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Arithmetical Problems in Number Fields, Abelian Varieties and Modular Forms*

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CONTRIBUTIONS to SCIENCE, 1 (2): 125-145 (1999) Institut d’Estudis Catalans, Barcelona Arithmetical problems in number fields, abelian varieties and modular forms* P. Bayer** et al. Abstract Resum Number theory, a fascinating area in mathematics and one La teoria de nombres, una àrea de la matemàtica fascinant i of the oldest, has experienced spectacular progress in re- de les més antigues, ha experimentat un progrés especta- cent years. The development of a deep theoretical back- cular durant els darrers anys. El desenvolupament d’una ground and the implementation of algorithms have led to base teòrica profunda i la implementació d’algoritmes han new and interesting interrelations with mathematics in gen- conduït a noves interrelacions matemàtiques interessants eral which have paved the way for the emergence of major que han fet palesos teoremes importants en aquesta àrea. theorems in the area. Aquest informe resumeix les contribucions a la teoria de This report summarizes the contribution to number theory nombres dutes a terme per les persones del Seminari de made by the members of the Seminari de Teoria de Teoria de Nombres (UB-UAB-UPC) de Barcelona. Els seus Nombres (UB-UAB-UPC) in Barcelona. These results are resultats són citats en connexió amb l’estat actual d’alguns presented in connection with the state of certain arithmetical problemes aritmètics, de manera que aquesta monografia problems, and so this monograph seeks to provide readers cerca proporcionar al públic lector una ullada sobre algunes with a glimpse of some specific lines of current mathemati- línies específiques de la recerca matemàtica actual. cal research. Key words: Number field, L-function, Galois group, quadratic form, modular form, modular curve, elliptic curve, Galois representation, abelian variety, Fermat curve, arithmetic variety, scheme, cycle, motive. Number theory, or arithmetic, is a fascinating area in mathe- Newton’s Principia [1686]. Nevertheless, from a set of arith- matics, and one of the oldest. It has changed dramatically metical problems proposed by Diophantus, Fermat, and since the 17th century, when Fermat was essentially working Gauss a mathematical treasure emerged: number theory as it in isolation in his careful study of Diophantus’ Arithmetica or, is understood today. needless to say, since the Babylonian period when an arith- Number theory covers a wide range of subjects: integer metician sculpted a stone tablet containing fifteen numbers, diophantine equations, number fields, arithmetical «pythagorean» triangles [Plimpton 322, ca. 1800 B.C.]. Dio- aspects of algebraic varieties, arithmetical functions, zeta phantus’ Arithmetica [ca. III A.C.] and Gauss’ Disquisitiones and L-functions, automorphic forms, and much more be- Arithmeticae [1801] are milestones of the past which never sides. A detailed outline of its advances is beyond the scope became as popular as Euclid’s Elementa [ca. IV-III B.C.] or of this article; suffice it to say that the progress made in the arithmetic of elliptic curves, modular curves and modular forms, combined with specific methods developed by G. *Dedicated to Albert, Carles, Carlos and Enrique, Elisabet, Pau Frey, B. Mazur, K. Ribet, and J-P. Serre, among others, and Mireia, Gabriel, Maria del Mar, Carolina, Javier, Maria dels Àn- gels, Antoni, Josep Maria and Joaquim, Marcel·lí, Enric, Judit, Sara, made possible the momentous study of A. Wiles [1995] on and Mireia. Fermat’s Last Theorem. **Autor for correspondence: Pilar Bayer, Departament d’Àlgebra i In Barcelona, in the mid-1970s, an appreciation at the in- Geometria. Universitat de Barcelona. Gran Via de les Corts Catalanes, 585. 08007 Barcelona, Catalonia (Spain). Tel. 34 93 402 herent beauty of arithmetic began to emerge; since then, an 16 16. Email: [email protected] increasing number of mathematicians have shared the plea- 126 P. Bayer et al. sure and pain of its unsolved problems, achievements, and Embedding problems over large fields (1998-99); Birch and methods, drawing on tools from many different fields in Swinnerton-Dyer conjecture (1998-99); Abelian varieties mathematics. The Seminari de Teoria de Nombres (UB- with complex muliplication (1999-00); Introduction to crys- UAB-UPC) began as an independent research group in talline cohomology (1999-00). 1986 and has organized activities every year since. Among the universities or research institutes where mem- Henceforth, for brevity’s sake, we will refer to this group as bers of the STNB have spent substantial periods of time are: STNB. Berkeley, Bonn, Essen, Freiburg, Harvard, Kraków, La Ha- bana, Luminy, Mar del Plata, McGill, Oberwolfach, Princeton, Research group Regensburg, Santander, Tokyo-Chuo, and Tokyo-Waseda. The STNB is made up of scientific personnel from three uni- The Journées Arithmétiques are an international meeting versities: Universitat de Barcelona (UB), Universitat with a long tradition. They take place every two years, held al- Autònoma de Barcelona (UAB), and Universitat Politècnica ternately in France and in other countries in Europe, and are de Catalunya (UPC). They are listed below. recognized as the most important meeting devoted to num- UB-team: A. Arenas, P. Bayer, T. Crespo, J. Guàrdia, G. ber theory. The STNB organized the 19èmes Journées Pascual, A. Travesa, M. Vela, N. Vila; UAB-team: F. Bars, S. Arithmétiques, with the support of the Generalitat de Comalada, J. Montes, E. Nart, X. Xarles; UPC-team: M. Catalunya, the Ministerio de Educación y Ciencia, and other Alsina, G. Cardona, J. Gonzàlez, J.-C. Lario, J. Quer, A. Rio; official institutions, in the summer of 1995. From July 16 to 20, as well as several graduate students writing their Ph. D. the- over 300 participants engaged in a forum for the presenta- ses under the direction of one of the doctors in the group. tion and discussion of recent number-theoretical develop- ments against a Mediterranean backdrop; cf. [e7: Ba-Cr 97]. Research area This article is organized as follows: in the following pages Classification codes will be given according to the we present an up-to-date overview of the main contributions Mathematics Subject Classification, 1991 revision. made by members of the STNB. This is divided in short sec- The research interests of the STNB lie mainly in: algebraic tions which include many references to pioneering studies. number theory (11R, 11S), field extensions (12F), quadratic A list of research publications is enclosed. In the main these forms (11E), automorphic forms (11F), diophantine geome- are quoted in the text, though additional references precede try (11G, 14G), and computational number theory (11Y). the main list. They include: textbooks, monographs, survey All aspects included in these interests are closely interre- articles, software packages and, in general, publications lated, and so those primarily engaged in one are inevitably supporting research and providing necessary background. involved in the others. In addition, most of the researchers of References are numbered in inverse chronological order, the STNB have worked together for a number of years on and so the lowest numbers correspond to the most recent joint projects supported by the DGICYT or the European items. Further details are available on request. Union. Co-authors of this report Research method A. Arenas, T. Crespo, J. Guàrdia, J.-C. Lario, E. Nart, J. The activities of the STNB can be divided into two broad Quer, A. Rio, A. Travesa, N. Vila, X. Xarles. groups: research and training. Research activities are nor- mally conducted via individual studies, interaction with col- leagues, and the exchange of scientific ideas with other spe- 1. Algebraic number theory: global and local fields cialists. Training activities, for their part, tend to take the form (11R, 11S) of lectures and since 1986 have been organized into 60- hour blocks of lectures per year. The STNB lectures are The arithmetical analysis of algebraic equations consists in open to all faculty members and graduate students, and solving the equations taking into account the minimum field serve to disseminate some of the major findings in current or ring to which solutions belong. Solutions lie in extensions research. The subject matter of the lectures is decided on by of finite degree of the field containing the coefficients of the the STNB each academic year. To date, the following sub- equations. Assuming this ground field to be the field Q of the jects have been addressed: rational numbers leads us to the study of their finite exten- Rational points on algebraic curves (1986-87); Serre’s sions, which are called number fields. conjecture on modular Galois representations (1987-88); The ring of integers OK of a number field K is a subring Modular curves and the Eisenstein ideal (1988-89); Galois which is able to support, at the level of ideals, arithmetical structures of Hodge-Tate type (1989-90); Algorithms in mod- properties analogous to those of the ring of integers Z. ular curves (1990-91); Arithmetic surfaces (1991-92); Galois These arithmetical properties are essential for the arithmeti- representations of dimension 2 (1992-93); Chow motives cal analysis of equations. (1992-93); Preliminaries to Fermat-Wiles theorem (1993-94); Among the fundamental challenges concerning the arith- Modular forms and Galois groups (1994-95); Torsion points metical structure of these rings or fields we find: of elliptic curves (1994-95); Modular elliptic curves (1995- (1) determining how the rational primes decompose as 96); Automorphic representations
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