L-invariants of Tate Curves

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1 Pure and Applied Mathematics Quarterly Volume 5, Number 4 (Special Issue: In honor of John Tate, Part 1 of , 2009 L-invariants of Tate Curves Haruzo Hida Dedicated to Professor John Tate on the occasion of his eightieth birthday. Abstract: We compute the Greenberg s L-invariant of the adjoint square of a Hilbert modular Galois representation, assuming a conjecture on the exact form of a certain universal Galois deformation ring (which is known to be valid for almost all cases. If the Galois representation is associated to an elliptic curve E with multiplicative reduction at all places over p, it is equal to log p (Q p p p ord, where Q p(q p p is the norm to Q p of the Tate period of E at the p-adic place p. This is an exact analogue of the conjecture of Mazur-Tate-Teitelbaum. Keywords: exceptional zero, L-invariant, Galois deformation, Tate curve, Tate period. Contents 1. Introduction Galois deformation and L-invariant Selmer Groups Greenberg s L invariant Hecke algebras for quaternion algebras 1358 Received October 21, The author is partially supported by the NSF grant: DMS DMS , DMS and

2 1344 Haruzo Hida 4. Extensions of Q p by its Tate twist Symmetric power of Tate curves A general conjecture and a proof of the theorem The global invariant L(m 1378 References Introduction In Sections 2 to 4 of this paper we describe a computation expressing the L-invariant of Greenberg in an explicit form for the symmetric n-th power for n 2 of the Tate module of an elliptic curve over a totally real field F. In the last section, we generalize our computation to the case where n > 2. Our result is conjectural if n > 2 in the sense that if we have the expected structure theorem of the nearly ordinary Galois deformation ring of the symmetric power, it is likely that the conjecture holds. Indeed, after having written this paper, the author found a proof (under mild assumptions of this implication: Conjecture 5.4 (on the structure of the deformation ring implies Conjecture 1.3, which can be found in [H09] and [H07b]. Though the discussion in [H09] and [H07b] of the material treated in Corollary 3.5 and Section 5 of this paper is simpler (and Galois cohomological, we keep our original approach of this paper because (i it is intriguing to have two essentially different proofs of Corollary 3.5 (one automorphic presented in this paper and the other Galois cohomological given in [H07b] Corollary 1.11 and (ii our approach in Section 5 to the factor L(m might indicate its independence of m, though the discussion in Section 5 is less elaborated than the computation in the proof of [H09] Theorem Our conjecture generalizes the conjecture of Mazur-Tate-Teitelbaum [MTT] in the case of n = 1 and F = Q, and our computation generalizes the one by Greenberg-Stevens [GS1] again in the case of n = 1 and F = Q. A principal point of the conjecture is that the L-invariant of a power of a Tate curve is basically independent of n. Let p be an odd prime and F be a totally real field of degree d < with integer ring O. Order the prime factors of p in O as p 1,..., p e. Throughout this paper, we study an elliptic curve E /F with multiplicative reduction at p j p > 2

3 L-invariants of Tate Curves 1345 for j = 1, 2,..., b and ordinary good reduction at p j p for j > b. When b = 0, as a convention, we assume that E /F has good ordinary reduction at every p-adic place of F. We assume throughout the paper that E does not have complex multiplication. Some cases of complex multiplication are treated in [HMI] Section Take an algebraic closure F of F. Writing ρ E : Gal(F /F GL 2 (Q p for the Galois representation on T p E Zp Q p for the Tate module T p E = lim E[p n ], at each n prime factor p p, we have ρ E ( Gal(F p/f p 0 α p for an unramified character αp. Let S be the set of prime ideals of O prime to p where E has bad reduction. Let K/Q p be a finite extension with p-adic integer ring W. We may take K = Q p, but it is useful to formulate the result allowing other choices of K. We consider the universal locally cyclotomic deformation ρ : Gal(F /F GL 2 (R with a pro- Artinian local universal K-algebra R having the residue field R/m R = K. Writing D p = Gal(F p /F p and I p for its inertia group, the couple (R, ρ is universal among the following (p-adically continuous deformations ρ A : Gal(F /F GL 2 (A for a local Artinian K-algebra A with A/m A = K for its maximal ideal m A such that (K1 unramified outside S, and p; (K2 for each prime p p, ρ A Gal(F p/f p = ( 0 α A,p for αa,p α p mod m A with α A,p Ip factoring through Gal(Fp ur [µ p ]/Fp ur for the maximal unramified extension Fp ur /F p (the local cyclotomy condition; (K3 det(ρ A = N for the p-adic cyclotomic character N ; (K4 ρ A (ρ E K mod m A. In other words, for any ρ A as above, there exists a unique K-algebra homomorphism ϕ : R A such that ϕ ρ = ρ A. Here we have written ρ E K for the K-linear Galois representation on T p E Zp K. The existence of the universal deformation ρ : Gal(F /F GL 2 (R can be shown in a standard manner. We write ρ Gal(F p/f p = ( 0 δ p with δp α p mod m R. If p = p j, we often write δ j (resp. α j for δ pj (resp. α pj. For simplicity, we write F j for the p j -adic completion F pj of F. Let Γ pj = Γ j N (I pj Z p be the group isomorphic (by N to the maximal p-profinite subgroup of the inertia subgroup of Gal(Fj ur [µ p ]/Fj ur. Choose a generator γ pj = γ j Z p of Γ j and identify W [[Γ j ]] with W [[X j ]] for a variable X j = X pj by γ j 1+X j. Since ρ Gal(F j /F j = ( 0 δ j with δ j α j mod m R, δ j αj 1 : Γ p R induces an algebra structure on R over W [[X j ]]. Thus R is an algebra over K[[X p ]] p p = K[[X j ]] j=1,...,e. If we write

4 1346 Haruzo Hida ϕ : R K for the morphism with ϕ ρ = (ρ E K, by our construction, Ker(ϕ (X 1,..., X e. Conjecture 1.1. We have R = K[[X 1,..., X e ]] = K[[X p ]] p p. When F = Q, by a result of Kisin [K] 9.10, [K1] and [K2] 3.4 (generalizing those of Wiles [W] and Taylor-Wiles [TaW], we always have R = K[[X p ]]. In general, assuming Hilbert modularity of E and the following condition: (ai The F p -linear Galois representation ρ = (T p E mod p is absolutely irreducible over Gal(F /F [µ p ]. Combining results of Fujiwara (see [F] and [F1] with that of Lin Chen [C], we can prove R = K[[X p ]] p p (see [HMI] Theorem 3.65 and Proposition Instead, by potential modularity proven by R. Taylor and C. Virdol [V], we have R = K[[X p ]] p p under the following condition stronger than (ai but without assuming modularity over F (see [H08] Proposition 3.2: (ns The image Im(ρ GL 2 (F p is non-soluble. We let σ Gal(F /F act on the three dimensional Lie algebra of SL(2 /K sl 2 (K = {x M 2 (K Tr(x = 0} by conjugation: x σx = ρ E (σxρ E (σ 1. This three dimensional representation is written as Ad(ρ E and called the adjoint square representation of ρ E. We have Ad(ρ E = Sym 2 (ρ E N 1 = Sym 2 (ρ E ( 1. By using a canonical isomorphism between the tangent space of Spf(R and a certain Selmer group of Ad(ρ E, we get Theorem 1.2. Assume Conjecture 1.1. Suppose that the Hilbert-modular elliptic curve E has split multiplicative reduction at p j for j = 1, 2,..., b (b e with Tate period q j at p j (in [T] for j b and has ordinary good reduction at p i with i > b. Then for the local Artin symbol [p, F j ] = F rob pj and the norm Q j = N Fj /Q p (q j, we have L(Ind Q F Ad(ρ E b ( log = p (Q j δi ([p, F i ] det ord p (Q j X j j=1 for local Artin symbol [p, F i ]. i>b,j>b X=0 i>b log p (γ i [F i : Q p ]α i ([p, F i ]

5 L-invariants of Tate Curves 1347 Here L(Ind Q F Ad(ρ E is Greenberg s L-invariant defined in [Gr], and all the assumptions in [Gr] made to define the invariant can be verified under Conjecture 1.1. The argument in [HMI] Section 3.4 essentially gives this claim, although in [HMI], we have some seemingly redundant assumptions (to make the book self-contained. The assumption in the theorem that E has split (multiplicative reduction at p j with j b is inessential, because Ad(ρ E = Ad(ρ E χ (for a K -valued Galois character χ and we can bring any elliptic curve with multiplicative reduction at p j to an elliptic curve with split multiplicative reduction at p j by a quadratic twist. We will prove a refined version of this theorem in Section 5.1 as Theorem 5.3. The following conjecture is a generalization of the above theorem for symmetric powers of ρ E : Conjecture 1.3. Let the notation and the assumption be as in Theorem 1.2. Suppose that the n-th symmetric power motive Sym n (H 1 (E( m with Tate twist by an integer m is critical at 1. Then if Ind Q F (Sym n (ρ E ( m has an exceptional zero at s = 1, we have L(Ind Q F (Sym n (ρ E ( m ( b log p (Q i i=1 ord = p(q i b i=1 L(m for L(m Q p if n = 2m with odd m, log p (Q i ord p(q i if n 2m. If b = e, we have L(m = 1, and the value L(1 is given by ( δi ([p, F i ] log p (γ i L(1 = det X j X1 =X 2 = =X e=0 [F i : Q p ]α i ([p, F i ] i>b,j>b for the local Artin symbol [p, F i ], where γ i is the generator of Γ i by which we identify the group algebra W [[Γ i ]] with W [[X i ]]. R. Greenberg proved the conjecture for his L-invariant of symmetric powers of Tate curves E over Q with multiplicative reduction at p. His proof is well hidden in his remark in page 170 of [Gr]. C.-P. Mok [M] has computed the analytic L-invariant of p-adic analytic L-functions (when n = 1 of elliptic curves E /F (while this paper was being written, following the method of [GS], and his result confirms the conjecture in some special cases. The expression of the L(ρ E/F in [M] and L(Ind Q F ρ E here appears to be different, but indeed the two formulas are equivalent for the following reasons: Though L(s, ρ E = L(s, Ind Q F ρ E, the i>b

6 1348 Haruzo Hida modification (vanishing Euler p-factor computed over Q following [Gr] (6 and the corresponding Euler p-factor over F are different if O/p F p. Indeed, the vanishing factor over Q p is of degree 1 for each p p (at which E has split multiplicative reduction and is given by p p,a p=1 (1 a p. However over F at p, it is of degree f p := [O/p : F p ], and is given by (1 a fp p = (1 a p 1 ζ µ (1 ζa f p. p Thus we have the following identity of the nonvanishing factor E + (Ind Q F ρ E = ( (1 ζe + (ρ E/F = ( f p E + (ρ E/F p p,a p=1 1 ζ µ f p p p,a p=1 under the notation in [H07a] Conjecture 0.1, because f p = 1 ζ µ f (1 ζ. p Thus the expression of the L-invariant L(ρ E/F over F is slightly different from L(Ind Q F ρ E over Q, and indeed, L(ρ E/F = ( p p,a p=1 f p L(Ind Q F ρ E, L since we have lim p(s,ρ s 1 (s 1 = L(ρE + (ρ L(1,ρ e c + (ρ(1 conjecturally for ρ = ρ E and Ind Q F ρ E. Then our formula as above formulated for L(ρ E/F takes the following shape: L(ρ E/F = p p log p (q p ord p (Q p ( L(Ind Q F ρ E = p p log p (q p ord p (q p, where 1 f p ord p (q p = ord p (Q p. C.-P. Mok [M] has computed the analytic L- invariant for some Tate curves E over F following the method of Greenberg Stevens and got the formula using ord p (Q p as above under some assumptions. His formula confirms the expression in Conjecture 1.3 for n = 1 if we take L(Ind Q F ρ E in place of L(ρ E/F in [M]. For the same reason, the L-invariant L(Ad(ρ E /F defined for Ad(ρ E : Gal(Q/F GL 3 (K has the following form (1.1 L(Ad(ρ E /F = ( p p f p L(Ind Q F Ad(ρ E. The motive Sym n (H 1 (E( m is critical at 1 if and only if the following two conditions are satisfied: 0 m < n; either n is odd or n = 2m with odd m. We will make in the text one more conjecture (with less ambitious goal in appearance similar to Conjecture 1.1 for the deformation ring of ρ n,0 = Sym n (ρ E

7 L-invariants of Tate Curves 1349 which is closely related to the above conjecture (see Conjecture 5.4. The value L(m with m > 1 can be conjecturally given by a formula similar to (but more complicated than the formula of L(1 in terms of the universal nearly ordinary deformation of ρ m,0. There is a wild guess asserting that L(m is independent of m and is given by L(1. We hope to present a general formula for L(m and some evidence for this wild guess in our future paper. Here are some additional remarks about the conjecture: (1 When n = 2m with even m, the motive associated to Sym n (ρ E ( m is not critical at s = 1; so, the situation is drastically different (and in such a case, we do not make any conjecture; see [H00] Examples 2.7 and 2.8. (2 The above conjecture is a generalization of the conjecture of Mazur-Tate- Teitelbaum (see [MTT] and applies to arithmetic and analytic p-adic L-functions. 2. Galois deformation and L-invariant Here is a theorem proved in [HMI] as Theorem 3.73: Theorem 2.1. Suppose R = K[[X p ]] p p. Then, if ϕ ρ = ρ E, for the local Artin symbol [p, F p ] = F rob p, we have the following formula of Greenberg s L-invariant: ( L(Ind Q F Ad(ρ δp ([p, F p ] log p (γ p E = det X p X=0 [F p : Q p ]α p ([p, F p ]. p,p The statement in [HMI] Theorem 3.73 is ( L(Ind Q F Ad(ρ δp ([p, F p ] E = det x p p,p p log p (γ p x=0 α p ([p, F p ]. The variable x p is defined by δ p ([γ p, F p ] = (1 + x p and has relation (1 + x p = (1 + X p [Fp:Qp], and thus we get [F p : Q p ] 1 X=0 X p = X=0 x p. Indeed, by local class field theory, [γ p, F p ] Q ab = [N Fp/Qp (γ p, Q p ] = [γ p, Q p ] [Fp:Qp] and for an element σ I p with σ F ab = [u, F p p ] and N Fp/Qp (u = γ p, we have = [γ p, Q p ]. Thus δ p ([u, F p ] [Fp:Qp] = δ p ([γ p, F p ], and on the other [u, F p ] Q ab p hand, δ p ([u, F p ] = (1 + X p by definition. In other words, we have δ p ([γ p, F p ] = δ p (γ [Fp:Qp] p = (1 + X p [Fp:Qp]. This explains the equivalence of the two formulas. Also some more conditions are assumed in [HMI] Theorem 3.73 to assure p

8 1350 Haruzo Hida R = K[[X p ]] p p, but the proof given there is valid only assuming R = K[[X p ]] p p. Indeed, the (seemingly redundant condition (vsl in [HMI] Theorem 3.73 is equivalent to R = K[[X p ]] p p by [HMI] Proposition We recall briefly an F -version (given in [HMI] Definition 3.85 of Greenberg s formula of the L invariant for a general p-adic totally p-ordinary Galois representation V (of Gal(F /F with an exceptional zero. This definition is equivalent to the one in [Gr] if we apply it to Ind Q F V as proved in [HMI] (in Definition When V = Ad(ρ E, the definition can be outlined as follows. Under some hypothesis, he found a unique subspace H H 1 (Q, Ind Q F Ad(ρ E of dimension e = {p p}. By Shapiro s lemma, H 1 (Q, Ind Q F Ad(ρ E = H 1 (F, Ad(ρ E, and one can give a definition of the image H F of H in H 1 (F, Ad(ρ E without reference to the induction Ind Q F Ad(ρ E ([HMI] Definition 3.85 as we recall the precise definition later. The space H F is represented by cocycles c : Gal(F /F Ad(ρ E such that (1 c is unramified outside p; (2 c restricted to the decomposition subgroup D p Gal(F /F at p p is upper triangular after conjugation, and c Dp modulo nilpotent matrices becomes unramified over F p [µ p ] for all p p. This subspace H F coincides with the locally cyclotomic Selmer group Sel cyc F (Ad (ρ E whose precise definition will be given in the following subsection. By the condition (2, c Dp with a prime p p (expressed in a matrix form taking a basis so that ρ E = ( 0 α p modulo upper nilpotent matrices factors through the cyclotomic Galois group Gal(F p [µ p ]/F p, and hence c Dp modulo upper nilpotent matrices becomes unramified everywhere over the cyclotomic Z p -extension F /F. In other words, the cohomology class [c] is in Sel F (Ad(ρ E but not in Sel F (Ad(ρ E. Take a basis {c p } p p of H F over K. Write c p (σ ( a p (σ 0 a p (σ for σ D p. Then a p : D p K is a homomorphism. We now have two e e matrices with coefficients in K: A = ( a p ([p, F p ] p,p p and B = ( log p (γ p 1 a p ([γ p, F p ] p,p p.

9 L-invariants of Tate Curves 1351 Under Conjecture 1.1, we can show that B is invertible. Then Greenberg s L- invariant is defined by (2.1 L(Ind Q F Ad(ρ E = det(ab 1. The determinant det(ab 1 is independent of the choice of the basis {c p } p. Let Q /Q be the cyclotomic Z p -extension, and put F /F for the composite of F and Q. Choose a generator γ of N (Gal(F /F Z p for the p-adic cyclotomic character N, and identify Λ = W [[Gal(F /F ]] with W [[T ]] by γ 1+T. The adjoint square Selmer group Sel F (Ad(T p E (Q p /Z p has its Pontryagin dual which is a Λ-module of finite type. Choose a characteristic power series Φ arith (T Λ of the Pontryagin dual. Put L arith p (s, Ad(ρ E = Φ arith (γ 1 s 1. This p-adic L-function corresponds to the Selmer group Sel F (Ad(ρ E, and hence we need to use the L(Ad(ρ E /F in (1.1 in place of L(Ind Q F (Ad(ρ E to describe its property at s = 1. The following conjecture for the arithmetic L- function L arith p (s, Ad(ρ E is a theorem (except for the nonvanishing L(Ind Q F Ad (ρ E 0 essentially under (mild but possibly restrictive conditions (see [Gr] Proposition 4 and Theorem 5.3 in the text for a precise statement: Conjecture 2.2 (Greenberg. Suppose (ds and that ρ is absolutely irreducible. Then L arith p (s, Ad(ρ E has zero of order equal to e = {p p} and for the constant L(Ind Q F Ad(ρ E K specified by the determinant as in the theorem, we have up to units. L arith p (s, Ad(ρ E lim s 1 (s 1 d = L(Ad(ρ E /F Sel F (Ad(ρ E 1/[K:Qp] p Note here that Ind Q F ρ E is not ordinary in the sense of [Gr] if p ramifies in F/Q; so, Greenberg made the above conjecture under the extra hypothesis of unramifiedness of p in F/Q. In the above conjecture, the modifying Euler factor at the p-adic places p j of good reduction (j > b: E + (Ad(ρ E = (1 αj 2 (F rob p N(p j (1 αj 2 (F rob p j>b does not appear. However, taking the Bloch-Kato Selmer group S F (Ad(ρ E over F (crystalline at p j for j > b in place of Greenberg s Selmer group Sel F (Ad (ρ E = Sel Q (Ind Q E Ad(ρ E, we have the relation Sel F (Ad(ρ E 1/[K:Qp] = E + (Ad(ρ p E S F (Ad(ρ E 1/[K:Qp] p

10 1352 Haruzo Hida up to p-adic units (as described in [MFG] page 284. Thus if one uses the formulation of Bloch-Kato, we do have the modifying Euler factor in the formula, and the size of the Bloch-Kato Selmer group is expected to be equal to the primitive archimedean L-values (divided by a suitable period; see Greenberg s Conjecture 0.1 in [H07a]. Though Sel F (Ad(ρ E = Sel Q (Ind Q F Ad(ρ E by [HMI] Proposition 3.80, the nonvanishing modification factor E + (Ind Q F Ad(ρ E and E + (Ad(ρ E could be different as explained above (1.1; so, we need to use L(Ad(ρ E /F in place of L(Ind Q F Ad(ρ E /F in the above conjecture Selmer Groups. First we recall Greenberg s Selmer groups. Write F (S /F for the maximal extension unramified outside S, p and. Put G = Gal(F (S /F and G M = Gal(F (S /M. Let V be a potentially ordinary representation of G on a K-vector space V. Thus V has decreasing filtration FpV i such that an open subgroup of I p (for each prime factor p p acts on FpV/F i p i+1 V by the i-th power N i of the p-adic cyclotomic character N. We fix a W -lattice T in V stable under G. Write D = D p G for the decomposition group of each prime factor p p. Put F + p V = F 1 p V and F p V = F 0 p V. We have a 3-step filtration: (ord V F p V F + p V {0}. Its dual V (1 = Hom K (V, K N again satisfies (ord. Let M/F be a subfield of F (S, and put G M = Gal(F (S /M. We write p for a prime of M over p and q for general primes outside p of M. We write I p and I q for the inertia subgroup in G M at p and q, respectively. We put and L p (A = Ker(Res : H 1 (M p, A H 1 (I p, A F + p (A, L q (A = Ker(Res : H 1 (M q, A H 1 (I q, A. Then we define the Selmer submodule in H 1 (M l, V/T by (2.2 Sel M (A = Ker(H 1 (G M, A q H 1 (M q, A L q (A p H 1 (M p, A L p (A for A = V, V/T. The classical Selmer group of V is given by Sel M (V/T, equipped with discrete topology. We define the minus, the locally cyclotomic and the

11 L-invariants of Tate Curves 1353 strict Selmer groups Sel M (A, Selcyc M (A and Selst M (A, respectively, replacing L p (A by L p (A = Ker(Res : H 1 (M p, V H 1 (I p, V F p (A L p(a L cyc p (A = Ker(Res : L p (A H 1 V (I p,, F p + (A L p (A L st p (A = Ker(Res : L p (A H 1 V (M p, F p + (A L p(a, where I p, is the inertia group of Gal(M p /M p [µ p ]. Then we have Sel cyc F (A = Res 1 F (Sel /F F (A. Lemma 2.3. Suppose R = K[[X p ]] p p. Sel cyc F (Ad(ρ E = Hom K (m R /m 2 R, K. Proof. Let V = Ad(ρ E. Then we have the filtration: V F p V F + p V {0}, Then we have Sel F (Ad(ρ E = 0 and where taking a basis so that ρ E Dp = ( 0 α p, F p V is made up of upper triangular matrices and F p + V is made up of upper nilpotent matrices, and on Fp V/F p + V, D p acts trivially (getting eigenvalue 1 for F rob p. We consider the space Der K (R, K of continuous K-derivations. Let K[ε] = K[t]/(t 2 for the dual number ε = (t mod t 2. Then writing each K-algebra homomorphism φ : R K[ε] as φ(r = φ 0 (r + φ (rε and sending φ to φ Der K (R, K, we have Hom K-alg (R, K[ε] = Der K (R, K = Hom K (m R /m 2 R, K. By the universality of (R, ρ, we have Hom K-alg (R, K[ε] = {ρ : Gal(F /F GL 2(K[ε] ρ satisfies (K1 4} = by Hom K-alg (R, K[ε] φ ρ φ = φ ρ = ρ E + ε φ ρ. Pick ρ = ρ φ as above. Write ρ(σ = ρ 0 (σ + ρ 1 (σε with ρ 1 (σ = ρ t = φρ(σ. Then c ρ = ( φ ρρ 1 E can be easily checked to be a 1-cocycle having values in M 2 (K V. Since det(ρ = det(ρ E Tr(c ρ = 0, c ρ has values in V = sl 2 (K. By the reducibility condition (K2, [c ρ ] vanishes in H1 (M p,a. By the local cyclotomy condition in L p (A (K2, [c ρ ] is unramified over F. For q S, first suppose that E has potentially good reduction at q. Then ρ E (I q is a finite group. Take a finite Galois extension L/F q over which E has good reduction. Then the inertia group I of Gal(F q /L acts trivially on T p E (see [ST], and H 1 (I q /I, V = 0 because I q /I is a finite group

12 1354 Haruzo Hida (and V is a Q p -vector space. We have the inflation-restriction exact sequence (made of F rob q -linear maps: 0 = H 1 (I q /I, V H 1 (I q, V Res H 1 (I, V = Hom(Z p (1, V. The Frobenius F rob q acts on I and its p-profinite quotient Z p (1 by conjugation. Restricting c ρ to I, we get a homomorphism of the p-profinite Tame inertia group c ρ : Z p (1 V compatible with the action of F rob q. Since E has good reduction over L, after restricting V to I, the action of F rob q on V and on Z p (1 does not match; so, Res(c ρ = 0, which shows that c ρ is unramified at q. If E has potentially multiplicative reduction at q S, the unramifiedness of c ρ follows from the following lemma. Thus the cohomology class [c ρ ] of c ρ is in Sel cyc F (V. We see easily that ρ = ρ [c ρ ] = [c ρ ]. We can reverse the above argument starting with a cocycle c giving an element of Sel cyc F (V to construct a deformation ρ c = ρ E +ε(cρ E with values in GL 2 (K[ε]. Thus we have {ρ : Gal(F /F GL 2 (K[ε] ρ satisfies the conditions (K1 4} = = Sel cyc F (V. The isomorphism Der K (R, K = Sel cyc F (V is given by Der K(R, K [c ] Sel cyc F (V for the cocycle c = c ρ = ( ρρ 1 E, where ρ = ρ E + ε( ρ. Since the algebra structure of R over W [[X p ]] p p is given by δ p αp 1, the K-derivation = φ : R K corresponding to a K[ε]-deformation ρ is a W [[X p ]]-derivation if and only if ρ ( Gal(F p/f p 0 0, which is equivalent to [c ] Sel F (V, because we have already shown that Tr(c = Tr(c ρ = 0. Thus we have Sel F (V = Der W [[Xp]](R, K = 0. Lemma 2.4. Let q be a prime outside p at which E has potentially multiplicative reduction. Then for a deformation ρ of ρ E satisfying (K1 4, the cocycle c ρ (defined in the above proof is unramified at q. Proof. By our assumption, ρ E ( Gal(F q/f q = ηn 0 η for a finite order character η. Since Ad(ρ E η 1 = Ad(ρ E, twisting by a character, we may assume that the restriction of ρ E to the inertia group I q has values in the upper unipotent subgroup; so, it factors through the tame inertia group = Ẑ(q (1. By the theory of Tate curves, ρ E ramifies at q. The p-factor of Ẑ(q is of rank 1 isomorphic to Z p (1. Then ρ(i q is cyclic, and therefore dim K ρ(i q = 1 = dim K ρ E (I q. Thus

13 L-invariants of Tate Curves 1355 the deformation ρ is constant over the inertia subgroup, and hence c ρ restricted to I q is trivial. Let ρ n,m = Sym n (ρ E ( m, and write V for the representation space of ρ n,m. For each prime q S {p p}, we put (2.3 Ker(H 1 (F j, V H 1 (F j, L q (V = L q (V V F + p j (V L p j (V if q = p j and j b, otherwise Once L q (V is defined, we define L q (V (1 = L q (V under the local Tate duality between H 1 (F q, V and H 1 (F q, V (1, where V (1 = Hom K (V, Q p (1 as Galois modules. Then we define the balanced Selmer group Sel F (V (resp. Sel F (V (1 by the same formula as in (2.2 replacing L p (V (resp. L p (V (1 by L p (V (resp. L p (V (1. By definition, Sel F (V Sel F (V. We have Lemma 2.5. If V = ρ n,m is motivic and critical at s = 1, (V Sel F (V = 0 H 1 (G, V = q S H 1 (F q, V L q (V p p H 1 (F p, V. L p (V Proof. Since Sel F (V Sel F (V, the assumption implies Sel F (V = 0. Then the Poitou-Tate exact sequence tells us the exactness of the following sequence: Sel F (V H 1 (G, V H 1 (F l, V Sel F (V (1. L l (V l S {p p} It is an old theorem of Greenberg (which assumes criticality at s = 1 that dim Sel F (V = dim Sel F (V (1 (see [Gr] Proposition 2 or [HMI] Proposition 3.82; so, we have the assertion (V. In [HMI], Proposition 3.82 is formulated in terms of Sel Q (Ind Q F V and Sel Q (Ind Q F V (1 defined in [HMI] (3.4.11, but this does not matter because we can easily verify Sel Q (Ind Q F? = Sel F (? (similarly to [HMI] Corollary Here we note that there is an error in the definition of U p (Ind Q F V in [HMI] page 265 at line 8 from the bottom. It has to be the pull back image of F + H 1 (Q p, Y Hom(D p /I p, K t 0 Hfl 1 (Q p, K(1 t 1 H 1 (Q p, Y decomposing Y = Y K t 0 K(1 t 1 for the product Y of nontrivial extensions of K by K(1 (see the errata list of [HMI] posted at the author s web page for more details.

14 1356 Haruzo Hida 2.2. Greenberg s L invariant. In this subsection, we let V = Ad(ρ E. Assuming for simplicity that F p /Q p is a Galois extension for all prime factors p p, we recall a little more detail of the F -version of Greenberg s definition of L(Ind Q F V (which is equivalent to the one given in [Gr] if we apply Greenberg s definition to Ind Q F V as explained in [HMI] without assuming the simplifying condition. Write G p = Gal(Q p /Q p and G p = Gal(F p /F p ; so, G p surjects down to D p. The long exact sequence associated to the short one Fp V/F p + V V/F p + V V/Fp V gives a homomorphism H 1 (F p, F p V F + p V Gp = Hom(G ab p, F p V F + p V Gp where G p acts on H 1 (F p, F p V regarding F p V F p + V F p + V ιp H 1 (F p, V /L p (V, as the trivial G p-module; so, its action on φ Hom(G ab p, F p V F + p V is given by φ τ φ(σ = φ(τστ 1. Note that canonically H 1 (F p, F p V F + p V Gp Hom(G ab p, F p V Res F p + V = Hom(Q p, F p V F p + V = K 2 by φ ( φ([γ,fp] log p (γ, φ([p, F p]. Here [x, F p ] is the local Artin symbol. Identifying H 1 (F p, F p V F + p V Gp with Hom(G ab p, F p V, a homomorphism φ : F p + Gab V p F p V F + p V in Ker(ι p is unramified; so, the image of ι p is one-dimensional (those ramified classes modulo unramified ones. In other words, the image of ι p is isomorphic to F p V/F + p V = K. Suppose R = K[[X p ]] p p. Then by (V in Lemma 2.5 (and Lemma 2.3, we have a unique subspace H F of H 1 (G, V projecting down onto Im(ι p p p H 1 (F p, V. L p (V Then by the restriction, H F gives rise to a subspace L of p Hom(G ab p, F p V/F + p V Gp = p Hom(G ab p, F p V/F + p V = p ( F p V F + p V isomorphic to p (F p V/F p + V. If a cocycle c representing an element in H F is unramified, it gives rise to an element in Sel F (V. By the vanishing of Sel F (V (Lemma 2.3, this implies c = 0; so, the projection of L to the first factor p (via φ (φ([γ, F p ]/ log p (γ p is surjective. Thus this subspace L is a graph of a 2 F p V F + p V

15 L-invariants of Tate Curves 1357 K linear map L : p F p V/F + p V p F p V/F + p V. We then define L(Ind Q F V = det(l K. This is a brief description of the direct construction of H F assuming normality of F p /Q p for all p p. The general non-galois case is treated in pages of [HMI] (or Section 1.2 of [H07b]. Now we return to the general setting allowing prime factors p p with non-galois extension F p /Q p. By R = K[[X p ]] p p, we have dim K Sel cyc F (V = e by Lemma 2.3. For cocycles c in H F, (c Dp mod F p + V is a G p -invariant homomorphism of G p into K; so, it extends to G p, and hence it factors through G ab p = Gal(Q p [µ p ]/Q p. Thus c is locally cyclotomic, and we have H F = Sel cyc F (V. Let ρ : G F GL 2 (R be the universal nearly ordinary deformation with ρ ( Dp =. Then c p = ρ X=0 X 0 δ p p ρ 1 E is a 1-cocycle (by the argument proving Lemma 2.3 giving rise to a class of H F (by (K2. The cocycles {c p } p give a basis of H F over K (by Lemma 2.3. We have δ p ([u, F p ] = (1 + X p log p (N F p /Qp (u/ log p (γ p for u O p (because N ([u, F p ] = N Fp /Q p (u 1. Writing ( a p c p = ρ 1 E 0 a over D p, p we have a p = δ 1 p dδ p dx p X=0 over D p, and from this and (2.1 we get the desired formula of L(Ind Q F Ad(ρ E in Theorem 2.1 (see the proof of Theorem 3.73 in [HMI] for computational details. If one restricts c H F to G = Gal(F (S /F, its ramification is exhausted by Γ = Gal(F /F (because of the definition of Sel cyc F (Ad(ρ E and H F giving rise to a class [c] Sel F (V. The kernel of the restriction map: H 1 (G, V H 1 (G, V is given by H 1 (Γ, H 0 (G, V = 0 because H 0 (G, V = 0. Thus the image of H F in Sel F (V/T gives rise to the order e exceptional zero of L arith (s, Ad(ρ E at s = 1. We have proved a weaker version of [Gr] Proposition 3: Proposition 2.6. Suppose R = K[[X p ]] p p. factors of p in F, we have Then for the number e of prime ord s=1 L arith p (s, Ad(ρ E e.

16 1358 Haruzo Hida 3. Hecke algebras for quaternion algebras We make some preparation for the proof of Theorem 1.2, gathering known facts. We assume that F Q (otherwise the theorem is known by [H04] and by Greenberg-Stevens [GS] and [GS1]. Take first a quaternion algebra D 0/F central over F unramified everywhere such that D 0 Q R = M 2 (R r H d r with 0 r 1 (so r d mod 2. Let S be an open compact subgroup of GL 2 (Ô with S = S p S (p such that S p = GL 2 (O p. For an ideal N O, we make the following specific choice of S (p : (3.1 S (p = { ( a b c d GL2 (Ô(p c 0 mod N for Ô(p = l p O l. Then we consider the automorphic variety (either a Shimura curve (r = 1 or a 0-dimensional point set (r = 0 given by where Z A = F A identity component of D 0, X 11 (p n = D 0 \D 0,A /S 11(p n Z A C, is the center of D A, C is a maximal compact subgroup of the all primes l, and S 11 (p n is given by { ( a b ( c d S a b c d and identifying D( 0,l = D 0 Q F l with M 2 (F l for ( 0 mod p n with N Fj /Q p (a/d 1 mod p n} for Ô = l O l. Let B (resp. Z be the upper triangular Borel subgroup B GL(2 and the center Z GL(2. Write S 11 (p = n=1 S 11(p n. Then B(O p /S 11 (p p = Im(N (Gal(F p /F p Z p by sending ( a 0 d to N Fp/Q p (ad 1 Z p. Consider M n = Hr (X 11 (p n, Z p which is the Pontryagin dual of H r (X 11 (p n, Q p /Z p. The Z p -module M n is a free module (if n 0 having the action of the following three type of linear operators: Hecke operator T (n for each integral ideals n outside p, the U(p-operator given by the double coset S 11 (p n ( p S11 (p n D 0,A (along with U(p p for the p-component p p O p of p associated with S 11 (p n ( p p S11 (p n D 0,A and U(p associated with S 11(p n ( ϖ p S11 (p n for (a choice of a uniformizer ϖ p O p, the diamond operator action z coming from ( z for z O p. By the above remark on B(O p /S 11 (p p, the action of z for z 1+pO p factor through Γ p Z p. Let e = lim n U(p n! as an operator acting on M n (U(p = }

17 L-invariants of Tate Curves 1359 p U(p p. Let Mn ord be the direct summand em n. We have natural trace map M m M n for m > n compatible with all Hecke operators and all diamond operators. By the diamond operator action, M ord = lim M ord n n naturally become a W [[Γ F ]]-module. Here is an old theorem of mine (see [HMI] 3.2.9: Theorem 3.1. The W [[Γ F ]]-module M ord is free of finite rank over W [[Γ F ]]. Let h be the W [[Γ F ]]-subalgebra of End W [[ΓF ]](M ord generated over W [[Γ F ]] by T (n for all n prime to p and U(p for all prime factors p of p. Then we have Corollary 3.2. The algebra h is torsion free of finite type over W [[Γ F ]] with h F /(X p p p h F pseudo isomorphic to the Hecke algebra of H r (X 11 (p, W. Actually if p 5, h is known to be free over W [[Γ F ]], and the pseudo isomorphism as above is actually an isomorphism (see [HMI] We take N to be the prime-to p part of the conductor of E. Let T be the local ring of the universal nearly ordinary Hecke algebra h acting nontrivially on the Hecke eigenform associated to E. Let P Spec(T(K corresponding to ρ E, that is, ρ T mod P (ρ E K. Let T P = lim T n P /P n T P for the localization T P of T at P. Since ρ E is absolutely irreducible, by the technique of pseudo representation, we can construct the modular deformation ρ T : G GL 2 ( T P which satisfies (K1 4; in particular, det ρ T = N, because the central character is trivial. Since E is modular over F, we have the surjective K-algebra homomorphism R T P for the localization-completion T P. Since T P is integral and of dimension e, we have Corollary 3.3. If R = K[[X p ]] p p, then R = T P. The isomorphism R = K[[X p ]] p p is often proven by showing R = T P first (see [HMI] Theorem 3.65 and Proposition Take a quaternion algebra D 1/F such that D 1, := D 1 Q R = M 2 (R q H d q with q 1 and D 1 is ramified only at p 1 (among finite places. At p 1, we have a unique maximal order R 1 in D p1. Identify D (p 1 1,A with M 2 (F (p 1 A. Then we define S 11 (pn to be the product of S 11 (p n (p1 and R 1 and define Y 11 (p n = D 1 \D 1,A /S 11(p n Z A C, where C is again the maximal compact subgroup of the identity component of D 1,. Let U(p(p 1 = j>1 U(p p j for the Hecke operator U(p p associated

18 1360 Haruzo Hida to S 11 (pn ( p p S 11 (p n, and define e 1 = lim n U(p (p1 n! acting on the dual N n = H q (Y 11 (p n, Z p of the cohomology group H q (Y 11 (p n, Q p /Z p. Put N = lim n N n. Then e 1 and U(p (p1 act on N. Let Γ 1 = p p 1 Γ p. We go through all the above process and define h 1 End W [[Γ1 ]](lim e n 1 N n by the W [[Γ 1 ]]- subalgebra generated by T (n for all n prime to p and U(p for all prime factors p of p. Here U(p 1 is associated to S 11 (pn ϖ 1 S 11 (pn D 1,A for (a choice of ϖ 1 R 1 whose reduced norm is equal to ϖ p1. Since ρ E (or more precisely, the corresponding Hilbert-modular automorphic representation π E with L(s, π E = L(s, E is Steinberg at p 1, by the Jacquet-Langlands correspondence (e.g., [HMI] combined with the Eichler Shimura isomorphism (e.g., [PAF] 4.3.4, we have a Hecke eigenvector f 1 in H q (Y 11 (p, Z p giving rise to E. Then we define T 1 to be the local ring of h 1 acting nontrivially on f 1. Let P 1 Spec(T 1 (W be the point associated to ρ E. We then have a deformation ρ T1 : G GL 2 ( T 1,P of ρ E. Since the central character is trivial, we have det ρ T1 = N. Theorem 3.4. We have (1 h 1 is torsion-free finite over W [[Γ 1 ]], and T 1,P1 = K[[X2,..., X e ]]; (2 ρ T1 restricted to Gal(F 1 /F 1 is isomorphic to ( εn 0 ε, where ε = ±1 is the eigenvalue of F rob p1 on the étale quotient of T p E; (3 There is a surjective algebra homomorphism T/X 1 T T 1 inducing an isomorphism T P /X 1 TP = T1,P1 ; (4 There is a surjective algebra homomorphism T/(U(p 1 εt T 1 sending T (n to T (n, where U(p 1 is given by the action of the double coset S 11 (p n ( ϖ S 11(p n for a prime element ϖ of p 1 O p1. Here is a sketch of proof (see [H00] Proposition 7.1 and [HMI] Corollary 3.57 for a detailed proof. The first assertion follows from construction; in other words, it can be proven by the same way as the proof of Corollary 3.2. By the Jacquet-Langlands correspondence, T covers T 1. Any Hilbert-modular automorphic representation π corresponding to a point of Spec(T 1 (Q p Spec(T(Q p is Steinberg at p 1 because D 1 ramifies at p 1. Since points corresponding classical automorphic representations are Zariski dense in Spec(T 1, the Galois representation has to have the form as in (2 (see [HMI] Proposition 2.44 (2. Thus the eigenvalue of U(p 1 of π is ±1 and the corresponding Galois representation has the form as in (2. The assertion (1 implies (3. By (2, U(p 1 is either

19 L-invariants of Tate Curves 1361 ±1. Since U(p 1 is a formal function on the connected Spf(T 1, U(p 1 = ε is a constant, which implies (4. Note that [ϖ, F 1 ] e 1 = [p, F 1 ] on any unramified abelian extension of F 1 for the ramification index e 1 of p 1 over Q p, and hence U(p p1 = U(p 1 e 1. By Theorem 3.4 (3 4 (and Theorem 2.43 in [HMI], we find U(p 1 = δ 1 ([ϖ, F 1 ] = ε + X 1 Φ(X 1,..., X e for Φ W [[X 1,..., X e ]], and hence, we get U(pp 1 X j X1 =0 = U(p 1 X j X1 =0 = 0 for all j 2. In other words, Thus we have ( δi ([p, F i ] det (0 X j ( δi ([p, F i ] (0 = X j i,j ( δ1 ([p,f 1 ] (0 0 X 1. = δ ( 1([p, F 1 ] δi ([p, F i ] (0 det (0. X 1 X j i 2,j 2 Inductively, we can continue choosing a quaternion algebra D i exactly ramifying at i p-adic places p 1,..., p i and D i Q R = M 2 (R r H d r with 0 r 1 and prove the theorem similar to Theorem 3.4. See [H00] Proposition 7.1 and [HMI] Corollary 3.57 for the exact statement, which yield the following result given in [HMI] as Proposition 3.91: Corollary 3.5. Suppose the Hilbert modular elliptic curve E /F has multiplicative reduction at p k for k = 1,..., b and ordinary good reduction at p j for j > b. Then we have det ( δi ([p, F i ] (0 X j = b k=1 ( δ k ([p, F k ] δi ([p, F i ] (0 det (0. X k X j i>b,j>b Although we have given an automorphic proof of the above factorization formula (over p-adic places of the L-invariant, there is another Galois cohomological proof, which is discussed in [H09] Section 1.3 and [H07b] Corollary Extensions of Q p by its Tate twist Let L be a p-adic field with p-adic integer ring W. We start with an extension of local Galois modules 0 K(1 T K 0

20 1362 Haruzo Hida over Gal(Q p /L for a finite extension L/Q p. This type of extensions (for K = Q p can be obtained by the p-adic Tate module T = T p E Zp Q p of a Tate elliptic curve E /L with multiplicative reduction. We prepare some general facts. The following is a slight generalization of [GS1] Section 2: Let L and K be a finite extension of Q p inside a fixed algebraic closure Q p /Q p and T be a two dimensional vector space over K on which D := Gal(Q p /L acts. We write H i (? for H i (D,?. By definition, H 1 (M = Ext 1 K[D] (K, M for a D-module M, and hence, there is a one-to-one correspondence: { nontrivial extensions of K by M } { 1-dimensional subspaces of H 1 (M From the left to the right, the map is given by (M X K δ X (1 for the connecting map K = H 0 (K δ X H 1 (M of the long exact sequence attached to (M X K. Out of a 1-cocycle c : D M, one can easily construct an extension (M X K taking X = M K and letting D acts on X by g(v, t = (gv + t c(g, t, and [c] (M X K gives the inverse map. By Kummer s theory, we have a canonical isomorphism: ( H 1 (K(1 = lim L /(L pn n Zp K. We write γ q H 1 (K(1 for the cohomology class associated to q 1 for q L. The class γ q is called the Kummer class of q. A canonical cocycle ξ q in the class γ q is given as follows. Define ξ n : D µ p n by ξ n (σ = (q 1/pn σ 1, which is a 1-cocycle. Then ξ q = lim ξ n n having values in Z p (1 K(1. Suppose we have a non-splitting exact sequence of D-modules K(1 T K with the splitting field n L[µ p n, q1/pn ] for q L with 0 < q p < 1. We have proven ( Proposition 4.1. If T is isomorphic to the representation σ N (σ ξq(σ, 0 1 then for the extension class of [T ] H 1 (K(1, we have K[T ] = Kγ q. In particular, Kγ q is in the image of the connecting homomorphism H 0 (K δ 0 H 1 (K(1 coming from the extension K(1 T K. Corollary 4.2. Let E /L be an elliptic curve. If E has split multiplicative reduction over W, the extension class of [T ] for the p-adic Tate module T is in Q p γ qe for the Tate period q E L. }.

21 L-invariants of Tate Curves 1363 Write D = Gal(Q p /Q p D. We consider V = Ind Qp L T := IndD D T. Then we have a D-stable exact sequence 0 F + T T T/F + T 0 such that D acts by N on F + T. Thus F + T is one dimensional. We then have the exact sequence of the induced modules: 0 Ind Qp L F + T Ind Qp L T IndQp L (T/F + T 0. We put F + V := Ind Qp L F + T, and define F 00 V by the maximal subspace of V stable under D such that D acts on F 00 V/F + V trivially. In other words, we have H 0 (D, V/F + V = F 00 V/F + V. Similarly, we define F 11 V V to be the smallest subspace stable under D such that D acts on F + V/F 11 V by N ; so, we have H 0 (D, F + V( 1 = (F + V/F 11 V( 1. Since we have Ind Qp L (T/F + T = Ind Qp L 1 and IndQp L F + T = Ind Qp L F + N = (Ind Qp L 1 N, we find dim K (F + V/F 11 V = dim K (F 00 V/F + V = 1, because H 0 (D, Ind Qp H 0 (D, Ind Qp L 1 = K. Thus we get an extension (4.1 0 F + V/F 11 V F 00 V/F 11 V F 00 V/F + V 0 of K[D]-modules. L 1 = Let K := K[ε] = K[t]/(t 2 with ε (t mod t 2. A K[D]-module T is called an infinitesimal deformation of T if T is K-free of rank 2 and T /ε T = T as K[D]- modules. Since the map ε : T T T given by v εv is Galois equivariant, we have an exact sequence of D-modules 0 T T T 0 if T is an infinitesimal deformation of T. Pick an infinitesimal character ψ : D K with ψ mod (ε = 1. Define K(ψ for the space of the character ψ. Obviously, dψ dε dψ : D K is a homomorphism; so, dε Since the extension T is split if and only if dψ dε Hom(D, K = H1 (K. = 0, we get Proposition 4.3. The correspondence K(ψ dψ dε H1 (K gives a one-to-one correspondence: { Nontrivial infinitesimal deformations of K and we have K[ T (ψ] = K dψ dε in H1 (K. } { } 1-dimensional subspaces of H 1, (K

22 1364 Haruzo Hida We have the restriction map Res : H 1 (D, K(m H 1 (K(m and the transfer map Tr : H 1 (K(m H 1 (D, K(m. They are adjoint each other under the Tate duality. Thus we have the cup product pairing giving Tate duality and the following commutative diagram:, : H 1 (K(1 H 1 (K H 2 (K(1 = K Tr Res, : H 1 (D, K(1 H 1 (D, K H 2 (D, K(1 = K. By Shapiro s lemma (and the Frobenius reciprocity; cf., [HMI] Section 3.4.4, we get Lemma 4.4. We have Tr([T ] = [F 00 V/F 11 V] H 1 (D, K(1 for the class [T ] H 1 (K(1 of the extension K(1 T K. Proof. Decompose D = σ Σ Dσ; so, Σ = Hom field (L, Q p. Then for τ D, we have στ = τ σ σ for σ Σ and τ σ D. We look at the matrix form of the induced representation. If the matrix form of T is given by ( N ξ 0 1 for a 1-cocycle ξ : D K(1, the cocycle giving the extension K(1 F 00 V/F 11 V K is given by τ σ Σ ξ(τ σ σ, which represents the class of Tr([ξ]. Here D acts on the right on Z p (1 = lim µ n p n following the tradition of right Galois action on roots of unity ζ ζ σ. Corollary 4.5. Let E /L be an elliptic curve with p-adic Tate module T. If E has split multiplicative reduction over W, the extension class of [F 00 V/F 11 V] for V = Ind Qp L T is in Q pγ NL/Qp (q E for the Tate period q E L. Proof. We keep the notation introduced in the proof of the above lemma. Consider the cocycle ξ n (τ = (q 1/pn E τ 1 of D with values in µ p n. Then we have Tr(ξ n (σ = σ Σ (q 1/pn E (τσ 1σ = σ Σ (q 1/pn E σ(τ 1 = (N L/Qp (q E 1/pn τ 1. Thus Tr([T ] = [F 00 V/F 11 V] is represented by the cocycle ξ given by lim n Tr(ξ n for Tr(ξ n (τ = (N L/Qp (q E 1/pn τ 1, which implies the identity Tr([T ] = γ NL/Qp (q E. Note that H 1 (D, K = Hom(D, K = Hom(D ab, K = K 2,

23 where the last isomorphism is given by L-invariants of Tate Curves 1365 Hom(D ab, K φ ( φ([γ, Q p] log p (γ, φ([p, Q p] K 2 for γ Z p of infinite order. This follows from class field theory. Since the Tate duality, is perfect, for any line l in H 1 (D, K, one can assign its orthogonal complement l in H 1 (D, K(1. Thus we have Proposition 4.6. Suppose L = Q p. The correspondence of a line in H 1 (D, K and its orthogonal complement in H 1 (D, K(1 gives a one-to-one correspondence: { } { } Nontrivial extensions nontrivial infinitesimal. of K by K(1 as K[D]-modules deformations of K over D Normalize the Artin symbol [x, Q p ] so that N ([u, Q p ] = u 1 for u Z p, [p, Q p ] is the arithmetic Frobenius element. Let σ q = [q, Q p ] 1. Then we have γ q, ξ = ξ(σ q for γ q H 1 (D, Q p (1 and ξ Hom(D, Q p = H 1 (D, Q p. Now we are ready to prove the following version of a theorem of Greenberg-Stevens (cf. [GS1] 2.3.4: Theorem 4.7. Let E /L be an elliptic curve with split multiplicative reduction, and let ψ : Gal(Q p /Q p Q p be a nontrivial character which is congruent to 1 modulo ε. Let T = T p E Q p for the p-adic Tate module T p E of E, V be the induced Galois representation Ind Qp L T and q E L be the Tate period of E. Then the following statements are equivalent: (a dψ dε (σ N L/Qp (q E = 0; (b W := F 00 V/F 11 V corresponds to Q p (ψ under the correspondence of Proposition 4.6; (c There is an infinitesimal deformation W of W and a commutative diagram: Q p (1 W Q p (ψ Q p (1 W Q p,

24 1366 Haruzo Hida in which the top row is an exact sequence of Q p [D]-modules and the vertical map is the reduction modulo ε. Proof. Since γ q, ξ = ξ(σ q for ξ H 1 (D, Q p and γ q H 1 (D, Q p (1, applying these formulas to ξ = dψ dε, we get (a (b by the definition of the correspondence in Proposition 4.6. The equivalence (b (c can be proven in exactly the same manner as in the proof of [GS1] Here is the argument proving (b (c. Let c be a 1-cocycle representing γ Q for Q = N L/Qp (q E. Then D D (σ, τ c(σ dψ dε (τ Q p(1 is the 2-cocycle representing the cup product γ Q [ Q p (ψ], which vanishes by (b. Thus it is a 2-coboundary: c(σ dψ (τ = ξ(σ, τ = ξ(στ N (σξ(τ ξ(σ dε for a 1-chain ξ : D Q p (1. Then defining an action of σ D on Q 2 p via the matrix multiplication by, the resulting Q p [D]-module W fits well in the diagram in (c. ( N (σ c(σ+ξ(σε 0 ψ(σ Conversely suppose we have the commutative diagram as in (c, which can be written as the following commutative diagram with exact rows and columns: Q p (1 W Q p 0 0 Q p (1 W Q p 0 0 Q p (1 W Q p The connecting homomorphism d : H 1 (D, Q p (1 H 2 (D, Q p (1 vanishes because the leftmost vertical sequence splits. On the other hand, letting δ ψ : H 0 (D, Q p H 1 (D, Q p stand for the connecting homomorphism of degree 0 coming from the rightmost vertical sequence, and letting δ i : H i (D, Q p H i+1 (D, Q p (1 be the connecting homomorphism of degree i associated to the bottom row (and also to the top row, by the commutativity of the diagram, we

25 get the following commutative square: L-invariants of Tate Curves 1367 H 0 δ (D, Q p = Q 0 p H 1 (D, Q p (1 δ d=0 ψ H 1 (D, Q p δ1 H 2 (D, Q p (1. Since δ ψ (1 = dψ dψ dε, we confirm dε Ker(δ 1. By Proposition 4.1, γ Q is the in the image of δ 0. Thus the assertion (b follows if we can show that Ker(δ 1 is orthogonal to Im(δ 0. Since V = Ind Q L T is the p-adic Tate module of the principally polarized abelian variety A = Res L/Qp E /L (the Weil restriction, V is self dual under the polarization pairing, which induces a self duality of W and also the self (Cartier duality of the exact sequence 0 Q p (1 W Q p 1. In particular the inclusion ι : Q p (1 W and the projection π : W Q p are mutually adjoint under the pairing. Thus the connecting maps δ 0 : H 0 (D, Q p H 1 (D, Q p (1 and δ 1 : H 1 (D, Q p H 2 (D, Q p (1 are mutually adjoint each other under the Tate duality pairing. In particular, Im(δ 0 is orthogonal to Ker(δ 1. Take a prime p p in F, and let D = Gal(F p /F p (L = F p and D = Gal(F p /Q p. We write I (resp. I for the inertia group of D (resp. D. Lemma 4.8. Let ρ A : Gal(F /F GL 2 (A be a deformation of ρ E for an artinian local K-algebra A with residue field K. Write ρ A D = ( 0 δ A with δa α p mod m A. Suppose that α p can be extended to a character α p : D K. If δ A I factors through Gal(F p [µ p ]/F p, the character δ A extends to a unique character δ A of D with values in A such that δ A α p mod m A. Proof. Let Fp ab (resp. Fp ur be the maximal abelian extension of F p (resp. the maximal unramified extension of F p. Then we have F p [µ p ] Fp ur [µ p ] = F p Q ur p [µ p ] = F p Q ab p. So Gal(F p Q ur p [µ p ]/F p is identified with the subgroup Gal(Q ab p /Q ab p F p of Gal(Q ab p /Q p of finite index. Since δ A is a character of Gal(F p Q ur p [µ p ]/F p, regarding it as a character of Gal(Q ab p /Q ab p F p, we only need to extend it to Gal(Q ab p /Q p. Since F p Q ab p /Q p is a finite Galois extension with an abelian Galois group, by the theory of the Schur multiplier, the obstruction of extending character lies in H 2 (, A (see [MFG] Section Since δ A α p mod m A,

26 1368 Haruzo Hida the obstruction class Ob(δ A Ob(α p = 0 mod m A. Thus Ob(δ A H 2 (, 1 + m A. Since 1 + m A is uniquely divisible (by log : 1 + m A = ma as K-vector spaces, we get the vanishing H 2 (, 1 + m A = 0 for the finite group. Then we can extend δ A to δ A with δ A α p mod m A as proven in [MFG] Section 5.4. If δ is another extension with δ α p 1 mod m A, we find δ A δ is a character of, which has to be trivial by the condition δ A = αp mod m A. Thus the extension is unique. 5. Symmetric power of Tate curves In this section, we state a conjectural formula of the L-invariant of the L- function of symmetric power p-adic L-functions of elliptic curves with semi-stable ordinary reduction at p. We prove the conjecture for the adjoint square arithmetic L-function under mild assumptions (Theorem A general conjecture and a proof of the theorem. Take an elliptic curve E with multiplicative reduction over the finite extension L /Qp. Let T = T p E Zp Q p. Let T (n, m be the symmetric n-th power of T with ( m-th Tate twist: We then put V(n, m = Ind Qp L T (n, m = (Sym n T ( m. T (n, m. Suppose 0 m < n. We have a decreasing filtration F k X of X = T (n, k and V(n, k stable under D = Gal(Q p /L so that an open subgroup of the inertia group I of D acts on F k X/F k+1 X by the k-th power of the p-adic cyclotomic character. We have F j V(n, m = F j T (n, m. We put F + X = F 1 X and F X = F 0 X. We define F 00 V(n, m Ind Qp L V(n, m so that F 00 V(n, m F + V(n, m and F 00 V(n,m = H 0 (Q F + V(n,m p, Similarly, we define F 11 V(n, m F + V(n, m so that Put D = Gal(Q p /Q p. Lemma 5.1. We have F + V(n, m/f 11 V(n, m = H 0 (Q p, F + V(n, m( 1. dim Qp F + V(n, m/f 11 V(n, m = dim Qp F 00 V(n, m/f + V(n, m = 1 and a Tate extension of D modules 0 Q p (1 F 00 V(n, m/f 11 V(n, m Q p 0. V(n,m F + V(n,m.