Research Programme ; Academic Year 2009

RESEARCH PROGRAMME IN ARITHMETIC GEOMETRY

An i-MATH Intensive Research Programme

Partially financed by the Barcelona Number Theory Research Group

First and third quadrimesters: From 1-9-2009 to 23-12-2009 and 19-4-2010 to 31-7-2010.

Key words: Global fields: number fields and function fields, Abelian varieties, Drinfeld modules, Galois representations, L-functions, Iwasawa theory, modular forms, Shimura varieties.

Already in the 19th century mathematicians had realized the striking similarities between number fields and function fields; later, prominent number theorists such as Artin and Weil made essential use of function fields of one variable over a finite field to reach a deeper understanding of L-functions and the Riemann hypothesis. Here we can see a first instance of a recurrent theme in the number fields/function field dialectic: the transposition into one setting of questions first asked over the other (often a far from trivial task) allows to gain new insight and, in many cases, to progress more than in the situation previously considered, thanks to a richer structure to be exploited. In particular, function fields have become a standard testing ground to get evidence for hard conjectures proposed over number fields.

In this Research Programme on Arithmetic Geometry, we will explore several key objects from the arithmetic of number fields/function fields which constitute a living proof of this fruitful cross-fertilization between the two worlds, such as Drinfeld and classical modular varieties, classical and p-adic L-functions, the Langlands programme, the Birch and Swinnerton-Dyer conjecture, Iwasawa theory and the Bloch-Kato conjecture.

The first and third quadrimesters of the programme will be focused on arithmetic geometry in the number field case, with an emphasis on modular forms and modularity questions, while the second quadrimester will be dedicated to the function field case in characteristic p, with an emphasis on Iwasawa theory and Characteristic p L-functions.

Below we provide a detailed description of the two sub-programmes: their background and specific objectives.

Background: During the last few years there have been fundamental breakthroughs in different aspects of the theory of modular forms and related subjects.

Several long-standing conjectures have been settled and fundamental progress has been achieved towards others. In addition, precise variants and generalizations of these conjectures have been raised, and these demand refinements of the existing methods as well as completely new ideas in order to be approached.

Some of these major achievements take the form of modularity results, which aim to establish a bridge between algebraic varieties or abstract geometric Galois representations on one side and modular or automorphic forms on a Shimura variety on the other. Prominent instances of this philosophy are Serre’s Modularity Conjecture and Fontaine-Mazur’s Conjecture, among others.

In parallel, the diophantine study of algebraic varieties has received a dramatic impulse over the last years provided they can be shown to be modular. Thanks to the celebrated result of Wiles and his collaborators, elliptic curves over the field of rational numbers provide the simplest (but highly non-trivial) examples of this phenomenon, and their arithmetic still offers deep questions which remain unsolved. The construction of rational points on them, the Birch and Swinnerton-Dyer Conjecture, its p-adic formulations and generalizations to higher dimensional abelian varieties over number fields are certainly one of them. So far, most insights to these problems exploit and are conditional to the fact that they appear as a factor of the Jacobian of a suitable modular or Shimura curve.

Computation has played a crucial role in the formulation of some of the most important conjectures involving modular forms and elliptic curves in the last decades. Nowadays a wide variety of algorithms, packages and software is available and, for this reason, the Research Programme will also focus on the computational side and the explicit approaches to the subject. Doubtlessly, number-theorists mainly interested in theoretical questions will benefit from a close interaction with those with a more computer-oriented background, and conversely.

Objectives: The main focuses of the first and third quadrimesters are the arithmetic of modular forms and modularity results. These are very broad and active areas of research, which see rapid developments in recent years. From our today’s perspective, it seems that the objectives of this program should at least include the following points:

-Generalizations of Serre's conjecture (to two-dimensional representations of the Galois group of a totally real number field and Galois representations of dimension greater than 2; local and global mod p Langlands correspondences),

-Potential modularity, modularity lifting theorems and applications (Fontaine-Mazur-Langlands conjecture; meromorphic continuation of L-functions; Sato-Tate conjecture; existence of compatible families; Cases of Tamagawa number conjectures;Tate conjecture and cases of Langlands functoriality),

-Galois Deformation Theory (Local problems: construction of potentially semistable deformation rings; framed deformations; local to global principles; construction of global deformations by purely Galois cohomological methods; deformation rings and base change),

- Arithmetic of abelian varieties (Theoretical and computational approaches to the Mordell-Weil group; Selmer and Tate-Shafarevic groups and the relation with the behaviour of the associated L-function at the critical point; Heegner points and Darmon’s conjectural constructions of Stark-Heegner points over non-CM number fields),

-Iwasawa theory of elliptic curves (Rational points over Z_p-extensions; p-adic L functions and p-adic formulations of the Birch and Swinnerton-Dyer Conjecture; Gross-Zagier like formulas and extensions of the theory to totally real number fields and higher dimensional abelian varieties),

-Arithmetic of Shimura varieties (André-Oort Conjecture; rational points (over finite fields, local and global fields, and the Hasse principle); integral models and moduli problems; complex and rigid analytic uniformisation; explicit approaches to Shimura curves; interaction with the arithmetic of abelian varieties),

Background: A new strand of function field arithmetic had its beginning in the 1930s, when Carlitz discovered an analogue of the exponential function over F_q[T]. Over a period of years, Carlitz and his students (most notably Hayes) expanded this analogy in a variety of ways, until the subject received tremendous impulse by Drinfeld's spectacular 1974 paper on elliptic modules (now called Drinfeld modules). Drinfeld's goal in introducing this theory was to advance the Langlands program for function fields, where his methods have proved extremely successful. However, looking at the mathematics originated with his paper one can see many striking examples of another motif in the relation between number and function fields: that is, the discovery of surprising phenomena and unexpected possibilities opened by the transposition of ideas from one setting to the other. An instance is provided by the theory of Drinfeld modular varieties, which show fascinating similarities with (and differences from) the classical modular curves; another is the development of a rich “analytic number theory in characteristic p”, which includes L- and Γ- functions, modular forms, Bernoulli numbers and transcendence results.

Objectives: The main focuses of the second quadrimester are Iwasawa theory and Characteristic p L-functions. Both topics have seen important advances in recent years, and of course they are strictly linked and help to the understanding of each other. Iwasawa theory techniques, which had among their sources the arithmetic of constant function field extensions, were later brought back to the characteristic p world: for example, Crew proved a geometric version of Iwasawa's main conjectures already twenty years ago and this result has been recently generalized to the non-commutative setting by Burns. Furthermore we mention Hayes' and Popescu's works on the Stark's conjectures. Initiated by Mazur, Iwasawa theory of abelian varieties is now also studied in the function field context. Ochiai-Trihan and Bandini-Longhi proved expected finiteness results on the Selmer group of elliptic curves. Function field analogues of the exceptional zero conjecture (work of Hauer-Longhi and of Pal), and various versions of Beilinson's conjecture on higher K-groups (work of Pal and of Kondo-Yasuda) have also been recently established. Iwasawa theory of abelian varieties is strictly linked to the Birch and Swinnerton-Dyer conjecture: among many recent advances in this direction, we mention Ulmer’s results on Gross-Zagier formulas and on the rank conjecture. As for L-functions, some of the most striking developments in this field over the past thirty years are the following: Goss' invention of “characteristic p-valued” L-functions, Anderson's generalization of Drinfeld modules to t-motives and his work with Thakur relating division points of such motives to special values of Goss' L-functions, the many spectacular results on transcendence of zeta-values (by Anderson, Brownawell, Denis, Papanikolas, Thakur, Yu), Thakur's gamma functions, Taguchi and Wan's work on meromorphic continuation of Goss' L-functions and its extension by Böckle. The theory of Drinfeld modules has been used by Consani and Marcolli to construct non-commutative spaces and investigate Goss' L-functions via quantum statistical mechanics. Another important recent result is Böckle's construction of an Eichler-Shimura isomorphism for Drinfeld modular forms: together with his previous work with Pink on “crystals”, this allows to move from cusp forms to L-series.

PERSPECTIVES OF THE PROGRAMME

The 2009-2010 Research Programme on Arithmetic Geometry at the Centre de Recerca Matemàtica (CRM) provides an excellent opportunity to bring together specialists and researchers in this field and germane lines of research, having expertise in different but always overlapping aspects of this beautiful branch of mathematics. The aim of the Research Programme is to learn about the details of major developments in the area, to explore new directions and perspectives, to promote fruitful collaborations among participants and to serve as a valuable centre for the advancement of young researchers.

The first quadrimester will be devoted to discuss the state of the art of the arithmetic of modular forms and modularity, whereas the third quadrimester will be focused on new horizons of the subject. This will be pursued through the organization of an intensive plan of activities, including a weekly seminar, a series of instructive and advanced courses each quadrimester, one workshop in December 2009 and a main conference in July 2010 around the topics above. The precise names of the lecturers will be proposed as soon as required.

As for the second quadrimester, we propose advanced courses by Gebhard Böckle, David Burns, David Goss, Dinesh Thakur and Douglas Ulmer as well as mini-courses by Urs Hartl, Fabien Trihan, Ignazio Longhi, and Ambrus Pal, which should cover, together with a weekly seminar, most of the developments sketched above. We also propose the organization of two workshops on Iwasawa theory and on characteristic p L-functions.

Since the subject of the proposal is a rapidly developing research area, it has the potential of being shaped considerably by young researchers. It is therefore believed that the program will have a broad international appeal and that the outcome will be of lasting scientific value.

In addition to the Number Theory group of Barcelona, a number of consolidated mathematicians, young researchers and post-doc students will be invited for short or long periods to the CRM. Long term (and if possible one-year-long) research stays will be specially encouraged.

Poster	A poster with information about the activity can be download here