Number Theory: New York Seminar 1989–1990 PDF
Preview Number Theory: New York Seminar 1989–1990
Number Theory New York Seminar 1989-1990 D. V. Chudnovsky G. V. Chudnovsky H. Cohn M.B. Nathanson Editors Number Theory New York Seminar 1989-1990 With 14 Figures Springer Science+Business Media, LLC David V. Chudnovsky Harvey Cohn Gregory V. Chudnovsky Department of Mathematics Department of Mathematics City University of New York Columbia University City College New York, NY 10027 New York, NY 10031 USA USA Melvyn B. Nathanson Provost and Vice President of Academic Affairs Lehman College City University of New York Bronx, NY 10468 USA AMS Classification: 10-06 Printed on acid-free paper. ©1991 Springer Science+Business Media New York Originally published by Springer-Verlag New York, Ine. in 1991. All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher ( Springer Science+Business Media, LLC ), except for brief excerpts in connection with reviews or scholarlyanalysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, ete., in this publication, even if the former are not especially identified, is not to be taken as a sign that sueh names, as understood by the Trade Marks and Merchandise Marks Act, may aeeordingly be used freely by anyone. Camera-ready copy provided by the editors. 987654321 ISBN 978-0-387-97670-9 ISBN 978-1-4757-4158-2 (eBook) DOI 10.1007/978-1-4757-4158-2 Preface New York Number Theory Seminar started its regular meeting in January, 1982. The Seminar has been meeting on a regular basis weekly during the academic year since then. The meeting place of the seminar is in midtown Manhattan at the Graduate School and University Center of the City Uni versity of New York. This central location allows number-theorists in the New York metropolitan area and vistors an easy access. Four volumes of the Seminar proceedings, containing expanded texts of Seminar's lectures had been published in the Springer's Lecture Notes in Mathematics series as volumes 1052 (1984), 1135 (1985), 1240 (1987), and 1383 (1989). Seminar co chairmen are pleased that some of the contributions to the Seminar opened new avenues of research in Number Theory and related areas. On a histori cal note, one of such contributions proved to be a contribution by P. Landweber. In addition to classical and modern Number Theory, this Semi nar encourages Computational Number Theory. This book presents a selection of invited lectures presented at the New York Number Theory Seminar during 1989-1990. These papers cover wide areas of Number Theory, particularly modular functions, Aigebraic and Diophantine Geometry, and Computational Number Theory. The review of C-L. Chai presents a broad view of the moduli of Abelian varieties based on recent work of the author and many other prominent experts. This provides the reader interested in Diophantine Analysis with access to state of the art research. The paper of D. V. and G. V. Chudnovsky deals with new and old continued fractions of classical functions and num bers. It presents results of extensive computations, new closed form contin ued fractions, and discussions of new transcendencies arising from the sim plest continued fractions. The first paper of H. Cohn presents a systematic, computer aided approach to classification of reduction of modular curves, initiated by Fricke. The second paper of H. Cohn is one of the first explicit works on singular moduli in complex multiplication of Hilbert modular functions. The work of J.1. Deutsch exarnines an appearance of a natural generalization of Ramanujan's tau-function in modular forms over special quadratic fieIds. R.R. Hall and G. Tenenbaum establish correct asymptotic behavior of the size of a set of multiples of a short interval, with a detailed analysis of a threshold behavior. In the paper of J. Huntley the first defini tive results on the equality of two non-holomorphic cusp forms based on comparison of Fourier coefficients are proved (unconditionally or assuming the Riemann hypotheses). A detailed paper by E. Kaltofen and N. Yui demonstrates the advantages of computer algebra to complete an explicit construction of Hilbert class fields of imaginary quadratic fields. This clas sical field of research is popular lately due to its primality proving implica- vi Preface tions. J. Lewittes paper provides interesting information on q-polynomials and finite fields. C. Meyer extends his work to present dass number formu las for pure imaginary quartic fields using complex multiplications. A.T. Vasquez's paper discusses in detail the desingularization of curves over finite fields. This process has proved to be effective in applications to co ding theory. The paper of N. Yui is focused on computations and conjectures concerning values of zeta-functions of Fermat varieties over finite fields at integral points. Mathematicians who plan to be in New York and would like to attend or lecture in the Seminar are encouraged to contact the organizers. The 1989- 1990 Seminar was partially supported by a grant from NSA Mathematical Sciences Program. Contents Preface . .............. ................... .... ....... ..... .. v Moduli of Abelian Varieties C-L. Chai 2 Classical Constants and Functions: Computations and Continued Fraction Expansions ....................................... 13 D. V. Chudnovsky and G. V. Chudnovsky 3 Some Special Complex Multiplications in '!Wo Variables Using Hilbert Singular Moduli .................................... 75 H. Cohn 4 A Numerical Survey of the Reduction of Modular Curve Genus by Fricke's Convolutions ...................................... 85 H. Cohn 5 Conjectures Relating to a Generalization of the Ramanujan lau Function .................................. 105 J. I. Deutsch 6 The Set of Multiples of a Short Interval 119 R. R. Hall and G. Tenenbaum 7 Comparison of Maaß Wave Forms 129 J. Huntley 8 Explicit Construction of the Hilbert Class Fields of Imaginary Quadratic Fields by Integer Lattice Reduction ................. 149 E. Kaltojen and N. Yui 9 On Certain q-Polynomials 203 J. Lewittes 10 A Gap Theorem for Differentially Algebraic Power Series 211 L. Lipshitz and L. A. Rubel 11 Class Number Formulas for Imaginary Pure Quadratic Number Fields .................................. 215 C. Meyer viii Contents 12 Rational Desingularization of a Curve Defined Over a Finite Field ..... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 229 A. T. Vasquez 13 Special Values of Zeta-Functions of Fermat Varieties Over Finite Fie1ds ......................................... 251 N. Yui 1 Moduli of Abelian Varieties Ching-Li ChaP This is the written version of my talk given on Oct. 11, 1990. Its very limited purpose is to outline a picture about the moduli of abelian varieties. Only a small portion of the theory of moduli of abelian varieties is covered. Many (if not aU) statements are imprecise and no proof is ofIered, in the hope that this makes the material more palatable. For the grumbling readers, the precise statements together with proofs can be found in the references. Recall that an abelian variety A is a smooth connected complete group variety. Over the field C of all complex numbers A( C) is a complex torus, identified with the quotient Lie(A)j H1(A(C), Z) of its Lie algebra by its first homology group. A polarization of an abelian scheme (Le. a farnily of abelian varieties) A -+ S is an S-homomorphism <p : A -+ At from A to its dual At which etale locally is induced from an arnple invert ible sheaf on A. <p is called a principal polarization if it is an isomorphism. Over C a polarization on a complex torus Vj A corresponds to a positive definite hermitian inner product h on V whose iniginary part Im(h) gives a Z-valued alternating form on A. The elementary divisors of Im(h) are caUed the type of the polarizationj if they are all equal to 1 the polar ization is principal. Our discussion will be concentrated on principally polarized abelian varieties. Over C it is easy to classify principally polarized abelian varieties of a fixed dimension gj this is the classical problem of classifying abelian integrals. Let Hg be the Siegel upper half space of genus g consisting of symmetrie g x g complex matrices whose imaginary parts are positive definite. The group SP2g(Z) of integral symplectic 2g x 2g matrices op erates on Hg via the classical formula Z ~ (AZ + B)· (BZ + D)-l. The quotient SP2g(Z)\Hg classifies principally polarized abelian varieties of dimension g. The same classification problem over Z has been solved. The result is as folIows: there is a smooth separated algebraic stack Ag over Z which classifies principally polarized abelian varieties of dimension g. Thus Ag is covered by some smooth scheme U -+ Spec(Z) in the etale topology, R =def U X Ag U together with the two finite etale projections Pl,P2 : R -+ U is a groupoid (or, an equivalence relation) over U, and Ag 'University of Pennsylvania, Philadelphia, PA 19003, USA. 2 c.-L. Chai is "the quotient of U by R". Over C, the attached analytic stack =Ag: an is the (stack) quotient of Hg by the groupoid R =def SP2g(Z) X Hg Hg given by the action of SP2g(Z) on Hg. (For instance, a vector bundle on Agan is (equivalent to) a vector bundle on Hg together with a SP2g(Z) action.) The algebraic stack Ag is not proper over Z, which reflects the fa miliar phenomenon that abelian varieties can degenerate. Intuitively, one should "add suitable degenerations of abelian varieties to Ag" in order to compactify Ag. The important semistable reduction theorem for abelian varieties, proved by Grothendieck and Mumford (c. [SGA 71], [DM]) sug gests that adding semiabelian degenerations would be enough. (Recall that a semiabelian scheme G --> S is a smooth group scheme such that every fiber is an extension of an abelian variety by a torus. When the base is a discrete valuation ring, the dimension of the toric part of the closed fibre is just the number of independent vanishing cycles of the degeneration.) The compactification problem for Ag has been solved. Many prominant mathematicians have contributed to this problem, no tably D. Mumford and G. Faltings. In the following several paragraphs we shall describe the main results about the arithmetic toroidal compact ifications of Ag. The readers can consult [Fe] for a detailed exposition, see also [F 2], [C 1], [C2]. Unlike the case for the moduli of curves, there are infinitely many good compactifications of Ag. This phenomenon already manifests itself over C. In [AMRT] a dass of toroidal compactifications for any arithmetic quotient of a bounded symmetrie domain was constructed. In order to construct a toroidal compactification, one needs to choose some combi natorial data called "admissible polyhedral cone decompositions". Dif ferent choices of admissible polyhedral cone decompositions give rise to different toroidal compactifications. The cone decompositions form a pro jective system with respect to sub divisions; corrspondingly the toroidal compactifications form a projective system, the transition maps being toroidal blowing-ups. For Shimura varieties which can be interpreted as moduli spaces for abelian varieties with specified dimension, polarization type, ring of endomorphisms and level structure (i.e. a Shimura variety oE PEL type), one can construct arithmetic version of the toroidal com pactifications, answering affirmatively a question Mumford raised in the introduction of [AMRT]. The case for Ag already contains the essential difficulty, which is why we concentrated in this case in the first place. For Ag the relevant cone is the cone Cg of all R-valued positive semi-definite quadratic forms on zg whose radicals are defined over Q; the existence of adrnissible polyhedral cone decomposition is a consequence of the re-
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