Completed Cohomology and Kato's Euler System for Modular Forms

COMPLETED COHOMOLOGY AND KATO’S EULER SYSTEM FOR MODULAR FORMS

YIWEN ZHOU

Abstract. In this paper, we compare two diﬀerent constructions of p-adic L-functions for modular forms and their relationship to Galois cohomology: one using Kato’s Euler system and the other using Emerton’s p-adically completed cohomology of modular curves. At a more technical level, we prove the equality of two elements of a local Iwasawa cohomology group, one arising from Kato’s Euler system, and the other from the theory of modular symbols and p-adic local Langlands correspondence for GL2(Qp). We show that this equality holds even in the cases when the construction of p-adic L-functions is still unknown (i.e. when the modular form f is supercuspidal at p). Thus, we are able to give some representation-theoretic descriptions of Kato’s Euler system. We expect this method to have some generalizations for groups other than GL2/Q.

Contents Introduction 1 Acknowledgement 4 1. Notations and conventions4 2. Special values of L-function in terms of modular symbols5 3. The p-adic Kirillov model, supercuspidal case 11 4. An explicit reciprocity law, supercuspidal case 15 ± 5. Images of zM (f) under dual exponential maps, supercuspidal case 21 6. Explicit p-adic Local Langlands correspondence, principal series case 25 ± 7. Images of zM (f) under dual exponential maps, principal series case 29 8. Comparison with Kato’s Euler system 32 References 34

Introduction

n Let f be a cuspidal newform of weight k ≥ 2 and level Γ0(p ) ∩ Γ1(N), Vf be the p-adic Galois representation attached to f. In [16], Kato constructed for every cuspidal Hecke eigenform f an ∗ Euler system zKato(f) in the global Iwasawa cohomology of the dual Galois representation Vf attached to f. The element zKato(f) is the key object for studying both the p-adic interpolation properties of classical L-functions and the p-adic arithmetic information of the modular form f. For example, an important property of zKato(f) we will be using in this paper is that the images of zKato(f) under various dual exponential maps compute the special values of the classical L-functions

Date: December 8, 2018. 1 2 YIWEN ZHOU of f and its twists by characters of p-power conductors. We will describe the precise statement in Theorem 0.3 below. Another approach to studying the p-adic interpolation properties of classical L-functions of modular forms is via modular symbols. In [11] and [12], Emerton rephrases this approach by regarding the modular symbol {0 − ∞} as a functional on p-adically complected cohomology of modular curves. Combining the construction of this functional with the p-adic local-global compatibility [13] and Colmez’s theory of p-adic local Langlands correspondence [9], we may regard {0 − ∞} as an element zM (f) (the letter “M” stands for modular symbols) in the local Iwasawa cohomology ∗ of Vf .

The precise deﬁnition of zM (f) is as follows: according to [7] and [9], there are isomorphisms ∼ ∗ 1 ∗ = ∗ ψ=1 Exp :HIw(Qp,Vf ) −→ D(Vf ) and p 0 =1 ∼ ∗ 0 1 = ∗ ψ=1 C : (Π(Vf ) ) −→ D(Vf ) , ∗ ∗ where D(Vf ) is the (φ, Γ)-module attached to Vf , Π(Vf ) is the p-adic Banach space representation 1 attached to Vf by the p-adic local Langlands correspondence . Using Emerton’s local-global compatibility [13], the modular form f gives a (pair of) GL2(Qp)- equivariant embedding (which is actually a GL2(Qp)-equivariant isomorphism) ± 1 p ±,f (0.1) Φf : Π(Vf ) ,→ Hec (K ) p 1 p p where K is the tame level of f, Hec (K ) is the completed cohomology of tame level K . We denote the composite

Φ± f 1 p ±,f {0−∞} Π(Vf ) −−→ Hec (K ) −−−−→ Cp ∗ ± as an element in Π(Vf ) by Mf . In fact, we have (Lemma 2.3) p 0 =1 ± ∗ 0 1 Mf ∈ (Π(Vf ) ) . ± ∗ −1 ± 1 ∗ + − We deﬁne zM (f) := (Exp ) ◦ C(Mf ) ∈ HIw(Qp,Vf ), and zM (f) := zM (f) − zM (f) ∈ 1 ∗ HIw(Qp,Vf ).

The cohomology classes zKato(f) and zM (f) arise from very diﬀerent perspectives of the p-adic arithmetic of the modular form f. On the other hand, both of them lie in the Iwasawa cohomology group attached to f, and in fact both encode special L-values of f and its twists. Thus it is natural to ask what the relationship between these two elements is. Our main result answers this question:

1 ∗ Theorem 0.1. If Vf |G is absolutely irreducible, then zM (f) = zKato(f) as elements in H ( p,V ). Qp Iw Q f

From the perspective of the special L-values they encode, zKato(f) comes from the Rankin- Selberg method, and zM (f) comes from the Mellin transforms, so the above theorem compares these two diﬀerent integral formulas for L-functions of modular forms. It would be interesting to have a direct comparison, but our approach is to use the equality of special values under dual exponential maps (computed either way for zKato(f) and zM (f)) as the basic input.

1Our notation is slightly diﬀerent from [9]: the Π(V ) in this paper is denoted by Π(V (1)) in [9]. COMPLETED COHOMOLOGY AND KATO’S EULER SYSTEM FOR MODULAR FORMS 3

The strategy of proving Theorem 0.1 is to show that the images of both zM (f) and zKato(f) under various dual exponential maps are the same, and then use Proposition II.3.1 of [4] to conclude 1 ∗ that zM (f) = zKato(f) as elements in HIw(Qp,Vf ).

1 ∗ Theorem 0.2. If Vf |G is absolutely irreducible, then the element zM (f) ∈ H ( p,V ) satisﬁes Qp Iw Q f the following property: ∗ ∼ n For any 0 ≤ j ≤ k − 2, any ﬁnite order character φ of Zp = ΓQp with conductor p , Z ! ∗ −j 1 Λe(p)(f, φ, j + 1) j exp φ(x)x · zM (f) = · · f φ · t , ∗ τ(φ) j! Zp where Γ(j + 1) L(p)(f, φ, j + 1) n Λe(p)(f, φ, j + 1) = j+1 · ± ∈ Q(f, µp ), (2πi) Ωf

L(p)(f, φ, j + 1) is obained by removing the Euler factor at p from the classical L-function of f twisted by φ, τ(φ) is the Gauss sum of the character φ, and ! j X a j j 0 ∗ f φ · t := φ(a)f ⊗ ζpn · t = f · eφ · t ∈ Fil DdR(Vf (φ)(−j)). a Two key ingredients used in proving Theorem 0.2 are the theory of p-adic Kirillov models of locally algebraic representations of GL2(Qp) introduced in [9] (Section VI.2.5), and an “explicit reciprocity law” introduced in [9] (Proposition VI.3.4), [10] (Th´eor`eme 8.3.1) and [14] (Theorem 5.4.3).

On the other hand, Kato in [16] showed that zKato(f) has the following interpolation property:

1 ∗ Theorem 0.3 (Kato, [16], Theorem 12.5). The element zKato(f) ∈ HIw(Qp,Vf ) satisﬁes the following property: ∗ ∼ n For any 0 ≤ j ≤ k − 2, any ﬁnite order character φ of Zp = ΓQp with conductor p , Z ! ∗ −j 1 Λe(p)(f, φ, j + 1) j exp φ(x)x · zKato(f) = · · f φ · t . ∗ τ(φ) j! Zp

Comparing the formulas in Theorem 0.2 and Theorem 0.3, and using Proposition II.3.1 of [4], we obtain Theorem 0.1.

Organization of the paper. We compute in Section 2 the evaluations of the modular symbol ± ˆ 1 p ±,f {0 − ∞} on locally algebraic vectors of Π(Vf ) under the embedding Φf : Π(Vf ) ,→ Hc (K ) given by the modular form f. Section 3 - 7 compute the images of zM (f) under various dual exponential maps, with the assumption that Vf |G is absolutely irreducible. These sections are divided into two parts: Section Qp 3 - 5 deal with the supercuspidal case (meaning that the smooth representation of GL2(Qp) attached to f is supercuspidal), and Section 6 & 7 deal with principal series case (meaning that the smooth representation of GL2(Qp) attached to f is a principal series). In Section 3, we introduce the theory of p-adic Kirillov models for locally algebraic representations of GL2(Qp), and use the explicit formulas of the p-adic Kirillov models to compute the images 4 YIWEN ZHOU

− of locally algebraic vectors of Π (Vf ) under the maps ιm. In section 4, we describe an “explicit reci- ± − procity law”, and use it to compute the image of zM (f) under the maps ιm. In section 5, we present and give a proof of the formulas for the images of zM (f) under various dual exponential maps in the supercuspidal case. In section 6, we describe explicitly the p-adic Local Langlands correspondence, ± + ∗ and then use results from [11] to describe explicitly the image of z (f) in B ⊗ Dcrys(V ). M rig,Qp Qp f In section 7, we deduce the formulas for the images of zM (f) under various dual exponential maps in the principal series case. In Section 8, we conclude zM (f) = zKato(f) in both the supercuspidal and the principal series cases, under the assumption that Vf |G is absolutely irreducible. Qp Acknowledgement. I’d like to thank my advisor, Matthew Emerton, for suggesting me working on this project and sharing with me his profond insight and ideas. I’d like to thank Robert Pollack for pointing out one key step of reaching the ﬁnal result of this paper, and for kindly sharing his drafts and notes with me. I’d like to thank Lue Pan, who has provided me endless help and encouragement on this project. I’d like to thank Gong Chen, Pierre Colmez, Yiwen Ding, Yongquan Hu, Joaqu´ınRodrigues Jacinto, Kazuya Kato, Ruochuan Liu, Xin Wan, Haining Wang, Jun Wang, Shanwen Wang for very helpful discussions.

1. Notations and conventions We denote Γ the Galois group Gal (Qp(ζp∞ )/Qp), and H the Galois group Gal Qp/Qp(ζp∞ ) . We ∼ ∗ = ∗ identify Γ with Zp via the cyclotomic character cycl : Γ −→ Zp. ∗ P ( ) is the subgroup of GL ( ) consisting of matrices of the form Qp Qp . Qp 2 Qp 0 1

Throughout this paper, f denotes a (normalized) classical cuspidal newform of weight k ≥ 2 and n level Γ0(p ) ∩ Γ1(N). Vf is the cohomological Galois representation of GQ attached to f. Thus the 2 restriction Vf |G has Hodge-Tate weight 0 and 1 − k. Qp Let F be a finite extension of Q which is sufficiently large, that is, containing all the Fourier coefficients of f and values of χ when a finite order character χ is chosen. We choose λ a place of F lying over p, and denote L := Fλ the localization of F at λ. For any integer n ≥ 0, we write n n Ln := L⊗Qp Qp(ζp ). Here, {ζp }n≥1 is chosen to be a compatible system of p-power roots of unity. We also write Qp,n := Qp(ζpn ) sm We denote πp (f) the p-adic smooth representation of GL2(Qp) attached to f, and Π(Vf ) the p-adic Banach space representation of GL2(Qp) attached to f. Our notation is slightly different from [9]: the Π(V ) in this paper is denoted by Π(V (1)) in [9]. We will only work with the cases when Vf |G is absolutely irreducible. This happens precisely Qp sm sm when πp (f) is supercuspidal, or when πp (f) is some twist of an unramified principal series. The modular form f is a holomorphic section of some line bundle on the modular curve, therefore 0 can be viewed naturally as an element in Fil DdR(Vf (k−1)). Similarly, the complex conjugate of f, 0 0 ∗ which we denote by f, can be viewed naturally as an element in Fil DdR(Vf (k−1)) = Fil DdR(Vf ).

2Here, the convention is that the cyclotomic character has Hodge-Tate weight 1. COMPLETED COHOMOLOGY AND KATO’S EULER SYSTEM FOR MODULAR FORMS 5

There is a natural pairing 1 [,] : D V ∗ × D (V (1)) → D (L(1)) =∼ L, dR dR f dR f dR t 0 ∗ 0 under which Fil DdR Vf and Fil DdR (Vf (1)) are orthogonal complement of each other. We ∗ 0 ∗ 1 deﬁne f the unique element in DdR (Vf (1)) /Fil DdR (Vf (1)) such that [f, f ]dR = t .

2. Special values of L-function in terms of modular symbols In this section, we compute the modular symbol {0 − ∞} evaluated at locally algebraic vectors of Π(Vf ) under the embedding (0.1).

n p Let f be a cuspidal newform of weight k ≥ 2 and level Γ0(p ) ∩ Γ1(N). We denote K the tame level of f, i.e. p a b p K := ∈ GL2(Zb ) c ≡ 0 mod N, d ≡ 1 mod N , c d and Y (Npn) the modular curve defined over Q whose C points are given by n / . n p × Y (Np )(C) = GL2(Q) GL2(AQ) Γ(p ) · K · C . n n ◦ n Let Y (Np ) denote the base change to (ζ n ) of Y (Np ), and Y (Np ) denote the con- Q(ζNpn ) Q Np nected component of Y (Npn) . We also write X◦(Npn) (resp. X(Npn)) the compactification Q(ζNpn ) of Y ◦(Npn) (resp. Y (Npn)). Fix an embedding ι : ,→ . There is an action of complex conjugation τ on Y (Npn) . Q C Q(ζNpn ) The action of τ on π Y (Npn) ⊂ ( /(Npn))× coincides with the one induced from multi- 0 Q(ζNpn ) Z × plication by −1 on (Z/(Npn)) . We let F be a finite extension of Q that contains all Fourier coefficients of f and values of χ when a finite order character χ is chosen in the future. Choose λ a place of F lying over p, and denote L := Fλ the localization of F at λ. The completed cohomology of the modular curve with tame level Kp and coefficient L is defined as 1 p 1 n m H (K ) := L ⊗O lim lim H Y (Np ) , /p . ec L ←− −→ ét,c Q Z m n p This is a p-adic Banach space equiped with a continuous action of GL2(Qp) × H × {Id, τ}.

◦ n n The modular symbol {0−∞} ∈ H1(X (Np )(C), cusps; Z) ⊂ H1(X(Np )(C), cusps; Z) for all n, n+1 n and are compatible with respect to the map H1(X(Np )(C), cusps; Z) → H1(X(Np )(C), cusps; Z) induced from the natural projection X(Npn+1) → X(Npn). Note that the homology theory we are using here is the singular homology. We then have for every positive integer m, ! {0 − ∞} ∈ lim H (X(Npn)( ), cusps; /pm) = Hom lim H1 (Y (Npn)( ), /pm) , /pm . ←− 1 C Z −→ c C Z Z n n Under the identiﬁcation H1 (Y (Npn)( ), /pm) =∼ H1 Y (Npn) , /pm , C Z ´et,c Q Z 6 YIWEN ZHOU

we can view ! (2.1) {0 − ∞} ∈ Hom lim H1 Y (Npn) , /pm , /pm −→ ´et,c Q Z Z n for every m, hence 0 1 p (2.2) {0 − ∞} ∈ Hec (K ) b 1 p where the right hand side is the bounded dual of the p-adic Banach space Hec (K ).

We denote Wk−2 the contragredient to the (k − 2)-nd symmetric power of the standard repre- 0 n k−2 1 sentation of GL2. Since f ∈ H Y (Np ), ω ⊗ Ω is a holomorphic cuspidal newform of weight k ≥ 2, we have a period map:

0 n k−2 1 1 n per : H Y (Np ), ω ⊗ Ω → H Y (Np )( ), V ˇ ⊗ . c C Wk−2(Q) Q C If we denote by i∗ the restriction

∗ 1 n 1 ◦ n 1 i : H Y (Np )( ), V ˇ ⊗ → H Y (Np )( ), V ˇ ⊗ = H Γ, Wˇ k−2( ) , c C Wk−2(Q) Q C c C Wk−2(Q) Q C c C then we have    Z  ∗ X i i j ˇ (2.3) i per(f) = γ ∈ Γ 7→ f(z)z dz · e1e2 ∈ Wk−2(C) .  i+j=k−2 γ  a b Here, we consider 2 having the standard basis e and e , and the action of any matrix ∈ Q 1 2 c d a b a b GL ( ) on 2 is given by formulas e = ae + ce and e = be + de . The basis 2 Q Q c d 1 1 2 c d 2 1 2 ˇ k−2 2 i j of Wk−2(Q) = Sym Q is chosen as e1e2 for any i + j = k − 2. a b Remark 2.1. If g = ∈ GL ( )+ normalizes Γ, then there is an action of g on the group c d 2 R 1 ◦ n ∼ 1 −1 H Y (Np )( ), V ˇ = H Γ, Wˇ k−2( ) given by formula: (g.c)(γ) = g. c(g γg) . c C Wk−2(C) c C In particular, we have: b i Z − a ∗ X k−2 az + b i j (2.4) (g. (i per(f))) ({0 − ∞}) = f(z)(cz + d) dz · e1e2. d cz + d i+j=k−2 − c

The complex conjugation τ acting on Y (Npn)(C) induces an action of complex conjugation on 1 n H Y (Np )( ), V ˇ , which we still denote by τ. We have c C Wk−2(Q)   Z ∗  X i i j  (2.5) i τ (per(f)) = γ ∈ Γ 7→ f(z)(−z) dz · e1e2.  i+j=k−2 τ(γ) 

Here, we are identifying Y ◦(Npn)(C) as a quotient of the complex upper half plane, and the action of τ on the upper half plane is given by reﬂecting along the imaginary axis. COMPLETED COHOMOLOGY AND KATO’S EULER SYSTEM FOR MODULAR FORMS 7

± 1 n We write H Y (Np )( ), V ˇ the ± eigenspace of the complex conjugation. We then c C Wk−2(Q) have the ± period maps: ±,f ± 0 n k−2 1f 1 n (2.6) per : H Y (Np ), ω ⊗ Ω → H Y (Np )( ), V ˇ ⊗ , c C Wk−2(Q) Q C one has for any 0 ≤ j ≤ k − 2,

 R f(z)zj dz+R f(z)zj dz  γ ∈ Γ 7→ P γ τ(γ) · ei ej , if (−1)j = ±1;  i+j=k−2 2 1 2 (2.7) i∗per±(f) = R f(z)zj dz−R f(z)zj dz  γ ∈ Γ 7→ P γ τ(γ) · ei ej , if (−1)j = ∓1.  i+j=k−2 2 1 2

± It is well known (see for example, [18], page 11) that there are complex numbers Ωf ∈ C such that for any 0 ≤ j ≤ k − 2, R j R j γ f(z)z dz ± τ(γ) f(z)z dz (2.8) ± ∈ F, 2Ωf where the signs in the numerator and the denominator are the same if (−1)j = 1, diﬀerent if otherwise (−1)j = −1. Therefore, we can deﬁne:

f ±,f (2.9) per± : H0 Y (Npn), ωk−2 ⊗ Ω1 → H1 Y (Npn)( ), V , F c C Wˇ k−2(F ) using the formula:

 R f(z)zj dz+R f(z)zj dz  γ ∈ Γ 7→ P γ τ(γ) · ei ej , if (−1)j = ±1  i+j=k−2 2Ω± 1 2 (2.10) i∗per±(f) = f F R f(z)zj dz−R f(z)zj dz  P γ τ(γ) i j j  γ ∈ Γ 7→ i+j=k−2 ± · e1e2 , if (−1) = ∓1  2Ωf where 0 ≤ j ≤ k − 2. Remark 2.2. We then have: + + − − per(f) = Ωf · perF (f) + Ωf · perF (f);(2.11) + + − − (2.12) τ(per(f)) = Ωf · perF (f) − Ωf · perF (f).

± Recall that λ is a place of F lying above p, and L = Fλ. Let perL denote the composite of isomorphism ± ± 1 n ∼ 1 n H Y (Np )( ), V ˇ ⊗F L = H Y (Np ) , V ˇ c C Wk−2(F ) ´et,c Q Wk−2(L) ± with perF .

Let Vf be the cohomological Galois representation of GQ attached to f with coeﬃcients in L. Thus the restriction Vf |G has Hodge-Tate weights 0 and 1 − k. Let Π (Vf ) (with convention same Qp as in [13]) be the p-adic Banach space representation of GL2 ( p) attached to Vf |G via the p-adic Q Qp local Langlands correspondence. 8 YIWEN ZHOU

As is mentioned in the introduction, according to Emerton’s local-global compatibility [13], the newform f gives a (pair of) GL2(Qp)-equivariant embedding, which are in fact isomorphisms: ± 1 p ±,f 1 p Φf : Π(Vf ) ,→ Hec (K ) ⊂ Hec (K ) by choosing a new vector at every place away from p. We denote the composite

Φ± f 1 p ±,f {0−∞} Π(Vf ) −−→ Hec (K ) −−−−→ Cp ∗ ± as an element in Π(Vf ) by Mf . 1 p Notice that if g ∈ GL2 (Q), then there is an action of g on Hec (K ) as a diagonal element in GL ( ) × Hp × {Id, τ} (induced from the action of GL ( ) on H1 := lim H1(Kp)), where the 2 Qp 2 AQ ec −→ ec Kp action of g through the last factor is Id if det(g) is positive, and τ if det(g) is negative. We have the following lemma: pZ Lemma 2.3. The actions of matrices of the form Z on H1(Kp) as a diagonal element in 0 1 ec p ± GL2 (Qp) × H × {Id, τ} and as an element in GL2 (Qp) coincide. In particular, we have Mf ∈ p 0 =1 ∗ 0 1 (Π(Vf ) ) . pZ ∗ ∗ Proof. The ﬁrst assertion follows from the fact that Z belongs to ⊂ GL ( ) for 0 1 0 1 2 Zl every l 6= p and has positive determinant. p 0 The second assertion follows from the fact that the modular symbol {0 − ∞} is ﬁxed by 0 1 p as an element in GL2 (Qp) × H × {Id, τ}.

We can then define ± ∗ −1 ± 1 ∗ zM (f) := (Exp ) ◦ C(Mf ) ∈ HIw(Qp,Vf ), where the isomorphisms ∼ ∗ 1 ∗ = ∗ ψ=1 Exp :HIw(Qp,Vf ) −→ D(Vf ) and p 0 =1 ∼ ∗ 0 1 = ∗ ψ=1 C : (Π(Vf ) ) −→ D(Vf ) ∗ ∗ are as described in [7] and [9]. Here, D(Vf ) is the (φ, Γ)-module attached to Vf as defined in [9]. There is an isomorphism: ∼ 1 n = 1 p (2.13) W (L) ⊗ lim H Y (Np ), V ˇ −→ H (K ) . k−2 L −→ ét,c Wk−2(L) ec Wk−2−lalg n This isomorphicm is equivariant for the complex conjugation τ, where the action of τ on the left is given by Id ⊗ τ. ± We still use perL to denote the following composition: (2.14) !± ± ± 0 n k−2 1 perL 1 n 1 n H Y (Np ), ω ⊗ Ω −−−→ H Y (Np ) , V ˇ → lim H Y (Np ), V ˇ . ét,c Q Wk−2(L) −→ ét,c Wk−2(L) n COMPLETED COHOMOLOGY AND KATO’S EULER SYSTEM FOR MODULAR FORMS 9

Classical Eichler-Shimura theory tells us that for any choice of sign ±, there is a unique GL2(Qp)- equivariant embedding

!±,f sm 1 n π (f) ,→ lim H Y (Np ), V ˇ p −→ ´et,c Wk−2(L) n

sm where πp (f) is the p-adic smooth representation of GL2(Qp) attached to f (or equivalently, attached to Weil-Deligne representation associated to Vf |G via the local Langlands correspodence), Qp sm ± under which the image of vnew ∈ πp (f) equals perL (f). Thus we have the following commutative diagram:

±,f ! ±,f 1 n ∼= 1 p {0−∞} W (L) ⊗ lim H Y (Np ), V ˇ H (K ) k−2 L −→ ´et,c Wk−2(L) ec Wk−2−lalg Cp n Φ± Id⊗Eichler-Shimura f ∼= Wk−2(L) ⊗L πp Π(Vf )Wk−2−lalg

± Mf

k−2 ∗ k−3 ∗ Write the weight vectors of Wk−2 to be v0 = vhw = (e2 ) , v1 = (e1e2 ) , ··· , vk−3 = k−3 ∗ k−2 ∗ (e1 e2) , vk−2 = vlw = (e1 ) . we then have the following lemma:

Lemma 2.4. For any 0 ≤ j ≤ k − 2,

 R 0 j i∞ f(z)z dz j ± ±  ± , if (−1) = ±1; Ωf (2.15) Mf (vj ⊗ vnew) = {0 − ∞}, vj ⊗ perL (f) (ej ek−2−j ) = 1 2 0 , if (−1)j = ∓1.

According to [18], for any ﬁnite order chraracter χ of conductor pn and any 0 ≤ j ≤ k − 2, we have the formula:

  j Z 0 k (−2πi) n( k −j−2) X j L(f, χ, j + 1) = · p 2 · τ(χ) · χ(a) · 2πi · f  (z) · z dz 1 −a j! n i∞ a mod p 0 pn − a (−2πi)j X Z pn = · pn(−j−1) · τ(χ) · χ(a) · 2πi · f(z) · (pnz + a)jdz. j! a mod pn i∞ 10 YIWEN ZHOU

a Notice that if we denote {− pn − ∞} by γ, then by equation (2.10), we have:

a a Z − pn Z + pn f(z) · (pnz + a)jdz + f(z) · (pnz − a)jdz i∞ i∞ Z Z = f(z) · (pnz + a)jdz + f(z) · (pnz − a)jdz γ τ(γ) * j ! + a X = 2Ω± · − − ∞ , Ctpntaj−tv ⊗ per±(f) , if (−1)j = ±1, f pn j t F t=0 and similarly,

a a Z − pn Z + pn f(z) · (pnz + a)jdz − f(z) · (pnz − a)jdz i∞ i∞ Z Z = f(z) · (pnz + a)jdz − f(z) · (pnz − a)jdz γ τ(γ) * j ! + a X = 2Ω± · − − ∞ , Ctpntaj−tv ⊗ per±(f) , if (−1)j = ∓1. f pn j t F t=0 Therefore,

− a (−2πi)j X Z pn L(f, χ, j + 1) = · pn(−j−1) · τ(χ) · χ(a) · 2πi · f(z) · (pnz + a)jdz j! a mod pn i∞ − a (−2πi)j 1 X Z pn = · pn(−j−1) · τ(χ) · χ(a) · 2πi · f(z) · (pnz + a)jdz j! 2 a mod pn i∞ a Z + pn + χ(−1) f(z) · (pnz − a)jdz i∞ * j ! + (−2πi)j 1 X a X = · pn(−j−1) · τ(χ) · χ(a) · 2πi · 2Ω± · − − ∞ , Ctpntaj−tv ⊗ per±(f) j! 2 f pn j t F a mod pn t=0 (Here the sign ± is chosen such that ±1 = χ(−1) · (−1)j.)

j+1 ± * n −1 n −1 ! + (2πi) Ωf X p a p a =(−1)j · · · τ(χ) · χ(a) · ·{0 − ∞} , · v ⊗ per±(f) Γ(j + 1) pn(j+1) 0 1 0 1 j F a mod pn j+1 ± n (2πi) Ωf X p a =(−1)j · · · τ(χ) · χ(a) · {0 − ∞} , v ⊗ per±(f) Γ(j + 1) pn(j+1) j 0 1 F a mod pn pn a (Here we are using Remark 2.1 about the action of on image(per).) 0 1 j+1 ± (2πi) Ωf X n =(−1)j · · · τ(χ) · χ(a) ·M± v ⊗ p a v . Γ(j + 1) pn(j+1) f j 0 1 new a mod pn COMPLETED COHOMOLOGY AND KATO’S EULER SYSTEM FOR MODULAR FORMS 11

Here in the last equality, we are using Lemma 2.3 to identify the (global) action of the matrix pn a on the modular symbols and the (local) action of it on Π(V ). 0 1 f

We write for all 0 ≤ j ≤ k − 2, Γ(j + 1) L(f, χ, j + 1) (2.16) Λ(e f, χ, j + 1) = j+1 · ± (2πi) Ωf where the sign ± is chosen so that χ(−1)·(−1)j = ±1. Thus we have obtained the following lemma: Lemma 2.5. For any ﬁnite order character χ of conductor pn and any integer 0 ≤ j ≤ k − 2, we have j n (−1) τ(χ) X ± p a (2.17) Λ(e f, χ, j + 1) = · χ(a) ·M vj ⊗ vnew , pn(j+1) f 0 1 a mod pn where the sign ± is chosen to satisfy χ(−1) · (−1)j = ±1.

2 We deﬁne X pn a F := χ(a) · v χ 0 1 new a mod pn when the conductor of χ is pn. Then Lemma 2.5 can also be stated as: Lemma 2.6. For any 0 ≤ j ≤ k − 2, any ﬁnite order character χ of conductor pn with n ≥ 0, let X pn a F := χ(a) · v , χ 0 1 new a mod pn then we have for all 0 ≤ j ≤ k − 2,

n(j+1) ± j p (2.18) M (vj ⊗ Fχ) = (−1) · · Λ(e f, χ, j + 1). f τ(χ) Here, the sign ± is chosen such that χ(−1) = ±(−1)j.

j ± 2 Remark 2.7. If the sign ± is such that χ(−1) 6= ±(−1) , then Mf (vj ⊗ Fχ) = 0.

3. The p-adic Kirillov model, supercuspidal case sm Throughout this section, we assume πp (f), the smooth p-adic representation of GL2 (Qp) attached to the modular form f, is supercuspidal. We recall the theory of p-adic Kirillov models developed in [9], chapter VI. 12 YIWEN ZHOU

If Π is a p-adic locally algebraic representation of GL2(Qp) with coefficient field L that can be a b written as Π = det ⊗Sym ⊗ π where a, b ∈ Z, b ≥ 0 and π is a smooth representation of GL2(Qp), then the p-adic Kirillov model of Π is the unique P (Qp)-equivariant embedding Y a a+b+1 K :Π → t L∞[t]/t , Z where the action of P (Qp) on the RHS is given by the formula a b n n (3.1) S = (1 ⊗ ε (bp )) · exp (bp t) · (1 ⊗ σ ∗ ) S 0 1 a v(a)+n n Q a a+b+1 ∗ for any S ∈ t L∞[t]/t and any n ∈ Z. Here, a := |a|p·a, and ε is defined in the following Z r r way: we let ε(x) = ζpn if x ≡ pn mod Zp, where {ζpn }n≥0 is the compatible system of p-power roots of unity chosen as in Section1. Remark 3.1. The usual definition of the Kirillov model of a p-adic locally algebraic representation (as a b [a,a+b] ∗ a a+b+1 in [9]) is a P (Qp) equivariant map K from Π = det ⊗Sym ⊗ π to LP Qp, t L∞[t]/t , [a,a+b] ∗ a a+b+1 ∞ where L∞ = L ⊗Qp Qp(ζp ). There is an action of Γ on LP Qp, t L∞[t]/t given by formula −1 (3.2) σu(S)(x) = (1 ⊗ σu) (S(ux)) ∗ ∗ [a,a+b] ∗ a a+b+1 for any u ∈ Zp and x ∈ Qp, which commutes with the action of P (Qp) on LP Qp, t L∞[t]/t . Therefore the uniqueness of the p-adic Kirillov model forces the image of K to be contained in [a,a+b] ∗ a a+b+1Γ Q a a+b+1 LP , t L∞[t]/t , which can then be identified as t L∞[t]/t by sending ev- Qp Z [a,a+b] ∗ a a+b+1Γ n ery S ∈ LP Qp, t L∞[t]/t to {S(p )}n∈Z. In the rest of this paper, we will always Q a a+b+1 regard a p-adic Kirillov function as an element in t L∞[t]/t . Z

lalg In the case when Π = Π(Vf ) is the collection of locally algebraic vectors of the p-adic Banach ∨ lalg k−2 2 sm space representation Π(Vf ) of GL2(Qp), we have Π(Vf ) = Sym L ⊗L πp (Vf ), thus its p-adic Kirillov model is a P (Qp)-equivariant embedding

lalg Y 1 K : Π(Vf ) → L∞[[t]]/tL∞[[t]], tk−2 Z where we continue using the notations F = Q(f), L = Fλ where λ is a place lying over p. sm Remark 3.2. Since in our case πp (f) is supercuspidal, the map K is in fact a P (Qp)-equivariant L 1 isomorphism onto k−2 L∞[[t]]/tL∞[[t]]. Z t sm We assume πp (f) has central character δ. Then the explicit formula of the Borel action on Q 1 k−2 L∞[[t]]/tL∞[[t]] is: Z t n n a b 2−k bp bp t S = d δ(d) ⊗ ε · exp · 1 ⊗ σ a ∗ Sv(a)−v(d)+n 0 d ( d ) n d d n 2−k h bp i = d δ(d) · (1 + T ) d · 1 ⊗ σ a ∗ Sv(a)−v(d)+n . ( d ) ∨ k−2 2 k−2 2 2−k If we write the weight vectors of Sym L = Sym L ⊗L det to be

v0 = vhw, v1, ··· , vk−3, vk−2 = vlw COMPLETED COHOMOLOGY AND KATO’S EULER SYSTEM FOR MODULAR FORMS 13 as in the previous sections, then by [9] Page 146, we have −nj sm −j (3.3) (K (vj ⊗ v))n = j! p · (K (v))n · t , where K sm is the p-adic Kirillov model for smooth representations. Q For every integer n, we let en ∈ L∞ be the vector whose coordinate at the n-th place is 1, Z and 0 everywhere else. sm sm Recall that we are in the case when π (f) is supercuspidal, hence (v ) = 1 ∗ = e . Thus p K new Zp 0   sm X a (3.4) K (Fχ) =  χ(a) ⊗ ζpn  · e−n, a mod pn n where χ is any ﬁnite order character of conductor p with n ≥ 0, and Fχ is deﬁned as in section2. Therefore, we have for all 0 ≤ j ≤ k − 2,   nj X a −j (3.5) K (vj ⊗ Fχ) = j! ·p ·  χ(a) ⊗ ζpn  · t · e−n. a mod pn

In the rest of this section, we will describe the p-adic Kirillov model of Π(Vf ) from another point of view.

Recall the following theorem from [9]: sm Theorem 3.3 (Colmez, [9], Corollaire II.2.9). If πp (f) is supercuspidal, then there is an isomorphism as P (Qp)-modules: ∼ . + = (3.6) De (Vf (1)) De (Vf (1)) −→ Π(Vf ).

. + Remark 3.4. The action of P (Qp) on De (Vf (1)) De (Vf (1)) is deﬁned as follows: p 0 . • The matrix acts on D(V (1)) D+(V (1)) via the operator ϕ; 0 1 e f e f ∗ 0 . • The matrix Zp acts on D(V (1)) D+(V (1)) via the operator Γ; 0 1 e f e f 1 a 1 . • The matrix ∈ Qp acts on D(V (1)) D+(V (1)) via multiplying (1 + T )a. 0 1 0 1 e f e f

We denote the inverse of the above isomorphism by η. We then have the following lemma: Lemma 3.5. lalg + h 1 i + The image of Π(Vf ) under the map η is contained in De (Vf (1)) ϕr (T ) , r ≥ 0 De (Vf (1)).

1 pr Proof. For any v ∈ πsm(f), v ⊗ v is ﬁxed by if r is suﬃciently large. This means sm p 0 sm 0 1 pr (1 + T ) · η(v0 ⊗ vsm) = η(v0 ⊗ vsm),

1 + . + that is, η(v0 ⊗ vsm) ∈ ϕr (T ) De (Vf (1)) De (Vf (1)). 14 YIWEN ZHOU . We then proceed by induction: if 0 ≤ j < k −2 and η(v ⊗v ) ∈ 1 D+(V (1)) D+(V (1)) i sm (ϕr (T ))j e f e f . for all 0 ≤ i < j and some r, then we will show that η(v ⊗v ) ∈ 1 D+(V (1)) D+(V (1)). j sm (ϕr (T ))j+1 e f e f To see this, notice that for suﬃciently large r, 1 pr 1 pr X j (v ⊗ v ) = v ⊗ v = v ⊗ v + pr(j−i)v ⊗ v , 0 1 j sm 0 1 j sm j sm i i sm 0≤i

Deﬁnition 3.6. We deﬁne a map + 1 + Y + I : De (Vf (1)) , r ≥ 0 De (Vf (1)) −→ De dif(Vf (1))/De (Vf (1)) ϕr(T ) dif Z as follows: − • The 0-th coordinate I0 := ι0 , which is the natural map − + 1 + + ι : De (Vf (1)) , r ≥ 0 De (Vf (1)) −→ De dif(Vf (1))/De (Vf (1)) 0 ϕr(T ) dif

+ h 1 i induced by the inclution Be ϕr (T ) , r ≥ 0 ⊂ BdR. − n • The n-th coordinate In is defined as ι0 ◦ ϕ if n ≥ 0. − • The (−n)-th coordinate In is defined as ι if n ≥ 0: we choose a compatible system of n i p-power roots of unity ζp i≥0 as before, and for any positive integer n and any function f(T ) ∈ Be +, we define t ι (f(T )) := f ζ n exp − 1 , n p pn

+ h 1 i The map ιn extends naturally to a map from Be ϕr (T ) , r ≥ 0 to BdR, which induces a + h 1 i − map ιn from De (Vf (1)) ϕr (T ) , r ≥ 0 to De dif(Vf (1)). The map ιn is the composite of the + natural projection from De dif(Vf (1)) to De dif(Vf (1))/De dif(Vf (1)) with ιn.

Q + We equip Ddif(Vf (1))/D (Vf (1)) with an action of P ( p) in the following way: Z e e dif Q ∗ 0 • The matrix Zp acts coordinate wise through the action of Γ; 0 1 1 b 1 • A matrix ∈ Qp acts on the n-th coordinate by multiplying ε (bpn)·exp (bpnt); 0 1 0 1 p 0 • acts by shifting to the left. 0 1 We then have the following observation: COMPLETED COHOMOLOGY AND KATO’S EULER SYSTEM FOR MODULAR FORMS 15

Proposition 3.7. The map + 1 + Y + I : De (Vf (1)) , r ≥ 0 De (Vf (1)) −→ De dif(Vf (1))/De (Vf (1)) ϕr(T ) dif Z is P (Qp)-equivariant. Moreover, the following diagram is commutative: + h 1 i + I Q + D (Vf (1)) r , r ≥ 0 D (Vf (1)) Ddif(Vf (1))/D (Vf (1)) e ϕ (T ) e Z e e dif

lalg K Q 1 Π(Vf ) k−2 L∞[[t]]/tL∞[[t]] Z t where the vertical map on the right on each coordinate is given by the composite of the following maps:

1 =∼ 1 . L [[t]]/tL [[t]] −−→ D (V (1)) ⊗ L [[t]] D+ (V (1)) ,→ D (V (1))/D+ (V (1)) tk−2 ∞ ∞ dR f L tk−2 ∞ dif f dif f dif f + ⊂ De dif(Vf (1))/De dif(Vf (1)).

∗ The first isomorphism above is defined by sending every g(t) to f ⊗ g(t). See Section1 for the ∗ 0 precise definition of f ∈ DdR(Vf (1))/Fil DdR(Vf (1)).

Proof. The P (Qp)-equivariance of the map I can be checked by direct computation. The commutativity of the diagram follows from the uniqueness of the p-adic Kirillov model.

n Corollary 3.8. Let χ be any ﬁnite order character of conductor p , where n ≥ 0. Let Fχ be as deﬁned in section2. Then Im ◦η (Fχ ⊗ vj) = 0 for any m 6= −n, and I−n ◦η (Fχ ⊗ vj) is contained + in Ddif,n (Vf (1)) /Ddif,n (Vf (1)), for any 0 ≤ j ≤ k − 2. In other words, we have: − – ιm ◦ η (vj ⊗ Fχ) = 0 for all integer m 6= n, − + – ιn ◦ η (vj ⊗ Fχ) ∈ Ddif,n (Vf (1)) /Ddif,n (Vf (1)). Proof. This follows immediately from Proposition 3.7, together with the explicit formula of the map K in equation (3.5).

4. An explicit reciprocity law, supercuspidal case

In this section, we continue assuming that the (p-adic) smooth representation of GL2 (Qp) attached to the modular form f is supercuspidal. We review an “explicit reciprocity law” proved in [10] and [14] as a generalization of Proposition VI.3.4 in [9].

We ﬁrst introduce two pairings that will be used later. The reference for the following deﬁnitions is Page 151 of [9]. 16 YIWEN ZHOU

The pairing “{ , }”.

The pairing ∗ Vf (1) × Vf → L(1) gives rise to the following pairing: ∗ ∼ [ , ]: De (Vf (1)) × De (Vf ) → De (L(1)) = BeL.

dT We deﬁne { , } to be ResT =0 [ , ] · 1+T . The key properties of the pairing { , } are summarized in the following proposition, whose proof can be found in [9]. Proposition 4.1. (1) D (V (1)) /D+ (V (1)) and D\(V ∗) := lim D\(V ∗) are topological dual to e f e f f Qp ←− f b ψ b + ∗ \ ∗ \ ∗ each other. We have De Vf = D (Vf ) Qp ⊂ D (Vf ) Qp , and the pairing pc b + \ ∗ + ∗ \ ∗ between De (Vf (1)) /De (Vf (1)) and D (Vf ) Qp restriced to De Vf = D (Vf ) Qp b pc is the pairing { , } deﬁned above. (2) We have the following compatibility:

+ h 1 i + ∗ ψ=1 { , } De (Vf (1)) ϕi(T ) , i≥0 /De (Vf (1)) × D(Vf ) L

∼ η C =

∗, p 0 =1 lalg 0 1 Π(Vf ) × Π(Vf ) L

where the pairing on the bottom row is induced by the usual pairing between Π(Vf ) and ∗ Π(Vf ) .

Proof. (1) is [9] Proposition I.3.20. and Proposition I.3.21.

The pairings “h , idif” and “h , idif,m”.

∗ The pairing Vf (1)×Vf → L(1) again gives rise to the following pairings (we still use the notation “[ , ]” here): ∗ ∼ H [ , ]: De dif (Vf (1)) × De dif Vf → De dif(L(1)) = BdR ⊗Qp L; and for every integer m ≥ 0, ∗ ∼ [ , ]m : Ddif,m (Vf (1)) × Ddif,m Vf → Ddif,m(L(1)) = Lm((t)). dt dt We deﬁne h , idif (resp. h , idif,m) as Tr◦Rest=0 [ , ] · t (resp. Tr◦Rest=0 [ , ]m · t ), where Tr is the normalized trace map. The key properties of the pairings h , idif and h , idif,m are summarized in the following proposition, whose proof can again be found in [9], Chapter VI, Section 3.4. Proposition 4.2.

(1) The pairings h , idif and h , idif,m are compatible for every integer m ≥ 0. In other words, we have the following commutative diagram: COMPLETED COHOMOLOGY AND KATO’S EULER SYSTEM FOR MODULAR FORMS 17

∗ h , idif,m Ddif,m (Vf (1)) × Ddif,m(Vf ) L

+ ∗ h , idif De dif (Vf (1)) × De dif(Vf ) L

+ + ∗ (2) De dif (Vf (1)) and De dif(Vf ) are orthogonal complement of each other under the pairing h , idif. + + ∗ (3) For every integer m ≥ 0, Ddif,m (Vf (1)) and Ddif,m(Vf ) are orthogonal compliment of each other under the pairing h , idif,m. (4) We have the following compatibility:

H ∗ H Be DdR(Vf (1)) ⊗Qp BdR × DdR(Vf ) ⊗Qp BdR L

∼= ∼=

∗ h , idif De dif (Vf (1)) × De dif(Vf ) L

where the pairing Be on the top row is deﬁned by the formula

Be(x ⊗ f(t), y ⊗ g(t)) := Tr ◦ Rest=0 ([x, y]dR · f(t)g(t)dt) ,

and [ , ]dR is the pairing [ , ] 1 D (V (1)) × D (V ∗) −−−−→dR D (L(1)) = L. dR f dR f dR t (5) Similarly, we have the following compatibility for every integer m ≥ 0:

∗ Bm DdR(Vf (1)) ⊗L Lm((t)) × DdR(Vf ) ⊗L Lm((t)) L

∼= ∼=

∗ h , idif,m Ddif,m (Vf (1)) × Ddif,m(Vf ) L

where the pairing Bm on the top row is deﬁned by the formula

Bm(x ⊗ f(t), y ⊗ g(t)) := Tr ◦ Rest=0 ([x, y]dR · f(t)g(t)dt) .

The following theorem, which we shall call the “explicit reciprocity law” relating the pairing h , idif and the pairing { , }, is proved by Dospinescu: Theorem 4.3 (Dospinescu [10], Théorème8.3.1). Let Π be a continuous Banach space representa- lalg lalg tion of GL2(Qp), Π be the collection of locally algebraic vectors of Π. Assume Π = W ⊗π with lalg W algebraic and π a supercuspidal smooth representation of GL2(Qp). Let v ∈ Π with Kj(v) = 0 + for all but finitely many j. Assume m is sufficiently large and Ij ◦ η(v) ∈ Ddif,m/Ddif,m for all j ∗,p 0=1 (using the notations as in Proposition 3.7). Then for any z ∈ Π 0 1 , we have X {v, z} = hIj ◦ η(v), ιm(C(z))idif,m . j∈Z In particular, the RHS of the above equation is independent of m (as long as m is sufficiently large).

2 18 YIWEN ZHOU

∗, p 0 =1 ± 0 1 ± ∗ −1 ± Recall that we have deﬁned the element Mf ∈ Π(Vf ) and zM (f) = (Exp ) C(Mf ) ∈ 1 ∗ ∗ ± ∗ ψ=1 ∗ ± HIw(Qp,Vf ). Since Exp (zM ) ∈ D(Vf ) , ιn is deﬁned on Exp (zM ) for all n ≥ 1 and ∗ ± + ∗ ιn ◦ Exp (zM ) ∈ Ddif,n(Vf ). Moreover, one has

1 Ln+1 ∗ ± ∗ ± (4.1) Tr ιn+1 ◦ Exp (z ) = ιn ◦ Exp (z ) p Ln M M for all n ≥ 1.

∗ n sm Corollary 4.4. Let χ be a ﬁnite order character of Zp with conductor p , n ≥ 0. Fχ ∈ πp (f) be as in section2. For every 0 ≤ j ≤ k − 2,

 − ∗ ±  ιn ◦ η(vj ⊗ Fχ) , ιn ◦ Exp (zM ) , if n ≥ 1; ± D dif,n E (4.2) Mf (vj ⊗ Fχ) = − 1 L1 ∗ ± ι0 ◦ η(vj ⊗ Fχ) , p−1 TrL ι1 ◦ Exp (zM ) , if n = 0.  0 dif,0 Proof. Using Theorem 4.4 and Corollary 3.8, we conclude that ± − ∗ ± Mf (vj ⊗ Fχ) = ιn ◦ η(vj ⊗ Fχ), ιm ◦ Exp (zM ) dif,m − + for all m suﬃciently large, where we view ιn ◦η(vj⊗Fχ) as an element in Ddif,m(Vf (1))/Ddif,m(Vf (1)) via the natural inclusion + + Ddif,n(Vf (1))/Ddif,n(Vf (1)) ,→ Ddif,m(Vf (1))/Ddif,m(Vf (1)). Since − ∗ ± − ∗ ± ιn ◦ η(vj ⊗ Fχ), ιm ◦ Exp (zM ) dif,m = ιn ◦ η(vj ⊗ Fχ) , TrLn ιm ◦ Exp (zM ) dif,n where TrLn denotes the normalized trace map to Ln, and ∗ ± ∗ ± TrLn ιm ◦ Exp (zM ) = ιn ◦ Exp (zM ) for all n ≥ 1, the conclusion follows.

∗ Notice that Vf has Hodge-Tate weights 0 and k − 1, we have + ∗ 0 ∗ ∗ k−1 0 ∗ Ddif,n Vf = Fil DdR(Vf ) ⊗L Ln((t)) = DdR(Vf ) ⊗L t Ln[[t]] + Fil DdR(Vf ) ⊗L Ln[[t]]. We make the following deﬁnition:

0 ∗ Definition 4.5. Let f ∈ Fil DdR(Vf ) be defined as in Section1. ± r For any 0 ≤ j ≤ k − 2 and any integer r ≥ 1, we write Cr,j ∈ Lr = L ⊗Qp Qp (ζp ) the coefficient ∗ ± j of ιr ◦ Exp (zM ) in front of t : k−2 ∗ ± X ± j ∗ k−1 (4.3) ιr ◦ Exp (zM ) = f ⊗ Cr,jt + “Something in DdR(Vf ) ⊗L t Lr[[t]]”. j=0 Further more, under the decomposition M −1 r L ⊗Qp Qp (ζp ) = L(φ ) φ COMPLETED COHOMOLOGY AND KATO’S EULER SYSTEM FOR MODULAR FORMS 19

× where the direct sum is taken among all characters φ of (Z/pr) , we write ± X ± (4.4) Cr,j = Cr,j,φ · eφ φ ± with Cr,j,φ ∈ L. −1 r Here, we are viewing each L(φ ) as a 1-dimensional L-subspace contained in L ⊗Qp Qp (ζp ), with basis X a r eφ = φ(a) ⊗ ζpl ∈ L ⊗Qp Qp (ζp ) , a mod pr where pl is the conductor of φ.

∗ ± Remark 4.6. Since Exp (zM ) is ﬁxed by ψ, we have 1 C± = TrQp,r+1 C± r,j p Qp,r r+1,j for all integers r ≥ 1 and j ≥ 0. Remark 4.7. It can be checked easily that 1 X C± = φ(a)σ (C± ) r,j,φ pr−1(p − 1) a r,j a mod pr × for all integers r ≥ 1, j ≥ 0 and φ a character of (Z/pr) .

We define 1 (4.5) C± = TrQp(ζp)C± 0,j p − 1 Qp 1,j for all j ≥ 0. Proposition 4.8. Let χ be any finite order character of conductor pn, n ≥ 0. We have for all 0 ≤ j ≤ k − 2, k−2 ! X ± i dt ± (4.6) Tr ◦ Rest=0 K−n (vj ⊗ Fχ) · C t · = M (vj ⊗ Fχ) , n,j t f i=0 where Tr is the normalized trace map to L. Proof. By Theorem 4.3, for any integer m sufficiently large, − ± ± hιn ◦ η(vj ⊗ Fχ), ιm ◦ C Mf idif,m = Mf (vj ⊗ Fχ) . Now by Proposition 3.7, 2−k − ∗ DdR(Vf (1)) t Ln[[t]] ιn ◦ η(vj ⊗ Fχ) = f ⊗ K−n (vj ⊗ Fχ) ∈ 0 ⊗L ; Fil DdR(Vf (1)) tLn[[t]] and by Definition 4.5, k−2 ∗ ± ± X ± j ∗ k−1 ιm ◦ Exp (zM ) = ιm ◦ C Mf = f ⊗ Cm,it + “Something in DdR(Vf ) ⊗L t Lm[[t]]” i=0 0 ∗ ∗ k−1 ∈ Fil DdR(Vf ) ⊗L Lm[[t]] + DdR(Vf ) ⊗L t Lm[[t]] 20 YIWEN ZHOU whenever m ≥ 1. ∗ k−1 ∗ Notice that anything in DdR(Vf )⊗L t Lm[[t]] paired with f ⊗K−n (vj ⊗ Fχ) equals 0 because the latter element has coefficient 0 in front of ti for all i ≤ 1 − k. Thus D − ±E D ∗ ±E ιn ◦ η(vj ⊗ Fχ), ιm ◦ C Mf = f ⊗ K−n (vj ⊗ Fχ) , TrLn ιm ◦ C Mf dif,m dif,n * k−2 + ∗ X ± j = f ⊗ K−n (vj ⊗ Fχ), f ⊗ Cn,it i=0 dif,n k−2 ! ∗ X ± j = Tr ◦ Rest=0 [f , f]dR · K−n (vj ⊗ Fχ) · Cn,it · dt i=0 k−2 ! X ± i dt = Tr ◦ Rest=0 K−n (vj ⊗ Fχ) · C t · . n,j t i=0

Here, TrLn denotes the normalized trace map to Ln, and Tr is the normalized trace map to L.

Corollary 4.9. Let χ, j be as in Propostion 4.8. We have:   nj ± X a ± j! ·p · Id ⊗ Tr Cn,j · χ(a) ⊗ ζpn  = Mf (vj ⊗ Fχ) a mod pn pn(j+1) = (−1)j · · Λ(e f, χ, j + 1) τ(χ) when the sign ± on the LHS are chosen so that χ(−1) = ±(−1)j, and   nj ± X a j! ·p · Id ⊗ Tr Cn,j · χ(a) ⊗ ζpn  = 0 a mod pn j if the sign ± doesn’t satisfy χ(−1) = ±(−1) . Here, Tr denotes the normalized trace map to Qp.

Proof. This follows directly from Proposition 4.8, equation (3.5), Lemma 2.6 and Remark 2.7.

n ± Corollary 4.10. Let χ be any finite order character of conductor p , n ≥ 1 as before, and Cn,j,χ be as defined in Definition 4.5. Then for every 0 ≤ j ≤ k − 2, p 1 Λ(e f, χ, j + 1) (4.7) C± = ± · · n,j,χ p − 1 τ(χ) j! when the sign ± are chosen so that χ(−1) = ±(−1)j. j ± If the sign ± doesn’t satisfy χ(−1) = ±(−1) , then Cn,j,χ = 0.

∗ Proof. For any character φ of (Z/pn) with conductor pl (l ≤ n), using the notation in Deﬁnition −1 4.5, we have a basis vector eφ of the 1-dimensional vector space L(φ ): X a −1 n eφ := φ(a) ⊗ ζpl ∈ L(φ ) ⊂ L ⊗Qp Qp (ζp ) . a mod pn COMPLETED COHOMOLOGY AND KATO’S EULER SYSTEM FOR MODULAR FORMS 21

We can check easily that ( pn−1(p − 1), If φ = 1 is the trivial character; (4.8) Id ⊗ Tr (eφ) = 0 , If φ has conductor pl with l ≥ 1 where Tr is the normalized trace map to Qp. Since ! ± X a X ± Cn,j · χ(a) ⊗ ζpn = Cn,j,φ · eφ · eχ a mod pn χ

(here χ is the character of conductor pn in the statement of Corollary 4.10), we have   ± X a Id ⊗ Tr Cn,j · χ(a) ⊗ ζpn  a mod pn ± = Id ⊗ Tr Cn,j,χ−1 · eχ−1 · eχ   ± X −1 a+b =Cn,j,χ−1 · Id ⊗ Tr  χ(ab ) ⊗ ζpn  n ∗ a, b∈(Z/p )   ± X X bc+b =Cn,j,χ−1 · Id ⊗ Tr  χ(c) ⊗ ζpn  n ∗ n ∗ c∈(Z/p ) b∈(Z/p ) n−1 ± =p (p − 1) · χ(−1) · Cn,j,χ−1 .

Combining this with Corollary 4.9 gives equation (4.7).

± 5. Images of zM (f) under dual exponential maps, supercuspidal case

In this section, we continue assuming that the local (p-adic) smooth representation of GL2 (Qp) attached to the modular form f is supercuspidal. Before we state our main theorem, let’s recall the deﬁnition of integrating classes in the local Iwasawa cohomology. 1 ∗ For any class c ∈ H (Qp,Vf ⊗Zp Λ) where Λ is the Iwasawa algebra viewed as the space of ∗ measures on Zp: ∗ n ∗ • If φ is a locally constant function on Zp which factors through (Z/p ) , we can deﬁne

Z ( Z ) ∗ 1 ∗ (5.1) φ · c := γ ∈ G 7→ φ(x) · c(γ) ∈ V ∈ H (ζ n ),V . Qp(ζpn ) f Qp p f ∗ ∗ Zp Zp

1 ∗ 1 ∗ n It is easy to check that the above map from H (Qp,Vf ⊗Zp Λ) to H (Qp(ζp ),Vf ) is ∗ well deﬁned for any locally constant function φ which factors through (Z/pn) . 22 YIWEN ZHOU

∗ • If χ is a continuous character of Zp, then we can define the integration which takes value 1 ∗ in H (Qp,Vf (χ)): Z ( Z ) ∗ 1 ∗ (5.2) χ · c := γ ∈ GQp 7→ χ(x) · c(γ) ∈ Vf ∈ H Qp,Vf (χ) . ∗ ∗ Zp Zp 1 ∗ 1 ∗ It is easy to check that the above map from H (Qp,Vf ⊗Zp Λ) to H (Qp,Vf (χ)) is well ∗ defined for any continuous character χ of Zp. ∗ • The above two definitions are compatible for finite order characters φ of Zp with conductor pn in the following sense: There is the following commutative diagram: R First definition of ∗ φ 1 ∗ Zp 1 ∗ n H Qp,Vf ⊗Zp Λ H Qp(ζp ),Vf

res R Second definition of ∗ φ Zp 1 ∗ H Qp,Vf (χ) In the rest of this paper, for notational convenience, we will not mention which of the above definitions we are using. 1 ∗ 1 ∗ Recall that Shapiro’s lemma gives an isomorphism between H (Qp,Vf ⊗Zp Λ) and HIw(Qp,Vf ). 1 ∗ We can regard the above integrations as defined on HIw(Qp,Vf ). 1 ∗ 1 ∗ Lemma 5.1. We denote cj,n the projection from HIw(Qp,Vf ) to H (Qp(ζpn ),Vf (−j)). Then for 1 ∗ n ∗ any z ∈ HIw(Qp,Vf ) and any a ∈ (Z/p ) : Z −j (5.3) x · z = σa (cj,n(z)) . n a+p Zp Therefore, for any locally constant function φ of conductor pn, Z −j X (5.4) φ(x)x · z = φ(a) · σa (cj,n(z)) . ∗ Zp a mod pn Proof. We first recall the explicit description of Shapiro’s lemma. Let G be a profinite group, and H ≤ G be a subgroup of finite index such that G/H is commutative, V a representation of the group G with coefficients in L. The isomorphism given by Shapiro’s lemma ∼ 1 = 1 S : H G, HomL[H](L[G], V ) −→ H (H, V ) has formula S(c) = {h ∈ H 7→ c(h) ([1])} ∈ H1 (H, V ), where “1” is the identity element in G. 0 0 Here, the action of G on HomL[H](L[G], V ) is given by formula (g.F )([g ]) = F [g g]. Since V a representation of the group G, there is an isomorphism of G-representations =∼ α : HomL[H](L[G], V ) −→ V ⊗L L [G/H] given by formula X α(f) = z−1 · f([z]) ⊗ [z−1], z∈G/H where z is any lift of z in G. Here, the action of G on HomL[H](L[G], V ) is still as described as 0 0 before, and the action of G on V ⊗L L [G/H] is given by formula g.(v ⊗ [g ]) = (g.v) ⊗ [gg ]. COMPLETED COHOMOLOGY AND KATO’S EULER SYSTEM FOR MODULAR FORMS 23

Thus the isomorphism α induces an isomorphism ∼ 1 = 1 α∗ : H G, HomL[H](L[G], V ) −→ H (G, V ⊗L L [G/H]) , so we have the composite: ∼ −1 1 = 1 α∗ ◦ S : H (H, V ) −→ H (G, V ⊗L L [G/H]) . If H0 is a subgroup of H such that G/H0 is commutative, then we have the following commutative diagram:

−1 H1 (H0, V ) α∗◦S H1 (G, V ⊗ L [G/H0]) ∼= L cores Proj −1 H1 (H, V ) α∗◦S H1 (G, V ⊗ L [G/H]) ∼= L where the vertical map on the right is induced by the natural projection from G/H0 to G/H. −1 The inverse limit of the map α∗ ◦S with respect to corestrictions and projections gives the iso- ∼ 1 1 1 = morphism between HIw (Qp,V ) and H Qp,V ⊗Zp Λ , which we will denote by S :HIw (Qp,V ) −→ 1 H Qp,V ⊗Zp Λ . To prove equation (5.3) for j = 0, we let G = Gal Qp/Qp and H = Gal Qp/Qp (ζpn ) . Notice that the following composite

R a+pn 1 H1 G, Hom (L[G],V ) α∗ H1 (G, V ⊗ L [G/H]) Zp H1 (H, V ) L[H] ∼= L

1 sends any class c ∈ H G, HomL[H](L[G],V ) to −1 1 h ∈ H 7→ σa · c(h)([σa ]) = σa ·{h ∈ H 7→ c(h)([1])} = σa · S(c) ∈ H (H, V ) 1 using the explicit formula of the map α∗ above. Here, the action of σa on H (H, V ) is through the usual action of G/H on H1 (H, V ). Thus we have Z Z −1 σa (c0,n(z)) = 1 · (α∗ ◦ S (c0,n(z))) = 1 · z. n n a+p Zp a+p Zp We can then deduce equation (5.3) for general j by using the following commutative diagram:

1 S 1 H p,V ⊗ Λ H ( p,V ) Q Zp ∼= Iw Q R j ∗ x φ(x) Zp ∼= ∼=

1 S 1 1 n H Qp,V (j) ⊗Zp Λ ∼ HIw (Qp,V (j)) R H (Qp (ζp ,V (j))) = ∗ φ(x) Zp

j where the vertical isomorphism on the left is induced by sending v⊗[σa] ∈ V ⊗Zp Λ to v⊗a ⊗[σa] ∈ n ∗ V (j) ⊗Zp Λ, and the locally constant function φ factors through (Z/p ) .

The follwing lemma will be useful later on: 24 YIWEN ZHOU

Lemma 5.2. Assume F is a ﬁnite extension of Qp. If V is a p-adic continuous representation of GF , and K/F is a ﬁnite extension, then following diagrams are commutative: ∗ ∗ 1 exp 1 exp H (K,V ) DdR,K (V ) H (K,V ) DdR,K (V )

cores K res TrF natural inclusion ∗ ∗ 1 exp 1 exp H (F,V ) DdR,F (V ) H (F,V ) DdR,F (V ) K Where the trace map TrF appeared above is the unnormalized one. Proof. Recall the deﬁnition of exp∗ is as the following commutative diagram:

^log(cycl) H0 (F,V ⊗ B ) H1 (F,V ⊗ B ) dR ∼= dR

exp∗ H1 (F,V )

1 where log(cycl) ∈ H (F, Qp). Therefore, Lemma 5.2 follows from the following two commutative diagrams for cup products, when F ⊂ K:

0 1 ^ 1 H (F,V ⊗ BdR) × H (F, Qp) H (F,V ⊗ BdR)

res res res

0 1 ^ 1 H (K,V ⊗ BdR) × H (K, Qp) H (K,V ⊗ BdR)

0 1 ^ 1 H (F,V ⊗ BdR) × H (F, Qp) H (F,V ⊗ BdR)

cores res cores

0 1 ^ 1 H (K,V ⊗ BdR) × H (K, Qp) H (K,V ⊗ BdR)

sm We can now state our main theorem in the case when πp (f) is supercuspidal:

± 1 ∗ Theorem 5.3. The element zM (f) ∈ HIw(Qp,Vf ) has the following property: ∗ n For any 0 ≤ j ≤ k − 2, any locally constant character χ of Zp with conductor p (n ≥ 0), Z ! ∗ −j ± 1 Λ(e f, χ, j + 1) j (5.5) exp χ(x)x · zM (f) = ± · · f χ · t , ∗ τ(χ) j! Zp where ! j X a j j 0 ∗ f χ · t := χ(a)f ⊗ ζpn · t = f · eχ · t ∈ Fil DdR(Vf (χ)(−j)). a ± j Here, the sign ± for zM (f) is chosen such that χ(−1) = ±(−1) . If the sign doesn’t match, then the LHS of equation (5.5) equals 0. COMPLETED COHOMOLOGY AND KATO’S EULER SYSTEM FOR MODULAR FORMS 25

1 ∗ 1 ∗ Proof. We denote cj,n the projection from HIw(Qp,Vf ) to H (Qp(ζpn ),Vf (−j)). ∗ ± 1 ∗ ± According to [9] Lemma VIII.2.1, exp cj,n(zM (f)) equals pn times the image of ιm Exp (zM (f)) j modulo γn − ε(γn) for all m suﬃciently large, where ε is the cyclotomic character. ∗ ± 1 j In other words, exp cj,n(zM (f)) equals pn times TrLn applied to the coeﬃcient of t of ∗ ± ιm Exp (zM (f)) , where TrLn is the normalized trace map to Ln. Since

k−2 ∗ ± X ± j ∗ k−1 ιm ◦ Exp (zM ) = f ⊗ Cm,jt + “Something in DdR(Vf ) ⊗L t Lm[[t]]”, j=0 we have

∗ ± 1 ± j ∗ (5.6) exp c (z (f)) = · C · f · t ∈ D (V (−j)) ⊗ (ζ n ). j,n M pn n,j dR f Qp Qp p

Using Lemma 5.1, we conclude that Z ! ∗ −j ± 1 X ∗ ± exp χ(x)x · zM (f) = n χ(a) · σa. exp cj,n(zM (f)) ∗ p Zp a mod pn 1 X = χ(a) · σ C± · f · tj pn a n,j a mod pn

1 X X ± 0 −1 j = C 0 · χ(a)χ (a) · f · e 0 · t pn n,j,χ χ a mod pn χ0 p − 1 = · C± · f · e · tj p n,j,χ χ (Using equation (4.7)) 1 Λ(e f, χ, j + 1) = ± · · f · tj. τ(χ) j! χ This ﬁnishes the proof. Remark 5.4. Since we are working with the case when the modular form f is supercuspidal at p, we know f necessarily has bad reduction at p. Hence Λ(e f, χ, j + 1) = Λe(p)(f, χ, j + 1) for all χ and j. Therefore, the formula in Theorem 5.3 can also be written as Z ! ∗ −j ± 1 Λe(p)(f, χ, j + 1) j (5.7) exp χ(x)x · zM (f) = ± · · f χ · t . ∗ τ(χ) j! Zp

6. Explicit p-adic Local Langlands correspondence, principal series case The main purpose of this section is to compare the notations used in [2], [3], [9] with the one used in this paper, and then describe explicitly the isomorphism (as representations of B(Qp))

∗ =∼ (6.1) :Π(V ) −→ lim D(V ∗) F f ←− f ψ b 26 YIWEN ZHOU

sm ∼ GL2(Qp) −1 k−1 ±1 in the cases when πp (f) = ind unr(α) ⊗ unr(βp ) with |αβ|= |p |, α/β 6= p and B(Qp) 0 < vp(α), vp(β) < k−1. This is equivalent to the condition that Vf |G is crystaline and absolutely Qp irreducible. Here, ind means the smooth induction, B(Qp) is the group of lower triangular matrices ∗ ∗ in GL2(Qp), and unr(λ) is the unramiﬁed character of Qp that sends p ∈ Qp to λ. The main reference for this section is [9] (Section II.3.3), [2] and [3].

Recall that the theory of intertwining operators gives an isomorphism ∼ GL2( p) = GL2( p) (6.2) I : ind Q unr(α) ⊗ unr(βp−1) −→ ind Q unr(β) ⊗ unr(αp−1) . B(Qp) B(Qp) We deﬁne πla(α) := IndGL2(Qp) unr(α) ⊗ unr(βp−1)x2−k B(Qp) and πla(β) := IndGL2(Qp) unr(β) ⊗ unr(αp−1)x2−k , B(Qp) where Ind means the locally analytic induction. ∨ la lalg k−2 sm It can be seen easily that π (α) contains Π(Vf ) := Sym ⊗ πp (f) as a subrepresen- la lalg tation. Using the intertwining operator, we know π (β) also contains Π(Vf ) . We thus have an inclusion (see Corollary 7.2.5 of [2]): la la lalg (6.3) JE : π (α) ⊕Π(Vf ) π (β) ,→ Π(Vf ), and in fact, Liu has proved in [17] that the above map identiﬁes the source with the space of locally analytic vectors of Π(Vf ).

Let LAc (Qp,L) be the space of locally analytic functions with compact support on Qp taking values in L, where L is the coefficient of the representation Vf . We define a map la iα : LAc (Qp,L) ,→ π (α) by sending every function h ∈ LAc (Qp,L) to a function iα(h) defined on GL2 (Qp), such that 1 x i (h) = h(x). β 0 1 la We define iβ : LAc (Qp,L) ,→ π (β) in the same way.

Recall that the induction from the lower triangular Borel B(Qp) and the induction from the upper triangular Borel B(Qp) are related by the following isomorphism: ∼ GL2(Qp) −1 2−k = GL2(Qp) −1 2−k (6.4) Wα : Ind unr(α) ⊗ unr(βp )x −→ IndB( ) unr(βp )x ⊗ unr(α) , B(Qp) Qp GL2(Qp) −1 2−k where for every h ∈ Ind unr(α) ⊗ unr(βp )x , Wα(h) is a function on GL2(Qp) such B(Qp) that for every g ∈ GL2 (Qp), 0 1 W (h)(g) = h g . α 1 0

We then deﬁne a map

GL2(Qp) −1 2−k jα : LAc ( p,L) ,→ Ind unr(βp )x ⊗ unr(α) Q B(Qp) COMPLETED COHOMOLOGY AND KATO’S EULER SYSTEM FOR MODULAR FORMS 27 by sending every function h ∈ LA (Qp,L) to a function jα(h) deﬁned on GL2 (Qp), such that 0 1 j (h) = h(x). β 1 x We then have the following commutative diagram:

iα GL2(Qp) −1 2−k LAc (Qp,L) Ind unr(α) ⊗ unr(βp )x B(Qp) jα ∼= Wα

IndGL2(Qp) unr(βp−1)x2−k ⊗ unr(α) B(Qp)

All the above deﬁnitions and discussions hold when we replace α by β.

2−k αβ Notice that the central character of Π (Vf ) is δ(z) = z unr( p )(z) , which is a unitary −1 ∗ character; while the determinant of Vf |G is (z|z|) δ(z) for z ∈ , where we are identifying Qp Qp ∗ ∗ QP with the Weil group W p by sending p ∈ Qp to the inverse of Frobenius. Hence we have Q ∗ ∼ 2 2 −1 ∗ ∼ −1 Vf (1) = Vf ⊗ z |z| δ (z) and Π Vf (−1) = Π(Vf ) ⊗ δ ◦ det .

In [2], Berger and Breuil deﬁned LA(α) := IndGL2(Qp) unr(α−1) ⊗ xk−2unr(pβ−1) B(Qp) and LA(β) := IndGL2(Qp) unr(β−1) ⊗ xk−2unr(pα−1) B(Qp) Similar to the inclusion (6.3), we have ∗ (6.5) BB : LA(α) ⊕ ∗ lalg LA(β) ,→ Π(V (−1)), J Π(Vf (−1)) f see Corollary 7.2.5 of [2]. (There is a diﬀerence of notations for Banach space representations between [2] and this paper, which will be explained in Remark 6.7 later in this section.) We now have the following commutative diagram, and we denote the map of the dashed arrow by sα:

iα la JE LAc (Qp,L) π (α) Π(Vf ) jα ∼= Wα

GL2( p) −1 2−k Ind Q unr(βp )x ⊗ unr(α) ∼ ϑ B(Qp)

sα multiplied by δ−1◦det

∗ LA(α) Π(Vf (−1)) JBB −1 Here, we write “multiplied by δ ◦ det” (which is not a GL2(Qp)-equivariant map) because we are viewing both the source and target of this map as some space of functions on the group GL2(Qp). We have similar diagram when we replace α by β, and the dotted arrow ϑ from Π(Vf ) to ∗ Π(Vf (−1)) is induced by the two diagrams for α and β. ∗ Remark 6.1. The map ϑ : Π(Vf ) → Π(Vf (−1)) is just an isomorphism of topological vector spaces. It is not GL2(Qp)-equivariant. 28 YIWEN ZHOU

We have the following lemma, whose proof is just a careful tracking of the diagram above:

∗ ∗ ∗ Lemma 6.2. For any z ∈ (Π(Vf )) , if we deﬁne να(z) := iα ◦ JE (z) ∈ D(Qp) and µα(z) := ∗ ∗ sα ◦ JBB ◦ ϑ∗(z) ∈ D(Qp), then να(f) and µα(f) are related by the following formula: Z Z (6.6) f(x)να(z) = f(−x)µα(z) Qp Qp for any function f(x) ∈ LAc (Qp,L). We have similar results when we replace α by β.

2 Remark 6.3. The distribution µα(z) is the distribution on Qp corresponding to z deﬁned in [2].

Definition 6.4. We equip lim D(V ∗) with an action of B( ) as follows: ←− f Qp ψ z 0 • The matrix in the center acts by the scalar δ−1(z) on each copy of D(V ∗). 0 z f p 0 • The matrix acts via ψ−1, that is, shifting to the right. 0 1 a 0 • The matrix where a ∈ ∗ acts by σ on each copy of D(V ∗). 0 1 Zp a f 1 b • The matrix where b ∈ acts via multiplying (1 + T )b on the last copy of D(V ∗). 0 1 Qp f Remark 6.5. If we take V = V ∗ and χ = δ−1, then lim D(V ∗) with the action of B ( ) as defined f ←− f Qp ψ in [3] (Definition 3.4.3) is isomorphic to lim D(V ∗) ⊗ (δ ◦ det) with the action of B ( ) as defined ←− f Qp ψ above in Definition 6.4.

In [2] and [3], Berger and Breuil proved the following theorem (See Theorem 8.1.1 in [2], or Th´eor`eme5.2.7 in [3]), which we present here using our notation:

sm ∼ GL2(Qp) −1 Theorem 6.6 (Berger & Breuil). Assume that πp (f) = ind unr(α) ⊗ unr(βp ) with B(Qp) k−1 ±1 |αβ|= |p |, α/β 6= p and 0 < vp(α), vp(β) < k − 1. Then under the B(Qp)-equivariant isomorphism of topological vector spaces (as deﬁned by Colmez in [9])

∗ =∼ (6.7) :Π(V ) −→ lim D(V ∗) , F f ←− f ψ b we have the explicit formula

−N −N (6.8) F (z) = α µα,N (z) ⊗ eα + β µβ,N (z) ⊗ eβ . N Here, we are identifying lim D(V ∗) with lim N(V ∗) (see Proposition 8.1.2 in [2]), and we view ←− f ←− f ψ b ψ ∗ + ∗ N(V ) as a subspace contained in B ⊗Dcrys(V ) (see Proposition 5.5.3 in [2]). Also, eα (resp. f rig,Qp f ∗ −1 eβ) is a Frobenius eigenvector in Dcrys(Vf ) with eigenvalue α (resp. β) such that f = eα + eβ ∈ COMPLETED COHOMOLOGY AND KATO’S EULER SYSTEM FOR MODULAR FORMS 29

0 ∗ + Fil DdR(V ), and µα,N (z), µβ,N (z) are distributions on p (viewed as elements in B via the f Z rig,Qp Amice transform3) defined as follows: ∗ For any z ∈ Π(Vf ) , let µα(z) and µβ(z) be as defined in Lemma 6.2. We define for every N ∈ Z≥0 two distributions µα,N (z) and µβ,N (z) on Zp using the formulas Z Z N f(x)µα,N (z) := f(p x)µα(z) 1 p p Z pN Z and Z Z N f(x)µβ,N (z) := f(p x)µβ(z) 1 p p Z pN Z for all f(x) ∈ LAc (Zp,L).

Remark 6.7. The notations we are using here is related to the notations used in [2] and [3] in the2 following way: for any continuous representation V of GQp , the Π(V ) in this paper is the Π(V (1)) in [2] and [3]. In [2] and [3], Theorem 6.6 is formulated as ∗ =∼ Π V ∗ −→ lim D(V ∗) , f ←− f ψ b with their notation of Π V ∗ and their deﬁnition of B ( ) acting on lim D(V ∗) . If we change f Qp ←− f ψ b both sides of the above isomorphism into our notations (see Lemma 6.5), we would get ∗ =∼ Π V ∗(−1) −→ lim D(V ∗) ⊗ (δ ◦ det) . f ←− f ψ b ∗ ∗ ∗ −1 ∼ ∗ ∗ ∼ Notice that we have Π Vf (−1) ⊗(δ ◦ det) = Π Vf (−1) ⊗ (δ ◦ det) and Π Vf (−1) = −1 Π(Vf )⊗(δ(z) ◦ det) , we see that our isomophism (6.7) coincides with the isomorphism introduced in [2] and [3]. The notation used in [9] is also compatible with the notation in [2] and [3], which has already been veriﬁed in [17].

± 7. Images of zM (f) under dual exponential maps, principal series case We continue using the same notations and assumptions as in the previous section. For example, sm πp (f) is still assumed to be a principal series. In [11], Emerton has proved the following result (Proposition 4.9 in [11]):

± Theorem 7.1 (Emerton, [11]). With the notations as in Lemma 6.2, we have να(M ) (resp. f ∗ Zp ± νβ(M ) ) coincides with the p-adic L-function Lp,α(f) (resp. Lp,β(f)) as deﬁned in [18]. f ∗ Zp 3 + Later in this and the next section, we will always identify D ( p) with B using the Amice transform without Z rig,Qp mentioning explicitly. This should not cause any confusion. 30 YIWEN ZHOU

sm We can now prove our main theorem in the case when πp (f) is an unramiﬁed principal series:2

± 1 ∗ Theorem 7.2. The element zM (f) ∈ HIw Qp,Vf has the following property: ∗ n For any 0 ≤ j ≤ k − 2, any locally constant character χ of Zp with conductor p (n ≥ 0), Z ! ∗ −j ± 1 Λe(p)(f, χ, j + 1) j 0 ∗ (7.1) exp χ(x)x · zM (f) = ± · · f χ · t ∈ Fil DdR(Vf (χ)(−j)). ∗ τ(χ) j! Zp ± j Here, the sign ± for zM (f) is chosen such that χ(−1) = ±(−1) . If the sign doesn’t match, then the LHS of equation (7.1) equals 0.

1 ∗ Proof. As in the proof of Theorem 5.3, we again denote cj,n the projection from HIw(Qp,Vf ) to 1 ∗ ∗ ± 1 H (Qp(ζpn ),Vf (−j)), and we use Lemma VIII.2.1 in [9] that exp cj,n(zM (f)) equals pn times ∗ ± j the image of ιn Exp (zM (f)) modulo γn − ε(γn) , where ε is the cyclotomic character. We have the following commutative diagram:

∗ ψ=1 ιn ∗ D(Vf ) Ln[[t]] ⊗L DdR(Vf )

F −n ιn⊗ϕ ψ=1 + ∗ B ⊗ Dcrys(V ) rig,Qp f where the vertical map F is induced from the isomorphism

+ ∗ 1 ∼ + 1 ∗ D (V )[ ] = B [ ] ⊗ Dcrys(V ) rig f t rig,Qp t Qp f ∗ ψ=1 ∗ + ∗ as described in [1], together with the fact that D(V ) ⊂ N(V ) ⊂ B ⊗ Dcrys(V ) (see f f rig,Qp f Theorem A.3 in [1] and Proposition 5.5.3 in [2], remember that we have always assumed Vf |G to Qp be absolutely irreducible).

We write ∗ ± + ∗ F Exp (z (f)) = Υα ⊗ eα + Υβ ⊗ eβ ∈ B ⊗ Dcrys(V ). M rig,Qp f −1 −1 Since Υα ⊗ eα + Υβ ⊗ eβ is ﬁxed by ψ, we have ψ (Υα) = α Υα and ψ (Υβ) = β Υβ.

∗ ± ± ± Notice that Exp (zM (f)) = C Mf = F Mf is ﬁxed by ψ, so if we write

± −N ± −N ± F Mf = α µα,N (Mf ) ⊗ eα + β µβ,N (Mf ) ⊗ eβ , N then

∗ ± ± ± ± ± F Exp (zM (f)) = µα,0(Mf ) ⊗ eα + µβ,0(Mf ) ⊗ eβ = µα(Mf ) ⊗ eα + µβ(Mf ) ⊗ eβ, Zp Zp

± ± hence Υα = µα(Mf ) and Υβ = µβ(Mf ) . Zp Zp COMPLETED COHOMOLOGY AND KATO’S EULER SYSTEM FOR MODULAR FORMS 31

t R tx x ± Now since ιn (Υα) = Υα ζpn exp n − 1 = exp n ⊗ ζ n · µα(M ), we have p Zp p p f

Z j 1 j x ± A := Coeﬃcient of t of ι (Υ ) = x ⊗ ζ n · µ (M ) j,n,α n α pjnj! p α f Zp +∞ Z 1 X j x ± = jn x ⊗ ζpn · µα(Mf ) p j! i ∗ i=0 p Zp +∞ Z 1 X i j x i ± = jn p x ⊗ ζpn−i · ψ · µα(Mf ) p j! ∗ i=0 Zp +∞ i Z 1 X p j x ± = jn i x ⊗ ζpn−i · µα(Mf ). p j! α ∗ i=0 Zp

∗ n Thus, for any locally constant character χ of Zp with conductor p (n ≥ 0),

  n +∞ i Z n X α X p j X ax ± α χ(a)σa (Aj,n,α) = jn i x  χ(a) ⊗ ζpn−i  · µα(Mf ). p j! α ∗ n ∗ n ∗ a∈(Z/p ) i=0 Zp a∈(Z/p )

P ax It is easily seen that the Gauss sum n ∗ χ(a) ⊗ ζ = 0 whenever i > 0 because χ has a∈(Z/p ) pn−i conductor pn. Therefore, if we pick the sign ± such that ±1 = (−1)jχ(−1), then we have

n P a Z n X α a χ(a) ⊗ ζpn j −1 ± α χ(a)σa (Aj,n,α) = jn x χ (x)µα(Mf ) p j! ∗ n ∗ a∈(Z/p ) Zp n P a Z j α a χ(a) ⊗ ζpn j −1 ± = (−1) χ(−1) jn x χ (x)να(Mf ) (Lemma 6.2) p j! ∗ Zp n P a Z α a χ(a) ⊗ ζpn j −1 ± = ± jn x χ (x)να(Mf ) p j! ∗ Zp n P a α χ(a) ⊗ ζ n = ± a p · L (f) xjχ−1(x) (Theorem 7.1) pjnj! p,α n P a α a χ(a) ⊗ ζpn n(j+1) −n −1 −(j+1) L(p)(f, χ, j + 1) = ± jn · j! ·p · α · τ(χ) · (2πi) · ± p j! Ωf (See for example, [16], Page 269)

n 1 Λe(p) (f, χ, j + 1) X a = ±p · · · χ(a) ⊗ ζ n . τ(χ) j! p a

n P n 1 Λe(p)(f,χ,j+1) P a Similarly, β n ∗ χ(a)σa (Aj,n,β) = ±p · · · χ(a) ⊗ ζ n , from which a∈(Z/p ) τ(χ) j! a p we can see that the RHS doesn’t depend on α or β. 32 YIWEN ZHOU

Hence, Z ! ∗ −j ± X ∗ ± exp χ(x)x · zM (f) = χ(a) exp cj,n(zM (f)) ∗ n ∗ Zp a∈(Z/p ) 1 X = · χ(a)σ Coeﬃcient of tj of ι Exp∗(z± (f)) · tj pn a n M n ∗ a∈(Z/p ) 1 X = · χ(a)σ (A ⊗ αne + A ⊗ βne ) · tj pn a j,n,α α j,n,β β n ∗ a∈(Z/p ) ! 1 Λe(p) (f, χ, j + 1) X a j = ± · · χ(a) ⊗ ζ n ⊗ (e + e ) · t τ(χ) j! p α β a ! 1 Λe(p) (f, χ, j + 1) X a j = ± · · χ(a) ⊗ ζ n ⊗ f · t τ(χ) j! p a

1 Λe(p) (f, χ, j + 1) = ± · · f · tj ∈ Fil0D (V ∗(χ)(−j)). τ(χ) j! χ dR f

8. Comparison with Kato’s Euler system n Let f be any cuspidal newform of weight k ≥ 2 and level Γ0(p ) ∩ Γ1(N), Vf the (cohomological) p-adic Galois representation of G attached to f. We assume throughout this section that Vf |G Q Qp is absolutely irreducible. sm We have exactly two cases when the above assumption can happen, namely, when πp (f) is supercuspidal (as in section 3 to 5) or a twist of unramiﬁed principal series (as in Section 6 and 7).

+ − 1 ∗ Corollary 8.1. Let f be as above. We deﬁne zM (f) := zM (f) − zM (f) ∈ HIw Qp,Vf . Then we ∗ n have for any 0 ≤ j ≤ k − 2, any locally constant character χ of Zp with conductor p (n ≥ 0), Z ! ∗ −j 1 Λe(p)(f, χ, j + 1) j (8.1) exp χ(x)x · zM (f) = · · f χ · t , ∗ τ(χ) j! Zp where

Γ(j + 1) L(p)(f, χ, j + 1) j 0 ∗ n Λe(p)(f, χ, j + 1) = j+1 · ± ∈ Q(f, µp ), f χ · t ∈ Fil DdR(Vf (χ)(−j)) (2πi) Ωf are deﬁned as in equation (2.16) and Theorem 5.3. Proof. This follows immediately from Theorem 5.3 (for the supercuspidal case) and Theorem 7.2 (for the principal series case).

Recall the following theorem by Kato: COMPLETED COHOMOLOGY AND KATO’S EULER SYSTEM FOR MODULAR FORMS 33

1 ∗ Theorem 8.2 (Kato, [16]). There exists a unique element zKato(f) ∈ HIw(Q,Vf )(in the global Iwasawa cohomology group), which is obtained by global methods using Siegel unites on modular ∗ ∼ curves, such that for any 0 ≤ j ≤ k − 2 and any locally constant character χ of Zp = ΓQp with conductor pn (n ≥ 0), we have

Z ! ∗ −j 1 Λe(p)(f, χ, j + 1) j (8.2) exp χ(x)x · zKato(f) = · · f χ · t . ∗ τ(χ) j! Zp

1 ∗2 We still use zKato(f) to denote its image in the local Iwasawa cohomology group HIw(Qp,Vf ). Combining Corollary 8.1 and Theorem 8.2, we can prove the following:

1 ∗ Theorem 8.3. If Vf |G is absolutely irreducible, then zM (f) = zKato(f) as elements in H ( p,V ). Qp Iw Q f

1 ∗ 1 ∗ Proof. We deﬁne HIw,e(Qp,Vf ) the collection of elements in HIw(Qp,Vf ) whose projection to 1 ∗ 1 ∗ 1 ∗ H (Qp(ζpn ),Vf ) belongs to He (Qp(ζpn ),Vf ) for all n ≥ 0, and HIw,g(Qp,Vf ) the collection of 1 ∗ 1 ∗ 1 ∗ elements in HIw(Qp,Vf ) whose projection to H (Qp(ζpn ),Vf ) belongs to Hg (Qp(ζpn ),Vf ) for all n ≥ 0, where

1 1 1 ϕ=1 n n n He (Qp(ζp ),V ) = ker H (Qp(ζp ),V ) → H Qp(ζp ),V ⊗Qp Bcrys ∗ 1 exp 0 = ker H (Qp(ζpn ),V ) −−−→ Fil DdR (V ) 1 1 1 n n n ⊂ Hg (Qp(ζp ),V ) := ker H (Qp(ζp ),V ) → H Qp(ζp ),V ⊗Qp BdR 1 ⊂ H (Qp(ζpn ),V ) for any continuous p-adic Galois representation V of GQp . By Corollary 8.1 and Theorem 8.2, we know that in both cases, we have for any 0 ≤ j ≤ k − 2 ∗ and any ﬁnite order character φ of Zp,

Z ! ∗ −j (8.3) exp φ(x)x (zM (f) − zKato(f)) = 0, ∗ Zp

This means 1 ∗ zM (f) − zKato(f) ∈ HIw,e(Qp,Vf ).

In the case when Vf |G is absolutely irreducible, Proposition II.3.1 of [4] says Qp

1 ∗ 1 ∗ (8.4) HIw,e(Qp,Vf ) ⊂ HIw,g(Qp,Vf ) = 0, hence zM (f) − zKato(f) = 0, which proves (1).

Remark 8.4. In fact, we only used equation (8.3) for j = 0 in order to conclude zM (f) = zKato(f). 34 YIWEN ZHOU

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Department of Mathematics, University of Chicago. E-mail address: [email protected]