P-adic modular symbols attached to CM forms - Vicentiu Pasol, Boston University

BOSTON UNIVERSITY GRADUATE SCHOOL OF ARTS AND SCIENCES Dissertation P-ADIC MODULAR SYMBOLS ATTACHED TO C. FORMS VICENTIU PASOL M., Universitatea Bucuresti, 1997 Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy 2006 UMI Number: 3186525 INFORMATION TO USERS The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleed-through, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted.

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ProQuest Information and Learning Company 300 North Zeeb Road P. Box 1346 Ann Arbor, MI 48106-1346 Approved by First Reader bn f= Glenn Stevens, Ph. Professor of Mathematics Second Reader David Rohrlich, Ph. Professor of Mathematics Third Reader Robert Pollack, Ph.

Assistant Professor of Mathematics Acknowledgments I dedicate this thesis to my advisor, Professor Glenn Stevens,who led me with patience and rigor through my doctorate years. I owe him the greatest part of my knowledge and the so far accomplishments in the mathematical field. My strongest sentiments are directed towards my family who took care and guided me in life. They are more than anything to me.

I owe a lot for the taste of research to my first advisor, Prof. Nae Popescu, who al- ways puts his great personality and mind into guiding new incomers into the mathematical research. I also have to express my deepest gratitude to Prof. Sasha Polishchuk who supported me and whose close didactic presence made me to appreciate even more the diverse wonderful aspects of mathematics.

/ Last, but not least, I want to thank from all my heart to my friends : Celia and Florin, for the long and insightful talks and all my other friends who made my life as good as it was so far. iii P-ADIC MODULAR SYMBOLS ATTACHED TO C. FORMS (Order No. ) VICENTIU PASOL Boston University Graduate School of Arts and Sciences, 2006 Major Professor: Glenn Stevens, Professor of Mathematics Abstract In this thesis I study the relation between the (CM) Elliptic Construction and the Mod- ular Construction of p-adic LZ-functions in two variables which interpolate special values of Hecke L-functions.

The first construction, due to N. Katz, and then R. Yager, is very explicit in terms of elliptic units using Iwasawa theory on formal groups. However, this construction does not imply the construction of a modular symbol! for which the attached L-function is exactly this p-adic L-function.

The Modular construction, developed by R. Stevens, is more general. They attach to any Hida family of p-stabilized modular forms a p-adic modular symbol with values in the space of two variable p-adic distributions. However, this construction is not explicit.

We use p-adic methods to prove the existence of a modular symbol whose L-function is the Katz-Yager p-adic L-function. The Weil representation constitutes a powerful tool in constructing modular forms. We use the Weil representation to construct a modular symbol with values in a big complex vector space, for which the attached L-function is the generating function of the special values of Hecke L-functions. This suggests the possibility of using the Weil representation to explicitly describe the p-adic modular symbol whose existence is established in this thesis.

iv Contents 1 Introduction 2 p-adic Distributions 2.1 Distributions on Zp.2 Locally analytic distributions .3 Operators and maps .1 The Norm map.2 Coleman Power series and Norm invariant Distributions .4 Weight Space and p-adic E-functions .1 The Melin Transform.5 Two variable distributions .1 Special values and convolution .2 Scalar equivalence for distributions on 2š x Zp .3 Trace compatible elements and the Frobenius twist .4 Existence of Trace compatible sequences .5 Trace compatible sequences and measures. 32 3 L-functions and special values 3.1 L-functions attached to modular forms.2 L-functions attached to elliptic curves .1 ee Formal group.1 Special values of Eisenstein series and L-function .4 2-variable p-adic measure attached toaC.2 The 2-variable p-adic measure wS-¥ 2. ee 48 Modular Symbols 51 41 Modular Symbols-Introduction .1 Modular Symbols attached to cusp forms .2 Two variable p-adic modular symbol .2 Elliptic construction-existence. uc ch gà ki àa 56 Weil representations 59 5.2 The cocycle of the Weil representation .1 The ZA isomorphism.

gà gà gà gà xà và va 63 5. cu kg v k k k k vo 67 vì Chapter 1 Introduction The goal of my thesis is to understand the relation between the Modular constructions and the CM (elliptic) constructions of p-adic L-functions in 2-variables. We start with an elliptic curve defined over Q and having complex multiplication by the ring of integers Ox of a quadratic imaginary field, K. Denote by W the associated Hecke character.

Fix a prime p and suppose p splits in K, p = @- @. There are two methods (CM, see [Kat76] and [Yag82] for various versions, and Modular, see [Gre93]) to attach a 2-variable p-adic L-function (we will call the CM one, LK ~Ÿ and the modular one, LE-s ), which interpolates the critical values L(W*, s) for 0 < s < k, s,k € Z. In the p-adic setting, L-functions are always attached to p-adic distributions by a Mellin transform (see 2.) analogous with the complex L-functions attached to cusp forms. We choose an embedding Q C Q), by choosing a prime P in Q above ø.

For simplicity, we make the assumption that a := W(Ø) generates Z>. Let’s denote by Gg the kernel of the surjective map Gal(Q77/Qp) — ZF sending Frob —+ a. Also put M := (QUT) Ge and. M , its p-adic completion.

Denote by D := (2š x Zip, M ), the set of M-valued p-adic distributions on 2 Xx Zp. There is an action of Gal(M/Q,) % Z> on D defined by: / f(a, v)dule (ø,y) = Frob / f(a, ay) dpe, y)). Denote by D© the spaces of Galois invariant distributions in D. One can easily show that the distribution „#~Y, for which the attached L-function is LX~Y, is actually an element of DŒ, Also, there is a natural action of the Iwahori group on D which commutes with the Galois action.

We have the following result: Theorem 1. Let Symbr,(D€) to be the space of DỢ -ualued modular symbols over Tg := To(Np), where N is the (tame) conductor of E. There exist a unique eigensymbol @X~Y := 6 € Symbr,(D°) with the same eigenvalues as the p-stabilized newform attached to E such that: This result is related to the modular symbol obtained by Ralph Greenberg and Glenn Stevens in [Gre93], using Hida’s deformation theory of p-stabilized newforms. Their con- struction, which we call the Modular Construction, was used to prove Mazur-Tate-Teitelbaum conjecture.

The downside of the modular construction is that this construction is not explicit. One should expect that there are global modular symbols to be deformed to p-adic modular symbols of type @X—Y, In this direction one seems to be directed towards the use of Weil representations. We introduce the space Z := S(C){[X,Y]] of formal power series with coefficients Schwartz functions. The group SLa(§) embeds in Sp(C), the symplectic group of C, in a natural way and it acts by unitary representations (Weil representation) on L?(C), pre- serving S(C).

The action of SL¿(R) on F is given by: +-h(z)Šg(X,Y) := (W()h)(2)Šg((X, Y3), where W is the Weil representation. A partial result (see 5.5) is the following: Theorem 1. Then, the map ộ: Ag — F defined by: $(D)(2) = 2ni- [ focyya(2)dr, is SLo(Q) invariant, therefore defines an element in Sụmbsr,(q)(7). Moreover, this modular symbol "remembers" (as explained in the mentioned theorems 5.5) all the critical values for all partial Hecke L-functions.

One should expect that from a such object, plus a Galois action (which would guarantee the algebraic nature), we can get by continuity in the p-adic direction (One must check this!), a p-adic modular symbol and by some existence-uniqueness theorem we should be : able to check that indeed we get ®X-Y, The hope we have is that this construction can be generalized for p-stabilized newforms, therefore one could get explicit construction for 6°-%, in particular, canonical choice for the p-adic periods. Chapter 2 p-adic Distributions 2. Locally convex vector spaces We use the standard notations from the theory of p-adic fields: 1. The p-adic valuation on Q, up = ordp; the corresponding (nonarchimidean) norm (absolute value) |z| := |a|p := pu’.

The completion Q, of Q relative to this norm; Z, the closed unit ball around the origin, which is a local, discrete valuation ring with the unique maximal ideal pp, which, also, is the open unit ball around the origin; the residue field F, := Z„/pổp. The algebraic closure Qp of Q, on which there exists a unique extension of the norm on Qp;. C, the completion of Q, relative to this norm. F will denote a fixed complete subfield F C Cy.

We denote by Or the ring of integers in , which is the same with the closed unit ball around the origin relative to the induced p-adic norm; mp its maximal ideal, which is the same with the open unit ball around the origin relative to the same norm; its residue field F := Op /mp. The residue field of C, is just the algebraic closure of Fp, i. We endow Œ with the topology induced by the norm : |(a1, đạ,. The underlined letters, a,i,X,z,.

The addition and multiplication are taken component-by-component. Also, by convention, X!:= Il, X7. Let V be a topological F-vector space. A seminorm on V is a function g: V —> Rso such that: 1.

For any À€ F and z€ V, we have q(A- x) = |À| - q()., g„} is a finite set of seminorms on V and e > 0, we define: U(q. If {q¡];er is a family of seminorms on V, then there exists a unique translation invariant topology on V for which a fundamental system of neighborhoods around the origin is given by the sets U(q,,.,i,} CI is a finite subset ofI œnde > 0. V is called locally conver if there exist a family of seminorms {q;}iez on V such that the topology on V is equivalent with the one induced by this family of seminorms. Assume that V is locally convex, with the topology induced from the family of seminorms {q}ier - 1.

A subset B C V is called bounded if g;(B) is bounded (in R) for all ¿ € J. A seminorm g : V — Rso is called bounded if q(B) is bounded for any bounded subset BC V. V is called bornological if the set of continuous seminorms on V is the same with the set of bounded seminorms on V. V is called Fréchet if it is metrizable and complete.

A locally convex topological space is Fréchet if and only if its topology is given by a countable family of seminorms and is complete with respect to this topology. Let V be any F-vector space. We also assume that we have a family {VWh}nem of locally conver F-vector spaces. If for each h € H is given a F-linear map fp: Vụ —> V, then there exists the finest locally convex topology on V for which all the maps fy, are continuous.

This topology is called locally convex final topology on V with respect to the family tfh}ncHn- 2. If for each h € H is given a F-linear map gụ : V — Vụ, then there exists the coarsest locally convex topology on V for which all the maps ƒn are continuous.