (Γ, Pk (K)). −
−
When N − = 1 and k = 0, LN (f ) was implicitly defined in [Dar]. When N − = 1, LN (f ) equals the − L-invariant of [O]. When k = 0 and the totally real field which is considered in [G] is Q, LN (f ) equals − the L-invariant of [G]. More generally, LN (f ) is the L-invariant of [RS], which recovers the L-invariants of [Dar], [O] and [G]. Before stating our main result, let us mention that, thanks to the work of several people - Greenberg and Stevens (unpublished), [CI], [IS], [Br], [Cz1], [Cz2], [BDI], [BD1] [DG] and [LRV] - we now know that − 0 LN (f ) = −2 (log ap ) (k) = L∗ (f ) with ∗ = C, F M or B, with the only possible exception of the indefinite case when N − 6= 1 and k > 0. The main result of this paper covers these missing cases. 3
Theorem 3. We have the equality −
0
LN (f ) = −2 (log ap ) (k) . −
In particular, in the indefinite setting, LN (f ) does not depend on the choice of the sign w∞ . When N − = 1, by means of modular symbols, one may reduce the above equality to H 0 -group cohomological computations. It is when N − 6= 1 that a real cohomological method is needed. While the proofs of [DG] and [LRV] are both based on a cohomological adaptation of the techniques developed in [Das], our technique is different and rather based on a cohomological adaptation of the techniques developed in [BDI], which seem more suited to handle the higher weight case, combined with Ash-Stevens results on slopes decompositions (see [AS]) and a p-adic Jacquet-Langlands correspondence which is investigated in [GSS]. We assume to be in the indefinite case and we may also uniformly treat the known N − = 1 or k = 0 cases. We finally remark that, as explained in §6.2, we may also deduce from the above theorem the stronger form [RS, Theorem 4.7] (that was indeed stated as a conjecture, in an earlier version of that paper). As explained in [RS], this makes the conjectural definition of so called Darmon cycles effective. Finally, as an application, we give a formula for the computation of the logarithm of the p-adic Abel-Jacobi maps of [RS]. This formula is crucial for the computations of the derivatives of the p-adic L-functions defined in [GSS], in connection with the logarithmic p-adic Abel-Jacobi image of the Darmon cycles. As explained in [GSS], combined with results from [GSS], it yields a generalization of the main result of [BD2] and [Se1] on the rationality of Darmon cycles. 2. Weights, distributions and families of distributions In order to avoid making notations heavier, let us assume p 6= 2 in the subsequent discussion. Let W := Homcont Z× , G be the weight space. It is a rigid analytic Qp -variety such that, for all p-adic fields m p × × K, W (K) = Homcts Zp , K . We write t ∈ Z× p as t = [t] hti, according to the canonical decomposition Z× p = µp−1 × (1 + pZp ), where µp−1 is the group of p − 1-roots of unity. If κ ∈ W (K), we usually write tκ := κ (t). The weight space W is the disjoint union of p−1 affinoid subvarieties Wi , 0 ≤ i < p−1, such that, i κ for all p-adic fields K, Wi (K) is the subset of those continuous homomorphisms κ such that tκ = [t] hti . There is a natural embedding of Z into W given by Z 3 n 7→ (t 7→ tn ) , t ∈ Z× p. For k ∈ Z such that k ≡ i mod (p − 1), there exists a Qp -affinoid neighborhood Ui ⊂ Wi of k ∈ W (Qp ) k s s such that, for every p-adic field K, κ ∈ Ui (K) is given by tκ = [t] hti , where hti := exp (s log (hti)) and × κ−k s−k s ∈ OK is uniquely determined by κ. Then, for t ∈ Qur , we set hti := hti = exp ((s − k) log (hti)). p κ−k × Then (κ · (−k)) (t) = hti if t ∈ Zp and, for fixed t, the expression is a rigid analytic function on U . From now on we fix a p-adic field K/Qp and we view the above rigid analytic objects as defined over K. We also fix k ∈ Z and let U ⊂ Ui be a K-affinoid neighborhood of k. Set W := Q2p − {0}, endowed with the left GL2 (Qp )-action obtained by viewing the elements of W as column vectors. The space W comes equipped with the natural continuous (for the p-adic topologies) projection: π : W → P1 (Qp ) , π (x, y) := [x : y] . Let Υ be either U or κ ∈ U (K) and let, respectively, RΥ := OW (U ) or K. For an open subset X ⊂ W , ∞ n −1 that we assume to be Z× (π (X)), which is then Q× p -stable, set p X := ∪n p X = π p -stable, and write Y ∞ b to denote either X or p X. Let RΥ ⊗K A (Y ) be the space of those functions F : Υ (K) × Y → K that are rigid analytic in the κ-variable and locally analytic in the (x, y)-variable. We are interested in the closed b K A (Y ) such that, respectively, RΥ -submodules AΥ (X) and AΥ,k (p∞ X) of those F ∈ RΥ ⊗ k ordp (t) ordp (t) (4) F (κ, tx, ty) = κ (t) F (κ, x, y) for t ∈ Z× , F (κ, tx, ty) = p κ t/p F (x, y) for t ∈ Q× p p. b K A (Y ) = A (Y ), the space of K-valued locally anaytic functions on Y , and Note that, when Υ = κ, RΥ ⊗ (4) reduces to F (tx, ty) = κ (t) F (x, y) and similarly for the second defining relation. Let Aφ,k (p∞ X) ⊂ A (p∞ X) be the subset of those functions such that F (px, py) = pk F (x, y). 4
We write D (X) := L (A (X) , K) and Dφ,k (p∞ X) := L Aφ,k (p∞ X) , K for the continuous K-dual, and let D Z× be the K-space of locally analytic distributions on Z× p p , defined in a similar way as for Y ⊂ W . φ,k ∞ × Then D (X), D (p X) are naturally D Zp -modules by the convolution product R R (rµ) (F ) := r (t) (µ (F t)) = Z× F (tx, ty) dµ (x, y) dr (t) . X p By the theorem of Amice-Velu D Z× p ' OW (W) (recall we view W over K). More precisely, note that every continuous homomorphism κ ∈ U (K) is locally analytic (and indeed analytic by our choice of U ). The composition of this identification with the restriction map ρW U : OW (W) → RU is given by the Fourier transform Z ρU : r 7→ αr (κ) := r (κ) = κ (t) dr (t) , for κ ∈ U (K) . Z× p
If κ ∈ U (K), let Iκ be the maximal ideal which corresponds to κ, and consider the exact sequence: (5)
0
→
Iκ
→ RU
κ
→ K
→ 0.
b ρ ,D Z× Dφ,k (p∞ X), where ρκ := κ◦ρU . D (X) and DΥ,k (p∞ X) := RΥ ⊗ (Z× ( p) p ) Υ × As we have Z× , Q ⊂ Z (GL (Q )), every g ∈ GL (Q ) induces R 2 p 2 p Υ -linear morphisms p p
bρ We define DΥ (X) := RΥ ⊗
Υ ,D
(6)
g : A∗,∗ (gY ) → A∗,∗ (Y ) and g : D∗,∗ (Y ) → D∗,∗ (gY ) ,
for ∗ = (Υ, φ), (φ, k), (Υ, k). In particular these spaces becomes RΥ [Σ]-modules if Σ ⊂ GL2 (Qp ) is a subsemigroup preserving Y . We will be interested in some specific choice of X and p∞ X. Write L (resp. L0 ) to denote the set of Zp -lattices in Q2p (resp. couples (L1 , L2 ) such that L2 ⊂p L1 , i.e. L2 ⊂ L1 with index p). Attached to the lattice L ∈ L (resp. (L1 , L2 ) ∈ L0 ) is the open compact set L0 := L − pL (resp. WL1 ,L2 := L01 ∩ L02 ). Recall the Bruhat-Tits tree Tp attached to PGL2 (Qp ), whose vertices V are the homothety classes [L] ∈ V represented by L ∈ L. We also denote by E the set of ordered edges of Tp . If e ∈ E we write s (e) (resp. t (e)) for the source (resp. the target) of e. Whenever v1 , v2 ∈ V, we write d (v1 , v2 ) for the distance between v1 and v2 . If e = ([L1 ] , [L2 ]), we may choose representatives L1 , L2 ∈ L such that (L1 , L2 ) ∈ L0 and we write e = [L1 , L2 ]. It follows that L → V and L0 → E are surjective. We write e = (s (e) , t (e)). Let L∗ := Z2p be the standard lattice and, for i = 0, ..., p−1 representatives of the classes Z/pZ and i = ∞, define Li := {(x, y) ∈ L∗ : x − iy ≡ 0 mod p} , L∞ = Zp × pZp . We set v∗ := [L∗ ], ei := [L∗ , Li ] for i = 0, ..., p − 1, ∞. The Bruhat-Tits tree is oriented as follows. We let V ± be the set of vertices such that d (v∗ , v) is even (resp. odd) for the + (resp. −) choice of the sign. We define E ± as the subset of those edges e such that s (e) ∈ V ± . Taking X = L0 and X = WL1 ,L2 is is easily checked that p∞ L0 = W and, respectively, p∞ WL1 ,L2 =: We which only depends on the homotethy class e = [L1 , L2 ]. Since for these X we have pm X ∩ pn X = φ ' ' for m 6= n, the restrictions yield isomorphisms Aφ,k (p∞ X) → A (X) and AΥ,k (p∞ X) → AΥ (X) and, by φ,k ∞ Υ Υ,k ∞ duality, D (X) = D (p X) and D (X) = D (p X) commuting with (6). Our interest in the spaces DΥ (X) and DΥ,k (p∞ X) is motivated by the fact that they naturally integrates, respectively, elements of AΥ (X) and AΥ,k (p∞ X). If µΥ,∗ ∈ DΥ,∗ (Y ) and F ∈ AΥ,∗ (Y ), we write Υ,∗ Υ,∗ µ (F ) := ΛΥ µ (F ) ∈ RΥ , where ΛΥ is provided by the subsequent lemma. Lemma 4. For ∗ = φ, k there is a unique continuous RΥ -linear morphism, valued in the continuous RΥ -dual, ΛΥ : DΥ,∗ (Y ) → LRΥ AΥ,∗ (Y ) , RΥ such that R b ρΥ µ (F ) = ααµ,F ∈ RΥ , αµ,F (κ) := µ (F (κ, ·)) = ΛΥ α⊗ F (κ, (x, y)) dµ (x, y) ∈ RΥ . X Furthermore, if Y = L0 , WL1 ,L2 , W or We := p∞ WL1 ,L2 for e = [L1 , L2 ], ΛΥ is an isomorphism. Proof. See [BDI, pag. 17] and [GSS, Lemmas 159 and 163]. 5
For a field F let Pk (F ) be the right GL2 (F )-module of degree k ∈ N homogeneus polynomials in two variables, the action being defined by the formula (7)
(P g) (X, Y ) = P (ag X + bg Y, cg X + dg Y )
making Pk = Pk (K) naturally a GL2 (Qp )-subspace of Ak,k (Y ). We let Vk (F ) be the F -dual, with the left action given by (gΛ) (P ) = Λ (P g), and set Vk = Vk (K). Remark 5. The maps F (x, y) 7→ fF (t) := F (t, 1) and f (t) 7→ Ff (x, y) := y k f (x/y) give rise to an isomorphism Ak,k (W ) = Ak P1 (Qp ) . By Lemma 4 and K-duality, Dk,k (W ) = Dk P1 (Qp ) . The sub GL2 (Qp )-module D0k P1 (Qp ) of those distributions that are zero on the polynomials of degree ≤ k is k,k (W ) ⊂ D0k (W ) identified with the subspace D0k,k (W ) of those distributions that are zero on Pk . We let D0,b k,k P1 (Qp ) ⊂ D0k P1 (Qp ) be the sub GL2 (Qp )-module which corresponds to the bounded distributions D0,b as defined, for example, in [RS, Remark 3.4], [T1] or [T2]. Every κ ∈ U (K) gives, by means of (5), b D Z× 1 : DU,∗ (Y ) → Dκ,∗ (Y ) ρY,κ := κ⊗ ( p) and, when κ = k, restricting the integration to polynomials yields ν Y,k : Dk,∗ (Y ) → Vk . Let η Y,k := ν Y,k ◦ ρY,k be the composition. Both ρY,κ and ν Y,k commutes with (6), so that they are K [Σ]morphisms, when Σ preserves Y . We have the following important result. Theorem 6. There is an exact sequence (set ρκ := ρWe ,κ ) (8)
ρ
κ 0 → Iκ DU,k (We ) → DU,k (We ) → Dκ,k (We ) → 0.
Proof. Using DU,k (We ) = DU (WL1 ,L2 ) it can be proved similarly as in [AS, Theorem 3.7.4] (see also [GSS]). We define C V, DΥ,k (resp. C E, D∗Υ,k ) as being the space of maps µ∗ from V (resp. E) to DΥ,k (W ) (resp. te∈E DΥ,k (We ) such that µe ∈ DΥ,k (We )). They are naturally GL2 (Qp )-modules, as explained in §7.1, by the rule (gµ)v := gµg−1 v for v ∈ V (resp. (gµ)e = gµg−1 e for e ∈ E). Lemma 7. Consider the morphisms ds : C V, DΥ,k → C E, D∗Υ,k , ds (µ)e := µs(e)|We and P L δ s : C E, D∗Υ,k → C V, DΥ,k , δ s (µ)v := s(e)=v µe ∈ s(e)=v DΥ,k (We ) = DΥ,k (W ) They induce a GL2 (Qp )-equivariant identification. Proof. This is a simple computation that we leave to the reader.
Let GL+ 2 (Qp ) ⊂ GL2 (Qp ) be the subgroup of matrices g such that ordp (det (g)) ≡ 0 mod 2. The identification of Lemma 7 induces a GL+ 2 (Qp )-equivariant identification ' ds : C V ± , DΥ,k → C E ± , D∗Υ,k , where these spaces are defined in a similar way with V ± (resp. E ± ) in place of V (resp. E). ± Note that, viewing the elements of Dk,k (W ) as constant functions on V (resp. GL2 (Qp ) V ) yields a k,k + k,k k,k equivariant (resp. a GL2 (Qp )-equivariant) inclusion ι : D (W ) ⊂ C V, D (resp. ι± : D (W ) ⊂ ± U,k p−new C V ± , Dk,k ). Define the GL+ by the following commutative diagram of 2 (Qp )-module C V , D 6
GL+ 2 (Qp )-modules with exact rows, where the first row is the pull-back of the second via ι± , which comes from (8) and Lemma 7: p−new ρ±,p−new k,k k 0 → Ik C V ± , DU,k → C V ± , DU,k → D0,b (W ) → 0 (9) k ∩ι± ∩ι± ρ± k 0 → Ik C V ± , DU,k → C V ± , DU,k → C V ± , Dk,k → 0 − Let ρk = ρ+ k ⊕ ρk be the specialization p−new the GL2 (Qp )-module C V, DU,k with exact rows 0 → Ik C V, DU,k (10) k 0 → Ik C V, DU,k
morphism obtained from (8) and Lemma 7. We may similarly define by means of the following commutative diagram of GL2 (Qp )-modules → →
p−new C V, DU,k ∩ι± C V, DU,k
ρp−new k
→ ρ
k →
k,k D0,b (W ) ∩ι C V, Dk,k
→
0
→ 0.
For any (L1 , L2 ) ∈ L0 , let TL1 ,L2 ⊂ L0 be the set of (L2 , L) such that L ⊂p L2 and L 6= pL1 . If e = [L1 , L2 ], we have e = [L2 , pL1 ] and define Te := {e0 : s (e0 ) = t (e) , e0 6= e} = {[L2 , L] : L ⊂p L2 , L 6= pL1 } . F F Note that we have decompositions WL1 ,L2 = (L2 ,L)∈TL ,L WL2 ,L , We = e0 ∈Te We0 . In particular, 1 2 L (11) DΥ,k (We ) = e0 ∈Te DΥ,k (We0 ) . Recall we set ei = [L∗ , Li ]. With these notations (12)
Tp−1 L∞ ,L∗ = {(L∗ , Li ) : i = 0, ..., p − 1} , Te∞ = {ei : i = 0, ..., p − 1} . 3. Hecke operators in cohomology
3.1. Hecke operators. Let G be a group, endowed with an anti-automorphism ι, i.e. a bijection of G onto ι iteself such that ι2 = 1, (g1 g2 ) = g2ι g1ι and 1ι = 1. Let (Γ, Σ) be a Hecke pair in G (meaning that ΓπΓ is a finite disjoint union of cosets for any π ∈ Σ) and suppose that the involution preserves Γ, inducing the inversion on it. If ππ ι ∈ CG (Γ), ι preserves π ι Γπ −ι , inducing the inversion on it. Let X and D be, respectively, a left Σ-module and a left Σι -module. We endow Hom (X, D) with a left (resp. right) Σι module (resp. Σ-module) structure by the rule (gf ) (x) := gf (g ι x) (resp. (f g) (x) := g ι f (gx)), so that Γ g ι f = f g for g ∈ Σ. As we assume F that ι induces the inversion on Γ, Hom (X, D) = HomΓ (X, D). If π ∈ Σ we may write ΓπΓ = i Γπ i . We define f | ΓπΓ :=
: Hom P ΓπΓ P Γ (X, D) → HomΓ (X, D)P ι ι π f = i i i f π i , i.e. (f | ΓπΓ) (x) := i π i f (π i x) .
Suppose that D is a Γ-module, ππ ι ∈ CG (Γ), π ι D is a π ι Γπ −ι -module and ϕ : D → π ι D is an equivariant morphism, i.e. such that ϕ (γd) = π ι γπ −ι ϕ (d). As ππ ι ∈ CG (Γ) and π ι Γπ −ι is conjugated to Γ, π ι Γπ −ι Hom (X, πD) = Homπι Γπ−ι (X, πD). We define ϕπ : HomΓ (X, D) → Homπι Γπ−ι (X, D) , f | ϕπ = ϕ ◦ f ◦ π. By applying the above formalism to a resolution X· of Z by projective Z [G]-modules, we may define in this way ΓπΓ : H · (Γ, D) → H · (Γ, D) , ϕπ : H · (Γ, D) → H · π ι Γπ −ι , πD . In particular, the Hecke algebra H (Γ, Σ) associated to the Hecke pair (Γ, Σ) acts on H · (Γ, D). Let B be an indefinite quaternion Q-algebra of discriminant N − ∈ N. For a place v let Hv (resp. M2 (Qv )) be the unique, up to isomorphism, division (resp. split) quaternion Qv -algebra and fix, once and for all, identifications ιv : Bv = Hv for v | N − , ιv : Bv = M2 (Qv ) for v - N − , where Bv := B ⊗ Qv . If 7
ag bg . If v = ∞, let GL+ 2 (R) be the group of cg dd those g such that det (g) > 0. If v = l | N − let OHl be the maximal order of Hv . If v = l - N − is a finite prime, we write R0 (ln Zl ) to denote Zl -order of those matrices in M2 (Zl ) such that cg ≡ 0 mod ln . We let Σ0 (ln Zl ) ⊂ R0 (ln Zl ) be the multiplicative subgroup defined by the further conditions ag ∈ Z× l and det (g) 6= 0 and define the subgroup Γ0 (ln Zl ) := Σ0 (ln Zl ) ∩ GL2 (Zl ). For an integer M prime to N − define the Eichler Z-order of level M in B by the formula Q Q −1 n R0 (M ) := B × ∩ l|N − ι−1 ln kM finite ιl (R0 (l Zl )) . l (OHl ) g ∈ M2 (Qv ), we define ag , bg , cg and dg by g =
We also define the semigroup Σ0 (M ) and the group Γ0 (M ) as Q Q −1 n Σ0 (M ) : = B × ∩ l|N − ι−1 ln kM finite ιl (Σ0 (l Zl )) , l (OHl ) Q Q + −1 × −1 n Γ0 (M ) : = B × ∩ ι−1 OH ∞ GL2 (R) l|N − ιl ln kM finite ιl (Γ0 (l Zl )) , l ×
so that Γ0 (M ) ⊂ Σ0 (M ) ⊂ R0 (M ). Note that Γ0 (M ) = R0 (M )1 , the subgroup of norm one elements × in R0 (M ) . The Eichler orders R0 (M ) are stable by the main involution ι of B × , as it follows from their local definition. It follows that Γ0 (M ) is stable under ι, which induces the inversion on it. This is not the case for the semigroup Σ0 (M ). Indeed Q Q ι ι −1 n Σ0 (M ) = B × ∩ l|N − ι−1 ln kM finite ιl (Σ0 (l Zl ) ) , l (OHl ) ι
where Σ0 (ln Zl ) is the semigroup of those g ∈ M2 (Zl ) such that cg ≡ 0 mod ln , dg ∈ Z× l and det (g) 6= 0. ι Then (Γ0 (M ) , Σ0 (M )) (resp. (Γ0 (M ) , Σ0 (M ) )) is a Hecke pair and we may consider the associ− − ι ι ated Hecke Q-algebra H0 (M ) = H0N (M ) := H (Γ0 (M ) , Σ0 (M )) (resp. H0 (M ) = H0N (M ) := ι H (Γ0 (M ) , Σ0 (M ) )). When M = pN + with p prime to N + , we also set R := R0 (N + ) [1/p], Γ = Γ1/p (N + ) := R× 1 , the subgroup of norm one elements in R× , and Γ0 := Γ0 (pN + ). It follows that both Γ and Γ0 are stable under ι, which induces the inversion on them. ι ι If D is a Σ0 (pZp ) -module (resp. a Σ0 (pZp )-module) we may consider it as a Σ0 (pN + ) -module (resp. a Σ0 (pN + )-module) and the cohomology groups H · (Γ0 , D) are endowed with an action of the Hecke algebra (resp. WL∗ ,L∞ = Z× H0 (pN + ) (resp. H0ι (pN + )). As the region Wp−1 L∞ ,L∗ = Zp × Z× p × pZp ) is stable p ι Υ under the action of Σ0 (pZp ) (resp. Σ0 (pZp )), D Wp−1 L∞ ,L∗ and DΥ,k (We∞ ) (resp. DΥ (WL∗ ,L∞ ) and DΥ,k (We∞ )) are example of such modules. From now on we view the elements of Bp as matrices via Bp = M2 (Qp ) and we simply write g = ιp (g). Of particular interest for our purposes is the action of the Hecke operator Up := Γ0 πΓ0 ∈ H0 (pN + ) attached Fp−1 to an element π ∈ Σ0 (pN + ) of reduced norm p. We write Γ0 πΓ0 = i=0 Γ0 π i and note that it is easily checked that the cosets Γ0 π i are characterized by the property (13)
δ (L∗ , Li ) = (L∞ , pL∗ ) for every δ ∈ Γ0 π i .
We let ω p ∈ R0 (pN + ) be an element of reduced norm p which normalizes Γ0 (pN + ). Then one checks that (14) δ p−1 L∞ , L∗ = (L∗ , L∞ ) and δ −1 p−1 L∞ , L∗ = p−1 L∗ , p−1 L∞ , for δ ∈ Γ0 ω p (or δ ∈ ω p Γ0 ). bp := Γ0 π We are also concerned with the action of the Hecke operator U bΓ0 ∈ H0ι (pN + ) attached to the F ι p−1 + bi with π bi = ω −1 element π b = ω −1 bΓ0 = i=0 Γ0 π p πω p ∈ Σ0 (pN ) . Since ω p normalizes Γ0 , Γ0 π p π i ω p . It follows from (13) and (14) that we have: (15) δ p−1 L∞ , ω −1 bi . p Li = (L∗ , L∞ ) for every δ ∈ Γ0 π ι −ι Set e+ := e∞ and e− := e∞ . Consider Wp := ϕωp , where ϕ = ω ιp . Since ω ιp Γω −ι p = Γ and ω p Γ0 ω p = Γ0 , ± there is a Wp operator on the Γ-cohomology of the spaces appearing in diagram (10). Let (9)H be the diagram obtained from (9) taking Γ-cohomology for the ± choice of the sign, and write f ± to denote any 8
∓
±
one of the morphisms appearing in (9)H and let f ∓ be the corresponding arrow in (9)H . We also remark that ω ιp induces a well defined Wp operator giving rise to a commutative diagram (16)
±
∓
Wp : (9)H → (9)H , i.e. f ± ◦ Wp = Wp ◦ f ∓ .
It is clear that these Wp -operators commute with the Shapiro’s isomorphisms ρe± of Proposition 27 and the identifications ds s. We may also consider the operators: bp : H · Γ0 , DΥ,k (We ) → H · Γ0 , DΥ,k (We ) . Up : H · Γ0 , DΥ,k (We∞ ) → H · Γ0 , DΥ,k (We∞ ) , U ∞ ∞ The relations (17)
bp = Wp−1 Up Wp : H · Γ0 , DΥ,k (We ) → H · Γ0 , DΥ,k (We ) , Wp2 = pk , U ∞ ∞
where Wp is any one of the above Wp -operators, are easily checked. 3.2. Lifting the operators to coefficients. We may define, using (11), V : C E, D∗Υ,k → C E, D∗Υ,k L P (µ∗ | V)e := t(e0 )=s(e),e0 6=e µe0 ∈ t(e0 )=s(e),e0 6=e DΥ,k (We0 ) = DΥ,k (We ) . Note that V = s ◦ V+ ⊕ V− , with V± : C E ± , D∗Υ,k → C E ∓ , D∗Υ,k defined by the same formula and s the switch of the factors. We also define V : C V, DΥ,k → C V, DΥ,k P L (µ∗ | V)v := d(v0 ,v)=1 µv0 |(v,v0 ) ∈ d(v0 ,v)=1 DΥ,k W(v,v0 ) = DΥ,k (W ) . Again V = s ◦ V+ ⊕ V− , with V± : C V ± , DΥ,k → C E ∓ , DΥ,k defined by the same formula and s the switch of the factors. Our notations is justified by the fact that, under the identification ds of Lemma 7, these V and V± operators correspond. Furthermore, one may easily check that V and V± are, respectively, GL2 (Qp ) and GL+ 2 (Qp )-equivariant. We remark that the inclusions ι and ι± satisfy the relations ι (−) | V = ι and ι± (−) | V± = ι∓ (−) on k,k k,k D0,b (W ). It follows that, setting V := 1 on D0,b (W ), V acts on the spaces in diagram (10) and the arrows in this diagram are V-equivariant, as well as the arrows obtained taking Γ-cohomology. Similarly, setting k,k V± := 1 on D0,b (W ), V± gives a commutative diagram ± ∓ (18) V± : (9)H → (9)H , i.e. f ± (−) | V± = f ∓ − | V± . The formalism of §7.1 applies to the Γ-modules C E, D∗Υ,k . We may lift the operators Wp−1 Up and bp at the level of coefficients as follows. p−k Wp U Proposition 8. The following diagrams commute, where the vertical arrows are Shapiro’s isomorphisms of Proposition 27: V+ H i Γ, C E + , D∗Υ,k → H i Γ, C E − , D∗Υ,k ρ e∞ k k ρ e∞ Wp−1 Up i Υ,k i H Γ0 , D (We∞ ) → H Γ0 , DΥ,k (We∞ ) and V− → H i Γ, C E + , D∗Υ,k H i Γ, C E − , D∗Υ,k ρ e∞ k k ρ e∞ −k bp p W U p H i Γ0 , DΥ,k (We∞ ) → H i Γ0 , DΥ,k (We∞ ) . Proof. Setting e+ := e∞ and e− := e∞ , from §7.1 we have the following diagram Γ V ± Γ C E ± , Hom X· , DΥ,k ∗ → C E ∓ , Hom X· , DΥ,k ∗ se± ↑ ↓ ρ e∓ Υ,k HomΓ0 X· , D W e± → HomΓ0 X· , DΥ,k We∓ . 9
Here, up to the identification provided by Lemma 28, V± is the operator P L s | V± (e) := e0 ∈Te s (e0 ) ∈ e0 ∈Te Hom X· , DΥ,k (We0 ) = Hom X· , DΥ,k (We ) . Suppose f ∈ HomΓ0 X· , DΥ,k (We∞ ) . By (33) and (12) Pp−1 Pp−1 f gei , ρe∞ se∞ (f ) | V+ = se∞ (f ) | V+ (e∞ ) = i=0 se∞ (f ) (ei ) = i=0 ge−1 i where gei ∈ Γ is any element such that gei ei = e∞ . It follows from (13) and (14) that ω −1 p π i ∈ Γ satisfies −1 −1 ω −1 π (L , L ) = (L , L ), so that ω π e = e and we may take g = ω π . Hence, i ∗ i ∗ ∞ i i ∞ ei i p p p Pp−1 −1 (19) ρe∞ (se∞ (f ) | V) = i=0 π i ω p f ω −1 p πi . −1 On the other hand, since π ιi ω −ι p = πi ωp , Pp−1 Pp−1 Pp−1 −1 −1 −1 (20) f | Wp−1 Up = i=0 π ιi f | Wp−1 π i = i=0 π ιi ω −ι p f ωp πi = i=0 π i ω p f ω p π i .
Comparing (19) and (20) yields the commutativity of the first diagram. The proof that the second diagram is commutative is similar, exploiting (15). 4. Slope decompositions and families of modular forms on indefinite quaternion algebras We recall the notion of slope decomposition from [AS]. Let M be a module over a K-Banach algebra A, with an A-linear morphism u : M → M . If Q (T ) ∈ A [T ] set Q∗ [T ] := T deg(Q) Q (1/T ). For every h ∈ R≥0 let Sh (u) ⊂ A [u] = R be the multiplicative subset of operators Q∗ (u), where Q (T ) ∈ A [T ] is a slope ≤ h polynomial whose leading coefficient is multiplicative with respect to the given the norm on A. Definition 9. A slope ≤ h decomposition of (M, u) is an R-module decomposition M = M ≤h ⊕ M >h such that: • M ≤h is a finitely generated A-module such that sM ≤h = 0 for some s ∈ Sh (u); • every element s ∈ Sh (u) acts invertibly on M >h . We call M ≤h (resp. M >h ) the slope ≤ h (resp. slope > h) part. It is clear that a slope ≤ h decomposition needs no to exist in general. However, whenever it exists, it enjoys good functorial properties. An u-morphism f : M → N is a morphism of A-modules such that f u = uf , where u acts A-linearly on M and N . Lemma 10. Let M , N , O, P and Q be A-modules endowed with u operators. (a) If the A-modules M and N have a slope ≤ h decomposition and f : M → N is an u-morphism, f M ≤h ⊂ M ≤h and f M >h ⊂ M >h . (b) Suppose that we have given an exact sequence of R-modules M → N → O → P → Q. If M , N , P and Q have a slope ≤ h decomposition, O has a slope ≤ h decomposition and the induced sequences M ≤h → N ≤h → O≤h → P ≤h → Q≤h , M >h → N >h → O>h → P >h → Q>h . are exact. Proof. See [AS, Proposition 4.1.2].
Remark 11. Suppose that m ∈ M is such that um = am with a ∈ A× a multiplicative element. If |a| ≥ p−h , then m ∈ M ≤h . Conversely, if m is R-torsion free and m ∈ M ≤h , then |a| ≥ p−h . 10
In our application we take U ⊂ Ui ' Sp (K hSi) and A = OW (U ). By [BGR, pag. 274 Prop. 3] and the fact that K hSi is anPID, a fundamental system of neighborhoods at k for the canonical topology on Ui is o k s −n given by the Un = κ (t) = [t] hti ∈ Ui : |s − k| ≤ p . Since the conclusion of the subsequent Theorem 12 holds up to shrinking U in a neighborhood of k, we may assume to take U small enough so that U = Un . In particular, U = Sp (K hpn Si) for n big enough and every non-zero element of A = K hpn Si is multiplicative (see [BGR, pag. 234 Prop. 2]). Our interest in the spaces of distributions DU,k (We ) is motivated by the following theorem. Set for bp , Up− := Up , e+ := e∞ , e− := e∞ , H+ (pN + ) = H0 (pN + )ι and H− (pN + ) = H0 (pN + ). shortness Up+ := U 0 0 Theorem 12. The following assertions hold, up to shrinking U in a neighborhood of k. (a) The OW (U ) H0± (pN + ) -module H · Γ0 , DΥ,k (We± ) , Up± has a slope ≤ h decomposition. The same holds for H · Γ0 , Ik DΥ,k (We± ) , Up± . (b) The exact sequence (8) induces a long exact sequence ≤h ≤h ρk i ≤h → .... → H Γ0 , Dk,k (We± ) → H i Γ0 , DU,k (We± ) ... → H i Γ0 , Ik DU,k (We± ) (c) If h < k + 1, ν k induces an isomorphism H · Γ0 , Dk,k (We± )
≤h
≤h
' H · (Γ0 , Vk (K))
, ≤h H 1 Γ0 , DU,k (We± ) is a free OW (U )-module and the above exact sequence reduces to the exact sequence ≤h ≤h ηk 1 ≤h 0 → H 1 Γ0 , Ik DU,k (We± ) → H 1 Γ0 , DU,k (We± ) → H (Γ0 , Vk (K)) → 0. ≤h
Furthermore, if c ∈ H · (Γ0 , Vk (K)) is a cuspidal new eigenvector (hence k/2 ≤ h), there exists ≤h U,k · U,k c ∈ H Γ0 , D (We∞ ) such that ck,k := η k cU,k = c which is an eigenvector for the Hecke algebra H0 (pN + ). (d) The lift provided in (c) is compatible with the Jacquet-Langlands correspondence in the following sense. If c has the same eigenpacket as f ∈ Sk (Γ0 (pN + N − )) (with respect to the embeddings σ p and σ ∞ described in the introduction) and f U is the Coleman family attached to f , cU,k and f U have the same eigenpacket. Proof. The analogous theorem with the spaces DΥ in place of the spaces DΥ,k can be proved as an application of the techniques developed in [AS] (see also [GSS] for a more down to earth discussion in this setting). The identitifcation DΥ,k = DΥ gives the result. By means of Shapiro’s isomorphism appearing in Proposition 8 and the identification provided by Lemma 7, we may define slope ≤h decompositions on H · Γ, C V ± , DΥ,k , H · Γ, C V, DΥ,k , H · Γ, Ik C V ± , DU,k and H · Γ, Ik C V, DU,k . Taking the long exact sequences from (9) yields the following diagram with exact rows: (21) p−new ρk±,p−new k,k → H i Γ, D0,b (W ) → ... → H i Γ, C V ± , DU,k ... → H i Γ, Ik C V ± , DU,k k ↓ ι± ↓ ι± ρ± k ... → H i Γ, Ik C V ± , DU,k → H i Γ, C V ± , DU,k → H i Γ, C V ± , Dk,k → ... p−new k,k We are going to define Hecke operators Up± on H i Γ, C V ± , DU,k and H i Γ, D0,b (W ) making ± · U,k · the above diagram Up -equivariant. We view the elements of H Γ, C V, D = H Γ, C V + , DU,k ⊕ H · Γ, C V − , DU,k as row vectors on which operators act from the right. By Proposition 8 and (17), bp 0 0 Wp−1 Up 0 Wp 0 V+ 0 Wp U (22) = = . bp Wp 0 V− 0 Wp 0 0 Up p−k Wp U 0 11
According to (16) and (18), the are Wp V∓ -equivariant. It follows that we may define op arrows in (21) p−new k,k ± ∓ · ± U,k i erators Up := Wp V on H Γ, C V , D and H Γ, D0,b (W ) that make the arrows in (21) Up± -equivariant. A similar argument appliesto the diagram obtained by taking the long exact sequences from (10). Indeed, ∗ 0 Wp setting Wp := (which is the operator Wp on the H i Γ, C V, DΥ,k s with ∗ = φ, p − Wp 0 p−new new) and Up := Wp V defines an operator on H i Γ, C V, DU,k making the analogous of (21) Up + − i Υ,k i equivariant, where Up := Up ⊕ Up on H Γ, C V, D and H Γ, Ik C V, DU,k and Up := Wp V = Wp k,k on H i Γ, D0,b (W ) . Let k,k R : D0,b (W ) → Char (E, Vk (K))
be the morphism considered in [RS, Theorem 3.5], i.e. R (µ)e (P ) := µ P χWe up to the identification of Remark 5. By [RS, Theorem 3.5 and Lemma 2.8] it induces, for e = e± with e+ := e∞ , e− := e∞ , p−new k,k (23) Re± = R± := ρe± R : H 1 Γ, D0,b (W ) ' H 1 (Γ0 , Vk (K))c . c
Here (−)c is the cuspidal part. Indeed R induces a Hecke equivariant identification in Γ-cohomology and, as explained in [RS, §2.4-5], H 1 (Γ, Char (E, Vk (K))) has an Eisenstein/cuspidal decomposition. We are now ready to investigate the p-newness at k of an element of ds (24) H 1 Γ, C V ± , DU,k ' H 1 Γ, C E ± , D∗U,k ' H 1 Γ0 , DU,k (We± ) . Write 1 ν± Γ, C V ± , Dk,k k :H
1 → H 1 (Γ0 , Vk (K)) , η ± Γ, C V ± , DU,k k :H
→ H 1 (Γ0 , Vk (K))
to denote the composition of ν k and η k with (24). Theorem 13. The following assertions hold. p−new ± k,k (a± ) The OW (U ) H0± (pN + ) -modules H · Γ, C V ± , DU,k , Up and H i Γ, D0,b (W ) , Up± have a slope ≤ h decomposition such that ≤h H i Γ, Dk,k (W ) h ≥ k/2 0,b k,k H i Γ, D0,b (W ) = >h k,k H i Γ, D0,b h < k/2. (W ) p−new (a) There is also a slope ≤ h decomposition for H · Γ, C V, DU,k , Up . (b± ) For k/2 ≤ h < k + 1, ι± induces an isomorphism p−new ≤h −1 1 k,k (25) ∩ ρ±,p−new H Γ, D (W ) → ι± : H 1 Γ, C V ± , DU,k k 0,b c ≤h −1 p−new → H 1 Γ, C V ± , DU,k ∩ η± H 1 (Γ0 , Vk (K))c . k (b) For k/2 ≤ h < k + 1, ι induces an isomorphism p−new ≤h −1 1 k,k (26) ι : H 1 Γ, C V, DU,k ∩ ρp−new H Γ, D (W ) k 0,b c ≤h −1 k,k 1 U,k 1 → H Γ, C V, D ∩ (ρk ) ι H Γ, D0,b (W ) . c ≤h p−new U,k 1 1 is such that c := η ± . (c± ) Suppose that cU,k Γ, C V ± , DU,k ∗,± ∈ H k c∗,± ∈ H (Γ0 , Vk (K))c ≤h p−new −1 1 Then cU,k cU,k Γ, C V ± , DU,k provided by (b± ) is such that ∗,± := ι± ∗,± ∈ H ±,p−new U,k −1 ck,k := ρ c ∗,± ∗,± = R± (c) . k 12
≤h k,k = ι (char ) ∈ ι H 1 Γ, D0,b (W ) ,e cU,k := := ρk cU,k ∈ H 1 Γ, C V, DU,k and ck,k (c) If cU,k ∗ ∗ ∗ ∗ c p−new ≤h ∈ H 1 Γ, C V, DU,k provided by (b) is such that ι−1 cU,k ∗ p−new e e ck,k cU,k = char . ∗ := ρk ∗ k,k Proof. (a± − a) We have Up± := Wp V∓ = Wp on H i Γ, D0,b (W ) . Since Wp2 = pk (by (17)) and pk/2 ∈ K k,k (k is even), H i Γ, D0,b (W ) is the direct sum of the subspaces where Wp = ±pk/2 . The assertions relative k,k to H i Γ, D0,b (W ) , Up± follow. As H i Γ, Ik C V ± , DU,k has a slope ≤ h decomposition, (21) and p−new ± Lemma 10 imply that H · Γ, C V ± , DU,k , Up has a slope ≤ h decomposition too. The analogous p−new of (21) imply that H · Γ, C V, DU,k , Up has a slope ≤ h decomposition. (b± ) Consider the following commutative diagram R k,k H 1 Γ, D0,b (W ) ' H 1 (Γ, Char (E, Vk (K)))c c (27) ι± ↓ ≤h ds ≤h H 1 Γ, C V + , Dk,k ' H 1 Γ, C E + , Dk,k
ρe±
⊂
ρe±
'
≤h
H 1 (Γ0 (pN + ) , Vk (K)) ↑ νk H
1
Γ0 , D
k,k
W e±
≤h
,
obtained from the corresponding commutative diagram at the level of coefficients, then taking cohomology and slope ≤ h parts for h ≥ k/2. Here we note that ν ± As we have h ≥ k/2, k = ν k ρe± ds . p−new
H 1 (Γ0 (pN + ) , Vk (K))c
⊂ H 1 (Γ0 (pN + ) , Vk (K))
≤h
; here the fact that R is an isomorphism and ρe± is p−new k,k an inclusion such that the composition ρe± ◦ R identifies H 1 Γ, D0,b (W ) ' H 1 (Γ0 (pN + ) , Vk (K))c c
follows from [RS, Theorem 3.5 and Lemma 2.8]. The commutativity of the diagram implies that ι± is an inclusion and, when h < k + 1 so that ν k is an isomorphism by Theorem 12 (c), that we have −1 1 p−new ≤h k,k + (28) ι± H 1 Γ, D0,b (W ) = ν± H Γ pN , V (K) ∩ H 1 Γ, C V ± , Dk,k . 0 k k c c ≤h U,k 1 Γ, C V ± , DU,k with k/2 ≤ h < k + 1 is such that η ± ∈ Suppose now cU,k := cU,k ∗ ∗,± ∈ H k c∗ p−new k,k U,k H 1 (Γ0 (pN + ) , Vk (K))c . By (28), ck,k := ρ± ∈ ι± H 1 Γ, D0,b (W ) . Let cU,k be a co∗ ∗ k c∗ c k,k U,k k,k cyle representing cU,k := ρ± represents ck,k + bk,k = ι± ck,k ∗ , so that c∗ ∗ . Then we have c∗ ∗ k c∗ k,k k,k for some ck,k ∈ Z 1 Γ, D0,b (W ) representing ck,k ∈ H 1 Γ, D0,b (W ) and bk,k ∈ B 1 Γ, C V ± , Dk,k . ∗ c
Since the morphism ρ± is surjective, it induces an epimorphism between the boundaries. It follows k in (9) k,k U,k ± := cU,k + bU,k is that we may write b∗ = ρk b∗ for some bU,k ∈ B 1 Γ, C V ± , DU,k . Then cU,k ∗ ∗ ∗ ∗ U,k ± U,k 1 ± U,k k,k a cocyle in Z Γ, C V , D which represents c∗ such that ρk c∗ = ι± c . By definition of ±,p−new U,k ± U,k p−new 1 ± U,k p−new such that ρk C V ,D , it represents c∗ ∈ H Γ, C V , D cU,k = ck,k and ∗ ≤h U,k U,k U,k,≤h U,k,>h U,k,≤h 1 ± U,k p−new ι± cU,k = c . By (a) we may write c = c ⊕ c with c ∈ H Γ, C V , D ∗ ∗ ∗ ∗ ∗ ∗ >h p−new k,k ∈ H 1 Γ, C V ± , DU,k . Since the slope > h part of ck,k (in H 1 Γ, D0,b (W ) ) and cU,k,>h ∗ p−new ≤h are zero, cU,k,≤h and cU,k ∈ H 1 Γ, C V ± , DU,k is such that ρk±,p−new cU,k,≤h = ck,k and ∗ ∗ ∗ = cU,k ι± cU,k,≤h ∗ . It follows that (25) is a surjection. ∗ ± By (a ) and Theorem 12 (c), (21) induces ≤h p−new ≤h ρ±,p−new k,k k ... → H 1 Γ, Ik C V ± , DU,k → H 1 Γ, C V ± , DU,k → H 1 Γ, D0,b (W ) k ↓ ↓ ι± ≤h ρ± ≤h ≤h k 0 → H 1 Γ, Ik C V ± , DU,k → H 1 Γ, C V ± , DU,k → H 1 Γ, C V ± , Dk,k 13
k,k We already remarked that ι± is an inclusion on H 1 Γ, D0,b (W ) for h ≥ k/2. The above diagram implies c
that the middle arrow is an inclusion on the source of (25). (b) The proof is similar to (b± ). We leave details to the reader. (c± ) This is clear from (27). U,k e cU,k = ι ρp−new c . By assumption (c) Since we have cU,k = ι e cU,k , ι (char ) = ρk ι e ∗ ∗ ∗ ∗ k p−new U,k k,k k,k e char , ρk c∗ ∈ H 1 Γ, D0,b (W ) and the injectivity of ι on H 1 Γ, D0,b (W ) that was proved c c p−new U,k e in (b) implies char = ρk c∗ .
k,k Lemma 14. H 0 Γ, D0,b (W ) = 0. k,k Proof. By [RS, Theorem 3.5], R induces an identification H i Γ, D0,b (W ) ' H i (Γ, Char (E, Vk (K))) for i = 0, 1. By [G, Lemma 24], the spaces H · (Γ, Char (E, Vk (K))) sit into the Hecke equivariant long exact sequence 2 ... → H i (Γ, Char (E, Vk (K))) → H i (Γ0 , Vk (K)) → H i (Γ00 , Vk (K)) → ..., where Γ00 := Γ0 (N + ). Hence we need to show that
H 0 (Γ0 , Vk (K))
Γ0
Γ0
0
0
coresΓ0 ,Wp coresΓ0
→
H 0 (Γ00 , Vk (K))
2
is injective. When k > 0, both the source and the target are zero. When k = 0, Vk (K) = K and the Γ00 Γ00 composition with resΓ0 , resΓ0 Wp is the multiplication by 2p map, which is injective. ≤h 1 is a cuspidal new eigenvector. Let cU,k Γ, C V − , DU,k ∗,− ∈ H U,k be an eigenvector for the Hecke algebra H0 (pN + ) such that η − = c, as granted by Theorem 12 k c∗,− p−new ≤h 1 Γ, C V − , DU,k such that (c − d). By Theorem 13 (b± ) and (c± ), there is a unique cU,k ∗,− ∈ H U,k U,k −1 cU,k ρ−,p−new ∗,− = Re∞ (c) and ι− c∗,− = c∗,− . In particular, k U,k − − (29) ck,k (c) . cU,k = ι− ρk−,p−new e cU,k = ι− Re−1 ∗,− := ρk c∗,− = ρk ι− e ∗,− ∗,− ∞ new
Suppose that c ∈ H 1 (Γ0 , Vk (K))c
k/2 Let ap ∈ OW (U ) be the eigenvalue of Up on cU,k with wp ∈ {±1}, ∗,− . Note that, as we have ap (k) = −wp p ap is invertible, up to shrinking U in a neighborhood of k. According to the discussion following Remark 11, we may further assume that ap is a multiplicative unit in OW (U ). U,k 1 Definition 15. With the above notations, set cU,k := cU,k Γ, C V, DU,k , where cU,k ∗ ∗,+ ⊕ c∗,− ∈ H ∗,+ := k/2 −1 U,k − −wp p ap c∗,− | V . ≤h −1 k,k (W ) and there is a unique e cU,k ∈ Corollary 16. cU,k ∈ H 1 Γ, C V, DU,k ∩ (ρk ) ι H 1 Γ, D0,b ∗ ∗ c ≤h −1 p−new k,k H 1 Γ, C V, DU,k ∩ ρkp−new H 1 Γ, D0,b (W ) such that ι e cU,k = cU,k ∗ . Furthermore, ∗ c e cU,k is an eigenvector for the V operator with eigenvalue −wp p−k/2 ap such that e ck,k := ρkp−new e cU,k = ∗ ∗ p−new U,k Re−1 (c). Whenever e c∗ ∈ Z 1 Γ, C V, DU,k is a representing cocyle, we may write ∞ e cU,k | 1 − V = 1 + wp p−k/2 ap e cU,k + dµU,k ∗ ∗ ∗
for some µU,k ∈ C V, Ik DU,k . ∗ 14
Proof. We have
U,k cU,k ∗,+ , c∗,−
0 V−
V+ 0
− U,k + . = cU,k ∗,− | V , c∗,+ | V
− −k/2 By definition cU,k ap cU,k ∗,− | V = −wp p ∗,+ . By Proposition 8 and (17)
cU,k ∗,+
| =
− + −k/2 −1 U,k bp Wp−1 Up V+ = −wp pk/2 ap−1 cU,k ap c∗,− | Wp U ∗,− | V V = −wp p U,k 2 −k/2 −wp p−k/2 a−1 ap cU,k p c∗,− | Up = −wp p ∗,− .
+ − −k/2 . Since cU,k ap cU,k | V = −wp p−k/2 ap cU,k | V − ) = ρ− It follows that cU,k ∗ ∗,− | V = −wp p ∗,+ andρk (− k (−) | ∗ U,k U,k U,k − + + −1 − − − V (by (18)), ρk c∗,− | V = ρk c∗,+ . By (29) and ι− (−) | V = ι+ (−), ρk c∗,+ = ι− Re∞ (c) | U,k = ι Re−1 V− = ι+ Re−1 (c) . It follows that ρ c (c) . We already know that cU,k ∗ k ∗,− is a slope ≤ h eigen∞ ∞ ≤h vector for the Up -operator with eigenvalue the multiplicative unit ap ; since H 1 Γ, C V, DU,k is OW (U )torsion free (by Theorem 12 (c)), by Remark 11 |ap | ≥ p−h . By (17), Wp2 = pk on H 1 Γ, C V, DU,k , which ± is the direct sum of the eigenspaces H 1 Γ, C V, DU,k on which Wp = ±pk/2 . By definition Up = Wp V and, using Proposition 8 as in (22), one may check that Wp V = VWp . Suppose that x is an eigenvec± tor for the V operator, say x | V = ax. If we write x = x+ + x− with x± ∈ H 1 Γ, C V, DU,k , the ± 1 U,k ± ± ± equality Wp V = VWp implies that x | V ∈ H Γ, C V, D and, hence, x | V = ax . Since Up = Wp V, x± | Up = ±pk/2 ax± . This remark, applied to x = cU,k and a = −wp p−k/2 ap , implies that ∗ ≤h U,k,± U,k,± −h c∗ | Up = ∓wp ap c∗ . Since |∓wp ap | = |ap | ≥ p , by Remark 11, cU,k,± ∈ H 1 Γ, C V, DU,k ∗ ≤h and, hence, cU,k = cU,k,+ + cU,k,− ∈ H 1 Γ, C V, DU,k . ∗ ∗ ∗ We may now apply Theorem 13 (b − c), that gives the existence of a unique e cU,k such that ι e cU,k = cU,k ∗ ∗ ∗ e (c). Using the relation Up V = VUp , it is easy to see that both the source and ρp−new cU,k = Re−1 ∗ k ∞
and the target of (26) are stable under the V operator. The V-equivariance of ι implies that e cU,k |V= ∗ U,k U,k −k/2 eU,k −k/2 −wp p ap c∗ . It follows that we have e c∗ | 1 − V = 1 + wp p ap e c∗ and, hence, that we may write e cU,k | 1 − V = 1 + wp p−k/2 ap e cU,k + dµU,k ∗ ∗ ∗ for some µU,k ∈ C V, DU,k ∗ 0, we see that dµk,k
p−new
k,k . As we have 1−V = 0 on D0,b (W ) (by definition) and 1+wp p−k/2 ap (k) = k,k U,k k,k 0 = 0, where µk,k := ρp−new µ . It follows that µ ∈ H Γ, D (W ) = 0 (Lemma ∗ k 0,b
14).
p−new ≤h For the rest of the paper we will write cU,k ∈ H 1 Γ, C V, DU,k to denote the lift provided by ∗ Corollary 16 and ρk = ρp−new . k 5. p-adic integration pairings If K 0 /K is a finite extension, we let AΥ,k (W, K 0 ) be the space obtained working over K 0 and we employ a similar notation for the corresponding modules of distributions. 5.1. Review of the pairings Φlog and Φord from [RS]. As explained in the introduction we may define GL2 (Qp )-invariant K-bilinear pairings Φlog , Φord : D0k,k (W ) ⊗ Div0 Hpur ⊗ Pk (K) → K as follows. Let τ 1 , τ 2 ∈ Hpur Qur p , P ∈ Pk (K) and let K (τ 1 , τ 2 ) be the field generated by K, τ 1 and τ 2 . Then x − τ 2y P (x, y) ∈ Ak,k (W, K (τ 1 , τ 2 )) . θτ 2 −τ 1 ,P := log x − τ 1y 15
By linearity the association (τ 2 − τ 1 , P ) 7→ θτ 2 −τ 1 ,P induces Div0 Hpur ⊗ Pk (K) → Ak,k (W, K) = AU,k (W ) d⊗P
7→ θd,P .
Setting, for d ⊗ P ∈ Div0 Hpur ⊗ Pk (K), Φlog (µ, d ⊗ P ) := µ θd,P ∈ K yields, up to the identification of Remark 5, the paring Φlog which is considered in [RS] (where the standard choice log0 := log (h−i) of the p-adic logarithm has been made). Up to the identification of Remark 5, the pairing Φord is given by P Φord (µ, τ 2 − τ 1 ⊗ P ) := e:red(τ 1 )→red(τ 2 ) µ P χWe . By cap product these pairings induce k,k Φlog , Φord : H 1 Γ, D0,b (W ) ⊗ H1 Γ, Div0 Hpur ⊗ Pk (K) → K. 5.2. The indefinite p-adic integration pairing. Let C V, D0U,k ⊂ C V, DU,k be the sub OW (U ) p−new U,k ∈ D0k,k (W ) for every v ∈ V, so that C V, DU,k module of those maps µU,k such that µk,k ∗ v := ρk µ∗ v ⊂ C V, D0U,k . We are going to define a GL2 (Qp )-invariant K-bilinear pairing Φind : C V, D0U,k ⊗ Div Hpur ⊗ Pk (K) → K as follows. Let τ ∈ Hpur Qur p , P ∈ Pk (K) and let K (τ ) be the field generated by K and τ . Then κ−k
Θτ ,P (κ, (x, y)) := hx − τ yi
P ∈ AU,k (W, K (τ )) .
Let D0U,k (W ) ⊂ DU,k (W ) the sub OW (U )-module of those µU,k such that µk,k := ρk µU,k ∈ D0k,k (W ). d [−]|κ=k . We first define the K-bilinear pairing Set ∇k := dκ Ψind : D0U,k (W ) ⊗ Div Hpur ⊗ Pk (K) → K d U,k d,P Ψind µU,k , d ⊗ P := ∇k µU,k Θd,P = µ Θ . |k dκ In the following discussion we evaluate µU,k ∈ D0U,k (W ) at F ∈ AU (W, K (τ )) by viewing it as the element 1 ⊗ µU,k ∈ D0U,k (W, K (τ )) = K (τ ) ⊗K D0U,k (W ). Hence, the expression Ψind µU,k , τ ⊗ P makes sense for any τ ∈ Hpur . P We now define, for d = τ nτ τ ∈ Div Hpur , P P U,k U,k τ ,P ind Φind µU,k , d ⊗ P := n Ψ µ , τ ⊗ P = ∇ n µ Θ . τ k τ ∗ W τ τ red(τ ) red(τ ) As in [BDI, Proposition 4.9] one checks that Ψind is GL2 (Qp )-invariant. Exploiting this fact the following lemma is easily established. Lemma 17. Φind µU,k , d ⊗ P ∈ K and Φind induces a GL2 (Qp )-invariant K-bilinear pairing ∗ C V, D0U,k ⊗ Div Hpur ⊗ Pk (K) → K. We end this section by recording two useful lemmas. The first result is [BDI, Lemma 4.11], while the second result is implicit in the proof of [BDI, Lemma 4.10]. Lemma 18. Let α ∈ OW (U ) and let µU,k ∈ DU,k (Y ), with Y ⊂ W open and Q× p -stable. Then U,k τ ,P 0 k,k U,k τ ,P ∇k αµ Θ χY = α (k) µ (P χY ) + α (k) ∇k µ Θ χY . ur Lemma 19. Let µU,k ∈ DU,k (Y ), with Y ⊂ W open and Q× p -stable, and let τ 1 , τ 2 ∈ Hp Qp . Then ∇k µU,k Θτ 2 ,P χY − ∇k µU,k Θτ 1 ,P χY = µk,k θτ 2 −τ 1 ,P χY . 16
6. Proof of the main results 6.1. Proof of the key proposition. In this subsection we prove the following key proposition, whose proof should be compared with [BDI, Proposition 4.10]. Proposition 20. The equality log (c) , z − 2wp p−k/2 a0p (k) Φord Re−1 (c) , z Φind cU,k Re−1 ∗ , iz = Φ ∞ ∞ p−new ≤h ∈ H 1 Γ, C V, DU,k is holds for z ∈ H1 Γ, Div0 Hpur ⊗ Pk , where i comes from (2) and cU,k ∗ given by Corollary 16. Let X· be a resolution of Z by projective Z [Γ]modules, that we use both to compute the homology and 0 ur the cohomology. We write z ∈ Z1 Γ, Div Hp ⊗ Pk to denote a cycle that represents z, that we may P write as z = h,i,j,l xh ⊗Γ τ i − τ j ⊗ Pl .with xh ∈ X1 , τ i − τ j ∈ Div0 H Qur and Pl ∈ Pk (K). We may p further assume that, for every (i, j), the vertices v := red (τ ) and v := red (τ ) i i j j are adjacent. U,k U,k 1 U,k p−new We write c∗ ∈ Z Γ, C V, D for a cocycle which represents c∗ , so that ck,k := ρk cU,k ∗ represents Re−1 (c) by Corollary 16. By definition of cap product ∞ P ind Φind cU,k = h,i,j,l Φind cU,k cU,k ∗ , iz ∗ (xh ) , τ i ⊗ Pl − Φ ∗ (xh ) , τ j ⊗ Pl P τ i .Pl τ j .Pl = h,i,j,l ∇k cU,k − ∇k cU,k vi (xh ) Θ vj (xh ) Θ P τ i .Pl τ j .Pl = h,i,j,l ∇k cU,k − ∇k cU,k vi (xh ) Θ vi (xh ) Θ P τ j .Pl τ j .Pl + h,i,j,l ∇k cU,k − ∇k cU,k . vi (xh ) Θ vj (xh ) Θ k,k By Lemma 19 applied to µU,k = cU,k (xh ), and the equality ck,k = vi (xh ), whose specialization is c (c), P U,k τ i .Pl τ j .Pl (c) , z . − ∇k cU,k = Φlog Re−1 vi (xh ) Θ h,i,j,l ∇k cvi (xh ) Θ ∞
Re−1 ∞
It follows that we have log Φind cU,k Re−1 (c) , z + A cU,k ∗ , iz = Φ ∗ , z , where ∞ P U,k τ j .Pl τ j .Pl A cU,k − ∇k cU,k . ∗ , z := vj (xh ) Θ h,i,j,l ∇k cvi (xh ) Θ Lemma 21. If e = (v1 , v2 ), µU,k ∈ C V, DU,k and F ∈ AU,k (W, K 0 ) where K 0 /K is a finite extension ∗ µv1 − µv2 F χW(v ,v ) = (µ | 1 − V)v1 F χW(v ,v ) , 1 2 1 2 µv1 − µv2 F χW(v ,v ) = − (µ | 1 − V)v2 F χW(v ,v ) . 2
1
2
1
P
Proof. We have (µ | V)v1 = µv2 |W(v ,v ) + d(v0 ,v1 )=1,v0 6=v2 µv0 |W v ,v0 . Since F χW(v ,v ) is supported on 1 2 ( 1 ) 1 2 W(v1 ,v2 ) , (µ | V)v1 F χW(v ,v ) = µv2 F χW(v ,v ) . Hence we find 1
2
1
2
µv1 − µv2 F χW(v ,v ) = µv1 F χW(v ,v ) − µv2 F χW(v ,v ) 1 2 1 2 1 2 = µv1 F χW(v ,v ) − (µ | V)v1 F χW(v ,v ) = (µ | 1 − V)v1 F χW(v 1
2
1
1 ,v2 )
2
.
We also have µv1 − µv2
F χW(v
2 ,v1 )
= − µv2 − µv1
F χW(v
2 ,v1 )
= − (µ | 1 − V)v2 F χW(v
2 ,v1 )
.
17
U,k p−new As we assume that vi and vj are adjacent, by Lemma 21 applied to cU,k and ∗ (xh ) ∈ C V, D e = (vi , vj ), P U,k U,k τ j .Pl A cU,k , z = ∇ c (x ) − c (x ) Θ k h h ∗ vi vj h,i,j,l P U,k = cvi (xh ) − cU,k Θτ j .Pl χW v ,v vj (xh ) h,i,j,l ∇k ( i j) P U,k U,k τ j .Pl + h,i,j,l ∇k cvi (xh ) − cvj (xh ) Θ χW v ,v ( j i) P τ j .Pl U,k = ∇ Θ χ c (x ) | 1 − V k h W ∗ h,i,j,l vi (vi ,vj ) P τ j .Pl − h,i,j,l ∇k cU,k . χW v ,v ∗ (xh ) | 1 − V vj Θ ( j i) | 1 − V (xh ) = cU,k By definition cU,k ∗ (xh ) | 1 − V, so that Corollary 16 yields ∗ A cU,k ∗ ,z = P τ j .Pl = h,i,j,l ∇k 1 + wp p−k/2 ap cU,k (xh ) Θτ j .Pl χW v ,v χW v ,v + ∇k dµU,k vi (xh ) Θ v i ( i j) ( i j) P −k/2 U,k τ j .Pl U,k τ j .Pl − h,i,j,l ∇k 1 + wp p ap cvj (xh ) Θ χW v ,v − ∇k dµvj (xh ) Θ χW v ,v (30) (
j
i)
−k/2
(
j
i)
−k/2
Note that α (k) = 0 with α = 1 + wp p ap . Applying Lemma 18 to α = 1 + wp p ap and µU,k = U,k k,k k,k (xh ) (resp. cvj (xh )), whose specialization is c (xh ) (resp. c (xh )) yields P −k/2 U,k τ j .Pl ∇ 1 + w p a c = (x ) Θ χ k p p h W(v ,v ) vi h,i,j,l i j P = h,i,j,l wp p−k/2 a0p (k) ck,k (xh ) Pl χ(vi ,vj ) , P −k/2 U,k τ j .Pl ∇ 1 + w p a c (x ) Θ χ = k p p h W(v ,v ) vj h,i,j,l j i P = h,i,j,l wp p−k/2 a0p (k) ck,k (xh ) Pl χ(vj ,vi ) . (31) Since ck,k (xh ) ∈ D0k,k (W ) we see that ck,k (xh ) Pl χ(vi ,vj ) = −ck,k (xh ) Pl χ(vj ,vi ) . Hence, by (30),
cU,k vi
(c), (31), the fact that we assume that vi and vj are adjacent and the equality ck,k = Re−1 ∞ P A cU,k = −2wp p−k/2 a0p (k) h,i,j,l ck,k (xh ) Pl χ(vj ,vi ) + ∆ dµU,k ∗ ,z ∗ ,z U,k −1 −k/2 0 ord = −2wp p ap (k) Φ Re∞ (c) , z + ∆ dµ∗ , z , where P P τ j .Pl τ j .Pl ∆ dµU,k := h,i,j,l ∇k dµU,k χW v ,v − h,i,j,l ∇k dµU,k (x ) Θ χ . ∗ ,z h W vi (xh ) Θ v j ( i j) (vj ,vi ) By Corollary 16 we have µU,k ∈ C V, Ik DU,k . As we are going to explain, we may realize ∆ dµU,k ∗ ∗ ,z as arising from a Γ-invariant pairing ∆ : C V, Ik DU,k ⊗ Div0 Hpur ⊗ Pk (K) → K. U,k Then we find ∆ dµU,k ∗ , z = ∆ µ∗ , dz = 0, thus proving our proposition. We define ∆ by the rule P U,k U,k τ t(e) ,P τ t(e) ,P , ∆ µU,k , τ − τ ⊗ P := ∇ µ Θ χ − ∇ µ (x ) Θ χ 1 2 k k h We We ∗ e:v1 →v2 s(e) t(e) where τ t(e) ∈ Hp Qur is any element such that red τ t(e) = t (e) and vi := red (τ i ) . p comes from this pairing, if well defined and ΓIt is then clear that the above quantity ∆ dµU,k ∗ ,z invariant. This is the content of the subsequent lemma. Lemma 22. The pairing ∆ is well defined, K-valued and GL2 (Qp )-invariant. Proof. To see that the pairing is well defined we have to show that: 18
A the definition does not depend on the choice of the τ t(e) s; U,k B ∆ µ∗ , τ 1 − τ 2 ⊗ P + ∆ µU,k ∗ , τ 2 − τ 1 ⊗ P = 0; U,k U,k C ∆ µU,k ∗ , τ 1 − τ 3 ⊗ P = ∆ µ∗ , τ 1 − τ 2 ⊗ P + ∆ µ∗ , τ 2 − τ 3 ⊗ P . To see A we may assume that v1 and v2 are adjacent and we write e = (v1 , v2 ). If τ and τ 0 both reduces to v2 , we have to show the equality 0 0 τ ,P U,k τ ,P τ ,P U,k τ ,P U,k Θ χ ∇k µU,k + ∇ µ Θ χ − ∇ Θ χ = 0. µ Θ χ − ∇ µ k k k We We We We v2 v1 v2 v1 Indeed, as we have µk,k vi = 0, by Lemma 19, 0 ∇k µU,k Θτ ,P χWe Θτ ,P χWe − ∇k µU,k v1 v1 0 Θτ ,P χWe ∇k µU,k − ∇k µU,k Θτ ,P χWe v2 v2
0 θτ −τ ,P χWe = 0, µk,k v1 0 θτ −τ ,P χWe = 0. = µk,k v2
=
To see B we may assume that v1 and v2 are adjacent and we write e = (v1 , v2 ). Then we have to show that ∇k µU,k Θτ 2 ,P χWe − ∇k µU,k Θτ 2 ,P χWe + ∇k µU,k Θτ 1 ,P χWe − ∇k µU,k Θτ 1 ,P χWe = 0. v1 v2 v2 v1 Indeed, as we have µk,k vi = 0, the claim follows by Lemma 19 similarly as above. Now C follows from the definition and A, as we may take the same lifts τ t(e) to compute the left and the right hand side. We leave to the reader to check that the pairing is K-valued and GL2 (Qp )-invariant. 6.2. Proof of the main result. We are now ready to prove the main result of the paper. Theorem 23. We have −
0
LN (f ) = 2wp p−k/2 a0p (k) = −2 (log ap ) (k) . Proof. Let c = cf be the new eigenvector which corresponds to f as explained in the introduction. By Proposition 20 and the exactness of (2) Φlog Re−1 (c) , ∂p − 2wp p−k/2 a0p (k) Φord Re−1 (c) , ∂p = Φind cU,k ∗ , i∂p = 0. ∞ ∞ The defining property (3) gives the claim.
We are now going to explain how [RS, Theorem 4.7], namely the subsequent (32), follows from Theorem − − 23. We first recall that [RS, Corollary 3.13] gives a unique LN ∈ H0N (pN + ) ⊗Q Qp such that (3) holds on − × HN (K). By the approximation Theorem, for every integer d ∈ N prime to N − , there exists ad ∈ B+ such − that det (ad ) = d and ad Γ0 (dM ) a−1 ⊂ Γ (M ), where M is an integer prime to N . It follows that, for any 0 d field F/Q, there are well defined operators − − | ad : Sk (Γ0 (M )) → Sk (Γ0 (dM )) , | ad : H · Γ0 (M ) , VkN (F ) → H · Γ0 (dM ) , VkN (F ) . We note that these morphisms commute with the Hecke correspondences Tl or Ul (belonging to the appropriate Hecke algebra) at the primes l - d. p−new Let us now focus on the space Sk (Γ0 (pN + )) of p-new cusp forms on B and let M + | N + . For every + new eigenvector g ∈ Sk (Γ0 (pM )) let us write [g] to denote the union of the eigenclasses of eigenvectors with eigenvalue σ ◦ λg , where λg is the eigenpacked attached to g and σ any automorphism of C. Write C [g] (resp. C [g]) to denote the C-vector space generated by {g | ad : g ∈ [g] , d | N + /M + } (resp. {g : g ∈ [g]}). By the Atkin-Lehner theory (see [AtLe]) and the Jacquet-Langlands correspondence, we may write p−new L new Sk Γ0 pN + = M + |N + proper,g∈Sk (Γ0 (pM + ))new C [g] ⊕ Sk Γ0 pN + , # d:d|N + /M + } where C [g] ' C [g] { as Hecke modules for Tl or Ul with l - N + /M + . There is a similar decomp−new position for the p-adic representation Vk (Γ0 (pN + )) attached to the space of p-new modular forms on p−new + Γ0 (pN ) (defined, for example, in [J] and [CI]). The associated monodromy module Dk (Γ0 (pN + )) at p 19
#{d:d|N + /M + } p−new ' Dk (Γ0 (pM + ))[g] .
p−new
decomposes as the sum of its new part and the sum of the Dk (Γ0 (pN + ))[g] Then one finds the following lemma.
Lemma 24. LDk (Γ0 (pN + ))p−new decomposes as the direct sum of its new part LDk (Γ0 (pN + ))new and the sum #{d:d|N + /M + } of the LDk (Γ0 (pN + ))p−new ' LD (Γ (pM + ))p−new . [g]
k
0
[g]
−
−
There is a similar statement for the L-invariant LN = LN (pN + ) (here we need to add the level in the notation). Indeed ad gives a morphism | ad : H 1 Γ1/p M + , Char (E, Vk ) → H 1 Γ1/p dM + , Char (E, Vk ) for every d | N + /M + and, as above, we write again | ad for the composition of this morphism with the restriction from Γ1/p (dM + ) to Γ1/p (N + ). There is a corresponding morphism in homology, induced by ad , going in the opposite direction. Exploiting the compatibility of these | ad with the cap product it is not difficult to show the following lemma. −
−
new
Lemma 25. LN (pN + ) decomposes as the direct sum of its new part LN (pN + ) #{d:d|N + /M + } − − LN (pN + )[g] ' LN (pM + )[g] .
and the sum of the
0
If g ∈ [g] is a newform of level pM + , Theorem 23 together with the equality LF M (g) = −2 (log ap ) (k) #{d:d|N + /M + } #{d:d|N + /M + } − − proved in [Cz1] implies that LF M (g) = LN (g). We deduce that LD (Γ (pM + ))p−new = LN (pM + )[g] k
0
[g]
and, by Lemmas 24 and 25, −
−
. LN = LN Dk (Γ0 (pN + ))p−new
(32)
−
p−new
6.3. Application to the p-adic Abel-Jacobi map. As explained in [RS, §4.2], (32) implies D ' DkN (Γ0 ) where D is the monodromy module over H0 (pN + ) ⊗Q Qp which is constructed in [RS, §4.2]. By construction − ∨,w its L-invariant is LD = LN . Its base change to K is built from the K-vector space H (K)c ∞ . As we have H1 (Γ, Pk (K)) = 0 (see [RS, Lemma 3.10]), (2) induces an identification H1 Γ, Div0 Hpur ⊗ Pk (K) ' i: → H1 Γ, Div Hpur ⊗ Pk (K) . Im ∂ AJ The p-adic Abel-Jacobi map Φ and its logarithm log ΦAJ are then defined by the following commutative diagram: ΦAJ : AJ
log Φ
H1 Γ, Div Hpur ⊗ Pk (K) k : H1 Γ, Div Hpur ⊗ Pk (K)
i−1
H1 (Div0 (Hur p )⊗Pk (K)) Im ∂
−Φlog ⊕Φord
i−1
H1 (Div0 (Hur p )⊗Pk0 ) Im ∂
Φlog −LΦord
'
→
k
'
→
D/F k/2 D k fD ∨,w∞
H (K)c
,
where fD is fD (x, y) = −x − LD y. Suppose for simplicity that f ∈ Sk (Γ0 ) is a new eigenvector. Then we − find D[f ] ' DkN (Γ0 )[f ] , where D[f ] is a quotient which is built from D as explained in [RS, §4.3]. We may therefore consider the [f ]-isotypic component of the above Abel-Jacobi map. As explained in [Se1, §2.3], assuming that K contains the eigenvalues attached to each eigenclass belonging to [f ], the tangent space ∨,w ∨,w D[f ] /F k/2 D[f ] = H (K)[f ] ∞ splits into f -components Df /F k/2 Df = H (K)f ∞ one for each f ∈ [f ]. Let ≤h ∨,w 1 U,k p−new cf a basis for the one dimensional K-vector space H (K)f ∞ and let cU,k ∈∈ H Γ, C V, D ∗,f c be a associated family as in Corollary 16. The following proposition is a direct consequence of Proposition 20, in view of Theorem 23. Proposition 26. For every y ∈ H1 Γ, Div Hpur ⊗ Pk (K) , log ΦAJ (y) (cf ) = Φind cU,k , y ∈ K. ∗,f 20
,
7. Appendix: base point free Shapiro’s Lemma Assume we have given the following data. SH1 A group G and a family of couples (Γe , Ae ) indexed by a set E such that Γe ⊂ G is a subgroup and Ae is a Γe -module. SH2 An action of G on E such that gΓe1 g −1 ⊂ Γge1 (then gΓe1 g −1 = Γge1 ). SH3 A family of group morphisms, for every g ∈ G and every e ∈ E, ϕg,e : Ae → Age . They are required to be functorial, i.e. we assume SH3a: ϕ1,e = 1Ae ; SH3b: ϕg2 g1 ,e = ϕg2 ,g1 e ◦ ϕg1 ,e . They are also required to be compatible with the conjugation γ g : Γe → Γge , i.e. we assume SH3c: ϕg,e (γa) = gγg −1 ϕg,e (a) for every γ ∈ Γe and every a ∈ Ae . Finally we assume that they are compatible with the action of Γe on Ae , i.e. we assume SH3d: ϕγ,e = γ acting on Ae for every e ∈ E and every γ ∈ Γe . Thanks to SH3a and SH3b, the module of sections F C (E, A∗ ) = s : E → e∈E Ae : s (e) ∈ Ae is a G-module by the rule (gs) (e) = ϕg,g−1 e s g −1 e ∈ Ae . Fix a distinguished element e∗ ∈ E and consider the map obtained by evaluting sections at e∗ : ρe∗ : C (E, A∗ ) → Ae∗ , ρe∗ (s) = s (e∗ ) . It is indeed a Γe∗ -module morphism, in light of SH3d. Proposition 27. The morphism ρe∗ induces in cohomology canonical morphisms ρe∗ : H · (G, C (E, A∗ )) → H · (Γe∗ , Ae∗ ) , that are isomorphisms if we assume SH4: the action of G on E is transitive and Γe = StabG (e) for some e ∈ E (thus Γe = StabG (e) for every e ∈ E). Proof. It is indeed easily checked that C (E, A∗ ) = IndG Γe∗ (Ae∗ ). For our purposes we need to make the isomorphism explicit. This is the aim of the subsequent discussion. Suppose we have given a G-module X and our data G, Γ∗ , A∗ , ϕ∗,∗ , E . Then we may consider the data G, Γ∗ , Hom (X, A)∗ , ϕX ∗,∗ , E defined as follows: • Hom (X, A)e := Hom (X, Ae ) as a Γe -module; −1 X . • ϕX g,e : Hom (X, Ae ) → Hom (X, Age ) is ϕg,e (f ) := ϕg,e ◦ f ◦ g Then G, Γ∗ , Hom (X, A)∗ , ϕX ∗,∗ , E satisfy SH1, SH2, SH3a, SH3b, SH3c and SH3d as well as SH4, if G, Γ∗ , A∗ , ϕ∗,∗ , E satisfy this further property. Lemma 28. For every G-module X there is a canonical G-isomorphism Hom (X, C (E, A∗ )) = C (E, Hom (X, A)∗ ) . Let X· be a resolution of Z by projective Z [G]-modules. By Lemma 28, the complex C (E, Hom (X· , A)∗ ) computes the cohomology groups H · (G, C (E, A∗ )). On the other hand, since X· is also a resolution of Z by projective Z [Γe∗ ]-modules, the complex Hom (X· , A)e∗ = Hom (X, Ae∗ ) computes the cohomology groups H · (Γe∗ , Ae∗ ). Assuming SH4, the isomorphism ρe∗ of Proposition 27 is then induced by the isomorphism of complexes G '
ρ = ρe∗ : C (E, Hom (X, A)∗ ) → HomΓe∗ (X, Ae∗ ) , whose inverse is given by '
G
s = se∗ : HomΓe∗ (X, Ae∗ ) → C (E, Hom (X, A)∗ ) , s (f ) (e) := ϕX (f ) = ϕge−1 ,e∗ ◦ f ◦ ge , g −1 ,e e
where ge ∈ G is any element such that ge e = e∗ . 21
∗
7.1. Application. The above formalism applies to the GL2 (Qp )-modules C E, D∗Υ,k (resp. C V, DΥ,k ) Υ,k ± (resp. C V ± , DΥ,k ). More precisely, for G ⊂ GL2 (Qp ) (resp. and the GL+ (Q )-modules C E , D ∗ p 2 Υ,k (resp. C E ± , D∗Υ,k ) as a G-module G ⊂ GL+ 2 (Qp )), the data G, Γ∗ , A∗ , ϕ∗,∗ , E that define C E, D∗ are defined by taking Ae := DΥ,k (We ), which is a Γe -module for any Γe ⊂ StabG (e), and setting ϕg,e := g : DΥ,k (We ) → DΥ,k (Wge ) , where g is (6) and we note that gWe = Wge . The data defining the G-modules C V, DΥ,k and C V ± , DΥ,k are defined in a similar way (Av := DΥ,k (W ) and ϕg,v := g). Υ,k ± In particular, when G = Γ (⊂ GL+ 2 (Qp ) by means of Bp = M2 (Qp )) and Γe = StabΓ (e), C E , D∗ satisfies SH4. For a projective resolution X· of Z by projective Z [Γ]-modules, we have ' Γ (33) s = se∗ : HomΓe∗ X· , DΥ,k (We∗ ) → C E ± , Hom X· , DΥ,k ∗ , s (f ) (e) := ge−1 ◦ f ◦ ge , where ge ∈ Γ is any element such that ge e = e∗ . References [AS] A. Ash and G. Stevens, p-adic deformation of arithmetic cohomology. Draft dated 29.09.2008. [AtLe] A. O. Atkin and J. Lehner, Hecke operators on Γ0 (m), Math. Ann. 185 (1970), 134–160. [BD1] M. Bertolini and H. Darmon, Hida families and rational points on elliptic curves, Invent. Math. 168 (2007), no. 2, 371-431. [BD2] M. Bertolini and H. Darmon, The rationality of Stark-Heegner points over genus fields of real quadratic fields, Annals of Math. 170 (2009), 343-369. [BDI] M. Bertolini, H. Darmon and A. Iovita, Families of automorphic forms on definite quaternion algebras and Teitelbaum’s conjecture, Ast´ erisque 331 (2010), 29-64. [BDIS] M. Bertolini, H. Darmon, A. Iovita and M. Spieß M., Teitelbaum’s exceptional zero conjecture in the anticyclotomic setting, Am. J. Math. 124 (2002), 411–449. [BGR] S. Bosch, U. Guntzer and R. Remmert, Non-Archimedean analysis, Springer-Verlang, 1984. [Br] C. Breuil, S´ erie sp´ eciale p-adique et cohomologie ´ etale compl´ et´ ee. Preprint available at: http://www.ihes.fr/ breuil/publications.html. [C] R. F. Coleman, A p-adic Shimura isomorphism and p-adic periods of modular forms. In: p-adic monodromy and the Birch and Swinnerton-Dyer conjecture (Boston, MA, 1991), 21-51, Contemp. Math. 165, Amer. Math. Soc., Providence, RI, 1994. [CI] R. F. Coleman and A. Iovita, Hidden Structures on curves. Submitted. [Cz1] P. Colmez, Invariants L et d´ eriv´ ees de valeurs propres de frobenius, Ast´ erisque 331 (2010), 13-28. [Cz2] P. Colmez, Repr´ esentations semi-crystallines et s´ erie principale pour GL2 (Qp ). In preparation. [Dar] H. Darmon, Integration of Hp × H and arithmetic applications, Ann. of Math. 154 (2001), no. 2, 589-639. [Das] S. Dasgupta, Stark-Heegner points on modular Jacobians, Ann. Sci. Ecole Norm. Sup. (4) 38 (2005), 427-469. [DG] S. Dasgupta and M. Greenberg M., L-invariants and Shimura curves. Submitted. [dS] E. de Shalit, Eichler cohomology and periods of modular forms on p-adic Schottky groups, J. Reine Angew. Math. 400 (1989), 3–31. [E] M. Emerton, p-adic L-functions and unitary completions of representations of p-adic reductive groups, Duke. Math. J. 130 (2005), no. 2, 353-392. [G] Greenberg M., Stark-Heegner points and the cohomology of quaternionic Shimura varieties, Duke. Math. J. 147 (2009), no. 3, 541-575. [GSS] Greenberg M., Seveso M. A. and Shahabi S., p-adic L-functions, p-adic Jacquet-Langlands, and arithmetic applications. Submitted. [GS] R. Greenberg and G. Stevens, p-adic L-functions and p-adic periods of modular forms, Invent. Math. 111 (1993), 407-447. [H] H. Hida, Elementary theory of L-functions and Eisenstein series, London Math. Soc. Student Texts 26 (1993). [IS] A. Iovita and M. Spieß M, Derivatives of p-adic L-functions, Heegner cycles and monodromy modules attached to modular forms, Invent. Math. 154, 333-384 (2003). [J] A. J. Scholl, Motives for modular forms, Invent. Math. 100 (1990), 419-430. [LRV] M. Longo, V. Rotger and S. Vigni, On rigid analytic uniformizations of Jacobians of Shimura curves, to appear in the Am. J. Math. [M] B. Mazur, On monodromy invariants occurring in global arithmetic, and Fontaine’s theory. In: p-adic monodromy and the Birch and Swinnerton-Dyer conjecture (Boston, MA, 1991), 1-20, Contemp. Math. 165, Amer. Math. Soc., Providence, RI, 1994. [MTT] B. Mazur, J. Tate, J. Teitelbaum, On p-adic analogues of the conjecture of Birch and Swinnerton-Dyer, Invent. Math. 84 (1986), 1-48. [O] L. Orton, An elementary proof of a weak exceptional zero conjecture, Can. J. Math. 56 (2004), no. 2, 373-405. 22
V. Rotger and M. A. Seveso, L-invariants and Darmon cycles attached to modular forms. To appear in J. European Math. Soc. [Se1] M. A. Seveso, p-adic L-functions and the rationality of Darmon cycles, Can. J. Math., published electronically on October 5, 2011, doi:10.4153/CJM-2011-076-8. [Se2] M. A. Seveso, Heegner cycles and derivatives of p-adic L-functions. To appear in J. Reine Angew. Math. [Sh] G. Shimura, Introduction to the Arithmetic Theory of Automorphic functions, Princeton Univ. Press, Princeton, N.J., 1971. [T1] J.T. Teitelbaum, Values of p-adic L-functions and a p-adic Poisson kernel, Invent. Math. 101 (1990), no. 2, 395-410. [T2] J. T. Teitelbaum, Modular representations of PGL2 and automorphic forms for Shimura curves, Invent. Math. 113 (1993), 561-580. E-mail address:
[email protected] [RS]
` degli studi di Milano, via Cesare Saldini 50, 20133 Dipartimento di Matematica Federigo Enriques, Universita Milano, Italy
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