Peter F. Stiller Automorphic Forms and the Picard Number of an Elliptic Surface Aspects d Mathematics Aspekte der Mathematik Editor: Klas Diederich Vol. E1: G. Hector/U. Hirsch, Introduction to the Geometry of Foliations, Part A Vol. E2: M. Knebusch/M. Kolster, Wittrings Vol. E3: G. Hector/U. Hirsch, Introduction to the Geometry of Foliations, Part B Vol. E4: M. Laska, Elliptic Curves over Number Fields with Prescribed Reduction Type Vol. E5: P. Stiller, Automorphic Forms and the Picard Number of an Elliptic Surface Vol. E6: G. Faltings/G. Wustholz et aI., Rational Points (A Publication of the Max-Planck·lnstitut fur Mathematik, Bonn) Vol. 01: H. Kraft, Geometrische Methoden in der I nvariantentheorie The .texts published in this series are intended for graduate students and all mathematicians who wish to broaden their research horizons or who simply want to get a better idea of what is going on in a given field. They are introductions to areas close to modern research at a high level and prepare the reader for a better understanding of research papers. Many of the books can also be used to supplement graduate course programs. The series comprises two sub-series, one with English texts only and the other in German. Peter F. Stiller Automorphic Forms and the Picard Number of an Elliptic Surface M Friedr. Vieweg & Sohn BraunschweiglWiesbaden Dr. Peter F. Stiller is Associate Professor of Mathematics at Texas A & M University, College Station, Texas 77843, USA. 1984 All rights reserved © Friedrich Vieweg & Sohn Verlagsgesellschaft mbH, Braunschweig 1984 Softcover reprint of the hardcover 1st edition 1984 No part of this publication may be reproduced. stored in a retrieval system or transmitted in any form or by any means. electronic. mechanical. photocopying, recording or otherwise. without prior permission of the copyright holder. Procuded by IVD, Walluf b. Wiesbaden ISBN 978-3-322-90710-3 ISBN 978-3-322-90708-0 (eBook) DOI10.1007/978-3-322-90708-0 INDEX Introduction ••.•.•.....•..•.••••..••••..••••••••••••••••••••••••••• 1 PART I. DIFFERENTIAL EQUATIONS fl. Generalities............................................. 6 §2. Inhomogeneous equations •••••••••••••••••••••••••••••••••• 8 §3. Automorphic forms •••••••••••••••••••••••••••••••••••••••• 11 §4. Periods •••••••••.•••••••••••••••••••••••••••••••••••••••• 16 PART II. K-EQUATIONS §1. Definitions 23 §2. Local properties •••••.•••.••.••••••.•••••••••••••.••..••. 25 §3. Automorphic forms associated to K-equations and parabolic cohomology ••••••••••••••••••••••••••••••••• 29 PART III. ELLIPTIC SURFACES §1. Introduction •••••••.••••••••••••.•••.•.•••.••••.••••••••• 41 §2. A bound on the rank r of Egen(K(X» ••••••••••••••••••• 45 §3. Automorphic forms and a result of Hoyt's ••••••••••••••••• 58 §4. Periods and the rank of Egen(K(X» ••••••••••••••••••••••• 67 §5. A generalization .••••••.•..•.••.•..•.•••••••••••••••.•••• 72 PART IV. HODGE THEORY §1. The filtrations 86 §2. Differentials of the second kind ••••••••••••••••••••••••• 98 PART V. THE PICARD NUMBER §I. Periods and period integrals ••••••••••••••••••••••••••••• 103 §2. Periods and differential equations satisfied by normal functions •••••••••••••••••••••••••••• 126 §3. A formula, a method, and a remark on special values of Dirichlet series •••••••••••••••••••••••••••••••••••••• 132 §4. Examples ••••••••••••••••••••••••••••••••••••••••••••••••• 151 APPENDIX I. THIRD ORDER DIFFERENTIAL EQUATIONS •••••••••••••••••••• 187 BIBLIOGRAPHY ••••••••••••••••••••••••••••••••••••••••••••••••••••••• 192 The author wishes to express his thanks to the Sonderforschungsbereich of the Universitat Bonn, West Germany for its invitation and financial support during all of 1981, and for providing a stimulating and rewarding atmosphere in which to work. Parts of this manuscript were also written during the 1982-1983 academic year while the author was a guest of the Institut des Hautes Etudes Scientifiques. Lastly, thanks must go to the referee for his many thoughtful comments which have improved the presentation of these results. Any remaining errors are the sole responsibility of the author. 1 INTRODUCTION In studying an algebraic surface E, which we assume is non-singular and projective over the field of complex numbers t, it is natural to study the curves on this surface. In order to do this one introduces various equivalence relations on the group of divisors (cycles of codimension one). One such relation is algebraic equivalence and we denote by NS(E) the group of divisors modulo algebraic equivalence which is called the N~ron-Severi group of the surface E. This is known to be a finitely generated abelian group which can be regarded naturally as a subgroup of 2 H (E,Z). The rank of NS(E) will be denoted p and is known as the Picard number of E. Every divisor determines a cohomology class in H2(E,E) which is of type (1,1), that is to say a class in HI(E,9!) which can be viewed as a subspace of H2(E,E) via the Hodge decomposition. The Hodge Conjecture asserts in general that every rational cohomology class of type (p,p) is algebraic. In our case this is the Lefschetz Theorem on (I,l)-classes: Every cohomology class 2 2 is the class associated to some divisor. Here we are writing H (E,Z) for 2 its image under the natural mapping into H (E,t). Thus NS(E) modulo torsion is Hl(E,n-~! ) n H2(E,Z) and p measures the a 1g e br ai c part 0 f t h e cohomology. In this paper we shall be interested in a certain class of elliptic surfaces and in the problem of extracting information about p directly from the Gauss-Manin connection for such families of elliptic curves. We shall develop a method to determine the Picard number (or more precisely, to establish the existence of sections of the elliptic surface E over its base curve X) by reducing the problem to the computation of certian mixed hypergeometric and abelian integrals on a punctured sphere and we shall interpret these integrals in numerous interesting ways -- namely, as the periods of an inhomogeneous differential equation; as the periods of a generalized automorphic form; in terms of the monodromy of the differential equation satisfied by a "normal function"; in terms of the special values of certain Dirichlet series; as extension class data for a locally split extension of flat vector bundles (see Stiller [36); and lastly from the standpoint of the representation theory of a certain parabolic subgroup of SL3 (see Vilenkin [35). Our primary tools will be the Gauss-Manin connection associated to the surface expressed as a homogeneous second order linear differential equation A with regular singular points on the base curve X and an inhomogeneous de Rham cohomology created from certain inhomogeneous differential equations formed with this operator A (along the lines of Atiyah and Hodge [41): locally exact modulo exact). Essential to understanding our point of view is the Eichler-Hoyt correspondence between normal functions and generalized automorphic forms, and the techniques inaugurated by Hoyt in [12). In addition it is useful 3 to have some knowledge of hypergeometric integrals. The most complete up- to-date source is Exton [40]. Our integrals are the analogues of those that arise in the theory of classical automorphic forms (see Stiller [36] and Chowla and Gross [44]). In all that follows E will be an elliptic surface having a global section over its base curve X. It is assumed throughout that the functional invariant J is non-constant and that E has no exceptional curves of the first kind in its fibers. We shall utilize the general theory of elliptic surfaces due to Kodaira [16] and [17], and properties of the Gauss-Manin connection associated with an elliptic surface (Stiller [28] and [29]). Letting K(X) be the function field of the base curve X, we denote by Egen(K(X» the group of K(X)-rational points on the generic fiber Egen of E over X which is an elliptic curve over K(X). By the Mordell-Weil -2en Theorem, ~ (K(X» is a finitely generated abelian group whose rank we denote by r. We refer the reader to Shioda [27] for a description of NS(E) and its relation to the group Egen(K(X». In particular recall p - r + 2 + I (m - 1) v v where v is an index running over the singular fibers of the elliptic surface and is the number of irreducible components of the fiber. We shall focus on the rank r which is also the rank of the group of sections of E over X, but as the above formula shows, any method to determine r allows us to determine p. In the first section we shall discuss differential equations in general and certain automorphic forms that can be attached to them. We then focus attention on a special class of differential equations known as K-equations which are the relevant ones to consider when dealing with 4 elliptic surfaces. In this case the associated automorphic forms are especially interesting and we can interpret the periods of these forms in a number of useful ways as mentioned above. In the third part we compare some bounds obtained on p by Stiller [29) and Hoyt [12), and give proofs of some of Hoyt's results by direct computation of the automorphic forms involved; including a generalization of a result in Shioda [27) identifying HO(E,--~E) wi t h a certai n spacef 0 automforphic or ms. Part f our relates these results, the differential equations, and the automorphic forms constructed from them to the Hodge theory of E over X. As a sidelight, we give some applications to determining the torsion in Egen(K(X». Finally, in part five we give some of the interpretations of the periods and a method for determining the Picard number -- illustrating it with some examples. That these examples and computations involve a great deal of the theory of special functions is easily understandable in light of the analogy with classical automorphic forms. For what we are really doing (in one interpretation) is finding the special values of certain Dirichlet series, and it thereby comes as no surprise when the values that we compute are ratios of products of values of the gamma function. (For some useful rationality properties of values of the gamma function see the appendix by Koblitz and Ogus in Deligne [42).) In fact our methods can be adapted to the classical case of the special values in the critical strip of the Dirichlet series associated to a cusp form of weight greater than two (Stiller [36). In this case one can sometimes make use of the formulas of Damerell and Chowla-Selberg -- for example when the cusp form is attached to a grossen-character. In all the examples we do, the integrands used to compute the periods resemble those of the classical beta functions times a hypergeometric function and are those which Exton [40) refers to as integrals of Euler type. These represent parts of weight two (specifically