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The Rank of the Jacobian of Modular Curves: Analytic Methods

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The Rank of the Jacobian of Modular Curves: Analytic Methods THE RANK OF THE JACOBIAN OF MODULAR CURVES: ANALYTIC METHODS BY EMMANUEL KOWALSKI ii Preface The interaction of analytic and algebraic methods in number theory is as old as Euler, and assumes many guises. Of course, the basic algebraic structures are ever present in any modern mathematical theory, and analytic number theory is no exception, but to algebraic geometry in particular it is indebted for the tremendous advances in under- standing of exponential sums over finite fields, since Andr´eWeil’s proof of the Riemann hypothesis for curves and subsequent deduction of the optimal bound for Kloosterman sums to prime moduli. On the other hand, algebraic number theory has often used input from L-functions; not only as a source of results, although few deep theorems in this area are proved with- out some appeal to Tchebotarev’s density theorem, but also as a source of inspiration, ideas and problems. One particular subject in arithmetic algebraic geometry which is now expected to benefit from analytic methods is the study of the rank of the Mordell-Weil group of an elliptic curve, or more generally of an abelian variety, over a number field. The beautiful conjecture of Birch and Swinnerton-Dyer asserts that this deep arithmetic invariant can be recovered from the order of vanishing of the L-function of the abelian variety at the center of the critical strip. This conjecture naturally opens two lines of investigation: to try to prove it, but here one is, in general, hampered by the necessary prerequisite of proving analytic continuation of the L-function up to this critical point, before any attempt can be made; or to take it for granted and use the information and insight it gives into the nature of the rank as a means of exploring further its behavior. This is justified by the trust put into the truth of the conjecture. Indeed, the first approach has been quite successful: in some cases, most notably large classes of elliptic curves over Q, analytic continuation is known, and partial results towards the conjecture have been obtained when the rank is 0 or 1. On the other hand, even when this is so, the second approach has had the aesthetic disadvantage that most studies of the rank, whether based on the assumption of the full statement of the conjecture or on known cases of it, have also assumed other analytic facts about the L-function, most notably that it satisfies the Generalized Riemann Hypothesis. This is somewhat unsatisfactory, inasmuch as this appears to be a much harder problem than even the Birch and Swinnerton-Dyer conjecture, although zeros of the L-function are of course very relevant to the problem. The contribution of this thesis is to show that analytic methods and techniques can indeed provide sharp, unconditional answers to some of the questions thus raised. This demonstrates that the implicit promise of the conjecture of Birch and Swinnerton-Dyer, of furnishing an effective way of answering questions about the rank through its analytic interpretation, can be kept without additional assumptions. The main results have been obtained in collaboration with Philippe Michel, and some auxiliary propositions had been proved earlier in the course of other work with ii iii William D. Duke. This volume is organized in six chapters. The first contains an introduction to the theory of abelian varieties and the Birch and Swinnerton-Dyer conjecture which is the motivating problem, and ends with the precise statements of the two principal theorems. The second chapter takes up the analytic side of the story. It recalls the results of Eichler-Shimura and Gross-Zagier which make the link between the algebraic geometry and modular forms, and provides an informal, but quite detailed, sketch of the proofs of the theorems. The extent of this first part, which is not original, stems from the fact that whereas the motivating problem lies in arithmetic algebraic geometry, an almost complete translation to a problem of analytic number theory is made, and this problem has intrinsic interest. Perchance, readers of both backgrounds will want to look at this document, and a goal of the text is to give to all an understanding of the other side of the story. The preliminaries over, at last, the process of proving is engaged with a stiff upper- lip. The third chapter contains a result about the “almost-orthogonality” of the sym- metric squares of modular forms which is crucial later for both results, and the fourth deals with another aspect of this kind of orthogonality principle. Then the last two chapters take each theorem in turn. There are a number of similarities in the prin- ciples and in some of the steps of both proofs, but since they seem to hold the same virtues of attraction and worth, the ordering is rather arbitrary and to accommodate the random-minded reader whose interest would lie in only one of the two, a certain amount of redundancy has been introduced, or cross-references sometimes inserted. A conclusion comes, not surprisingly, to conclude all this with some reflections about the meaning of the results and possible developments. iii iv Table of Contents Preface ......................................... ii Notations ....................................... v 1. Context and statements ............................ 1 1.1. Abelian varieties . 1 1.2. The Jacobian of an algebraic curve . 11 1.3. The modular curves and their Jacobians . 18 2. The analytic side ................................ 27 2.1. Reducing to modular forms . 27 2.1.1. Hecke theory, primitive forms . 27 2.1.2. Eichler-Shimura theory and corollaries . 30 2.1.3. The Gross-Zagier formula and consequences . 32 2.2. Sketch: the upper bound . 37 2.3. Sketch: the lower bound . 40 3. Mean-value and symmetric square ..................... 46 3.1. The symmetric square of modular forms . 46 3.2. The mean-value estimate . 48 3.3. Proof of the mean-value estimate . 51 3.4. Notational matters . 59 3.5. Removing the harmonic weight: the tail . 60 3.5.1. Sketch of the idea . 62 3.5.2. The tail of the series . 63 4. The Delta symbol ................................ 68 4.1. The Delta symbol for primitive forms . 68 4.2. The Delta symbol for odd primitive forms . 70 4.3. The Delta-symbol without weight . 71 Appendix: Multiplicativity . 72 5. The upper bound ................................ 75 5.1. The explicit formula: reduction to a density theorem . 75 5.2. The density theorem . 82 5.3. The harmonic second moment . 86 5.3.1. The square of the L-function . 87 5.3.2. Computation of the harmonic second moment . 90 5.3.3. Diagonalization . 91 5.3.4. Estimation of the harmonic second moment . 92 iv v 5.4. Removing the harmonic weight: the head, I . 96 6. The lower bound ................................. 105 6.1. Non-vanishing in harmonic average . 105 6.1.1. Preliminary: a refined statement . 105 6.1.2. Computation of the first moment . 107 6.1.3. Computation of the second moment . 110 6.1.4. The preferred quadratic form, I . 126 6.1.5. Harmonic non-vanishing . 134 6.2. Removing the harmonic weight: the head, II . 136 6.2.1. Computation of the first moment . 138 6.2.2. Computation of the second moment . 139 6.2.3. Mutations of the second moment . 139 6.2.4. The preferred quadratic form, II . 143 6.2.5. Optimization of the preferred form . 144 6.2.6. The second part of the main term . 148 6.2.7. The residual quadratic forms . 149 6.2.8. Conclusion . 150 Appendix: Extending the mollifier . 152 Conclusion ...................................... 156 References ....................................... 160 v vi Notations We introduce here some basic notations. • For any set X, |X| denotes its cardinality, a natural number if X is finite, and +∞ if it is infinite. • If A is an algebraic variety defined over a field k and K is an extension of k, A(K) is the set of points of A with coordinates in K, also called K-rational points. n • Fq denotes a field with q elements (q = p is the power of a prime number p), with Fp = Z/pZ for a prime p. • If k/Q is a number field, Ok will denote its ring of integers. The norm of an ideal a ⊂ Ok is written Na. • In any group G, the subgroup generated by a subset H ⊂ G is denoted by <H>; if H = {h} is a singleton, we also write simply <h>. • For any ring A, A× is the group of units of A. • For any two objects X and Y of any category, X ' Y means that X and Y are isomorphic. • The notation χD designates the Kronecker symbol of a quadratic field with dis- criminant D. • If σ is an automorphism of a field K, the action of σ on x ∈ K is denoted by σ(x) as well as by xσ. The following pertain to analytic number theory. • The notation n ∼ N, for real numbers n and N, means N < n 6 2N. • For any integer q > 1, εq is the trivial Dirichlet character modulo q, and ζq(s) = L(s, εq) is the corresponding L-function, the Riemann Zeta function with the Euler factors for p | q removed. • We denote by τ, ϕ, µ, Λ the classical arithmetic functions, namely (respectively) the divisor function, the Euler function, the M¨obiusfunction and the Van Man- goldt function, so X X ζ(s − 1) τ(n)n−s = ζ(s)2, ϕ(n)n−s = ζ(s) n>1 n>1 X X ζ0(s) µ(n)n−s = ζ(s)−1, Λ(n)n−s = − .
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