Congruences Among Automorphic Forms on U(2,2) - Luận án Klosin

Congruences among automorphic forms on the unitary group (2,2) by Krzysztof Klosin A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Mathematics) in The University of Michigan 2006 Doctoral Committee: Professor Christopher M. Skinner, Chair Professor Brian D. Conrad Professor Andrzej 5. Nowak Associate Professor Stephen M.

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Box 1346 Ann Arbor, MI 48106-1346 © Krzysztof Klosin 2006 All Rights Reserved ACKNOWLEDGEMENTS Many people, some of them probably unaware of it, have contributed to this thesis. The list begins with Chris Skinner, my advisor here at Michigan, who generously shared with me many of his ideas and guided me through the process of turning them into theorems. I would like to thank him for his patience and encouragement, needed especially in the all too frequent moments of skeptisism. This thesis would have not come to exist if it was not for my wife, Marta, who agreed to come with me to America and supported me every step of the way.

Her love and companionship made my stay in Ann Arbor so much brighter. Brian Conrad deserves special thanks for devoting a lot of his time to instill understanding and mathematical taste in me. I would also like to thank Trevor Arnold, Tobias Berger, Jim Brown, Jim Cogdell, Stephen DeBacker, Muthukrishnan Krishnamurthy and Eric Urban for useful and inspiring discussions. I am indebted to Grzegorz Banaszak and Wojciech Gajda for encouraging me to choose this career path.

I would also like to take this opportunity to thank Janusz Migda for generously dedicating his time to teach me mathematics when I was a student in Poznan and Wiodzimierz Strózniak, who is responsible for my addiction to mathematics. Finally, I have always felt privileged to have parents who have created a world of opportunities for me. Along with my sisters, whose friendship has been invaluable over the years, they own an especially large share in this enterprise. ii TABLE OF CONTENTS ACKNOWLEDGEMENTS.

Notation and terminology. Q Q Q cu ee eee ee ee 8 2.1 Number fields and Hecke characf€T§. HH HQ va 8 2.2 The unitary group. LH nu gà và gà Và kg và va 10 2.

Q Q Q HQ HQ ng gà v k kg kg va 14 3. LG Q HH nà g k kg kg TK xa 20 "90-5.ốẽ ẽẼ Ẽ Ha ynxIlR qa aannRH <4 20 3.1 Siegel and Klingen Eisenstein series.2 Borel Eisenstein series .3 Holomorphic Eisenstein series on SLz(2).4 Siegel Eisenstein series with positive weight .3 Poles and residues of Eisenstein series. pe ee ee 36 3. The Petersson norm of a Maass lift.

Q nu ng g g v v v v v kg kg va 43 4.2 The Petersson normof Fp. Q Q HQ HQ HQ ee, 45 4.3 Residue of the Klingen Eisenstein §@Ties.4 Eo(g,s) as a residual Eisenstein series .5 Inner product formula for Jacobi forms. 1 HQ ng Quà ng kg gà gi kg sa 65 5.1 Classical Elliptic Hecke algebra.2 Classical Hermitian Hecke algebra.1 Hermitian Hecke operators.3 Integral structure of the hermitian Hecke algebra .4 Action on the Maassspace. 0000 cee eee eae 80 5.5 Lifting Hecke operators to the Maass space .38 Adelic Hecke operators.

ee va ga ga kg va 84 5. HH gà ee 84 ili 5.4 Hecke Eigenvalues of Siegel Eisenstein §eTl@€S. The standard L-function of a Maass lift.1 The inner product. LH HQ HH vn ee 93 6.2 The standard l-fInctilon.

uc cv và ee 96 7.1 Fourier coefficients of Eisenstein SeTÌS.2 Fourier coefficients of theta series .4 Main congruence result .2 Algebraicity of Cp, 6.3 Congruence between Fy and anon-Maass form .4 Congruence between Fy and a non-Maasseigenform. Hecke algebras and deformation rings .1 Congruences and weak congruences. LH HQ HQ ee eee 123 8.2 Deformations of Galois representations .1 Universal deformation ring.2 Hecke algebras as quotients of deformation rings .3 2ariski tangent space of the deformation functor .4 Isomorphism between TH, 0.3 Hida’s congruence modules.2 Maass lifts revisited .3 CAP representations for general imaginary quadratic fields. Galois representations and Selmer groups.

1 Hà gà va na 160 10.3 The adjoint representation. ng gu gà ga 165 10.4 A non-semisimple reduction of a Galois representation.00 00 2 ee 174 iv CHAPTER 1 Introduction Relating algebraic and analytic objects has been a common theme in number theory for nearly a century. Among the oldest examples of a connection of this kind is the analytic class number formula relating the residue of the Dedekind zeta function (studied by Hecke) to the size of the class group of a number field. Another instance, the Birch and Swinnerton-Dyer conjecture (or its generalization, the conjecture of Bloch and Kato) remains open and a lot of effort has been put to prove important special cases of it.

In these examples the role of an analytic invariant is played by an L-function, and the role of an algebraic invariant by a (piece of) a class group or its generalization, the Selmer group. In this thesis we prove that (under certain conditions), if a special value of the symmetric square L-function attached to a modular form f (of appropriate weight, level and character) vanishes mod a prime ý, then the Selmer group of the adjoint of the mod £ Galois representation attached to f is of size divisible by @ (Theorem 7. The idea of the proof can be traced back to the celebrated 1976 paper of Ribet [Rib76]; we explain it below. Here é; denotes a primitive /-th root of unity, and is the mod £ cyclotomic character.

His idea can roughly be summarized in the following sequence of implications: (11) 2| L(-1,x) => dcusp form g = Eisenstein series E, (mod) => => /| # Gv: The congruence in (1.1) is to be understood as a congruence between the Fourier coefficients of g and those of Ey. In the language of representation theory, x can be thought of as an automorphic form on the adelic points of GL; x GL, which is the Levi subgroup of a maximal parabolic B of GLa. Any elliptic modular form f gives rise to an automorphic form ys : GL2(A) > C, and yz, is a function occurring in the automorphic representation obtained by inducing y from B to GL2(A). Here A denotes the ring of adeles of Q.

The first implication of (1.1) can be rephrased as follows: IfÊ divides a special L-value of an automorphic form x on the Levi subgroup of the standard mazimal parabolic B of GLa, then the Fourier coefficients of the automor- phic form yp, obtained from x by inducing it to GL2(A) are congruent (mod £) to the Fourier coefficients of a cusp form on GLa(A). Even though this statement appears more complicated than the original formulation of Ribet, it suggests ways to generalize it to higher rank groups. Let K be an imaginary quadratic field. To the extension K/Q one can associate the quasi-split unitary group U(n,n) = {A € Resxjq GL, | AJA’ = J}, where J =|, ~TM], and J, denotes the n x n identity matrix.

In this thesis the role of x will be played by an elliptic modular form ƒ, the role of E, by a modular form F; on G := U(2,2) “lifted” from f, and the role of Cla ¢e,) by the Selmer group of the adjoint of ø;, the mod £ Galois representation attached to ƒ. The group G possesses two standard maximal parabolics, one of which - called the Siegel parabolic, denoted by P - has Levi subgroup M isomorphic to Resx/q(GLa)x. As in the case of elliptic modular forms, there are two ways of thinking of automorphic forms on G, one - as hermitian modular forms, i., functions F on an analogue H of the complex upper half-plane transforming appropriately under the action of G(Z) on H, and the other one as (adelic) automorphic forms, i. For precise definitions we refer the reader to section 2.

An analogue of Ribet’s theorem (1.1) in our situation, can be summarized as the following two implications: 1. If £ divides a special [-value associated to a cusp form ys on M(A) = GLo(Ax), then the form yp, obtained by “lifting” yy to G(A) has Fourier coefficients congruent (mod £) to the Fourier coefficients of a “proper” cusp form on G(A). The existence of the congruence described in (1) implies that £ divides the order of the Selmer group associated to (the twist of) the mod ý Galois representation Here Gx = Gal(K/K), Ax denotes the ring of adeles of K’, J; is the (mod £) Galois representation attached to f by Deligne, et al., and adf,|c, denotes the adjoint representation of Ø;|øx: We still need to explain how we lift ys to G(A). A standard method is to con- sider the representation induced from the automorphic representation 7; in which yr occurs.

This way one obtains Eisenstein series in a fashion analogous to the approach of Ribet. However, in the case of the group U(2,2) there is yet another way of lifting wy» to a form on G(A) which yields not an Eisenstein series, but a cusp form. Namely, one first considers an automorphic representation of = ©, Ofp obtained by inducing certain twists of Z; (by specific Hecke characters) from P(A) to G(A). One then studies a cuspidal representation II = @, H; such that for all but finitely many p, II, and of, share the same absolute values of Hecke eigenvalues.

Such representations were first studied by Piatetski-Shapiro [PS83] for the symplec- tic group Sp, and he called them CAP representations (Cuspidal, but Associated to Parabolic subgroups). We will call the Hecke eigenforms occurring in such represen- tations CAP forms. By a “proper” cusp form in implication (1) we mean a cusp form which is not CAP. A similar procedure is impossible for the group GL¿, as the work of Jacquet-Shalika [JS81a], [JS81b] and Moeglin-Waldspurger [MW89} shows that no CAP representations exist on GL,.

Thus our approach has no classical analogue. Even though our method should be applicable for any imaginary quadratic field, our current result is valid only when K = Q(. The reason for this restriction is that for this field we know the existence of “Hecke-equivariant” CAP lifts by the work of Kojima [Koj82], Krieg [Kri91] and Gritsenko [Gri90a]. Generalizing their methods and our current result to other class number one imaginary quadratic fields should not pose any serious difficulty.

The problem with fields of class number higher than one lies in the fact that the symmetric space on which the automorphic forms are defined is disconnected, and one either has to deal with Hecke operators which permute the connected components (which complicates computations) or prove the existence of CAP lifts using representation-theoretic methods. For some conjectural statements phrased in the language of representation theory, see section 9. The method used to prove implication (1) differs substantially from the one ap- plied in [Rib76]. The level of the form in implication (1) equals the discriminant of the field AK, hence our current result concerns only cusp forms of level 4.