School and research conference
Modular Forms and Beyond

Scientific organisers:

The school and research conference are organised by EIMI and Laboratory of Mirror Symmetry and Automorphic Forms of NRU HSE and supported by the Simons Foundation (via PDMI RAS).

The school will take place at Euler International Mathematical Institute. The institute address is: 10 Pesochnaya nab. Saint Peterburg 197022

The goal of the school-conference is to bring together PhD students and young researchers using automorphic forms in different research areas. We plan three or four mini-courses in the morning sessions and 30 minutes research talks of young participants in the afternoon.

This school-conference will be the first joint event of the French-German Aachen-Bonn-Cologne-Lille-Siegen automorphic seminar working from 2003, and the new Automorphic seminar of the International laboratory of mirror symmetry and automorphic forms of NRU HSE in Moscow.

The school has some funds to cover local expenses of participants and a rather limited travel grant.

This short course is an invitation to the theory of modular forms. All definitions will be given in the course. We discuss the classical SL_2(Z)-modular forms and the Jacobi modular forms from the point of view of the Siegel modular forms of genus 2. We discuss also different types of commutative and non-comutative Hecke rings. Then we define the global L-functions of modular forms. The final result of the course if a construction of the analytic continuation of the Spin (Andrianov) L-function of Siegel modular forms using the lifting of Jacobi modular forms. The course is related to the lectures of Niels Skoruppa and it is an introduction to the course of Rainer Weissauer.

Adeles on algebraic curves and number fields were introduced by C. Chevalley and A. Weil in the middle of XX-th centure. Adeles were applicable as for class field theory as, for example, for description of vector bundles on algebraic curves. Adeles on higher-dimensional algebraic varieties were introduced by A. N. Parshin, in 1976, for smooth algebraic surfaces, and by A. A. Beilinson, in 1980, in higher-dimensional case. Adeles connect local and global properties of algebraic varieties, where local object is a higher-dimensional local field. On an algebraic surface, a two-dimensional local field is the field of iterated Laurent series and is constructed by a point and a formal stalk of algebraic curve containing this point on the algebraic surface. Two-dimensional local fields are important for two-dimensional class field theory and for various geometric applications.
In my lectures I will describe how two-dimensional local fields appear from algebraic surfaces. I will talk on adelic complexes on algebraic curves and surfaces, and how these complexes calculate cohomology of coherent sheaves. I will talk on adelic interpretation of divisors and intersection index of divisors on algebraic surfaces, and on adelic description of vector bundles and their Chern numbers. Also I will talk on two-dimensional residues and reciprocity laws for residues on algebraic surfaces. It is expected that participants will know basic elements from algebraic geometry: algebraic curves and surfaces, coherent sheaves and corresponding commutative algebra.

After a steep review of the basics of the arithmetic theory of elliptic modular forms and Jacobi forms (Eichler-Shimura isomorphism, L-Series, Hecke operators, new form theory, liftings from Jacobi forms to elliptic modular forms, Waldspurger's theorem for Jacobi forms) we shall concentrate on explicit descriptions and formulas for modular and Jacobi forms. By the latter we understand finite closed formulas for the Fourier coefficients of these forms. It is indeed possible to derive such formulas in a surprisingly easy and non-technical way. The starting point for this are the Schreier cosets graphs associated to subgroups of the modular group. We shall accordingly study these graphs and show how to relate them to spaces of modular and Jacobi forms. To minimize technical considerations we shall concentrate mainly on the important case of modular and Jacobi forms of weight 2. At the end of this mini course every participant should be able to compute by hand from a given elliptic curve over the rationals with, say, conductor m below 100, an explicit closed formula for the Fourier coefficients of its L-series and for the values at the critical point of the L-series of its twists with discriminants which are squares modulo 4m.

The Langlands philosophy predicts a strong relationship between finite dimensional representations of the absolute Galois group of number fields $K$ and automorphic representations of reductive groups $G$ over the field $K$. Similarly, one expects analogous local results for the completions of the number fields. In all cases known so far, there exist global $L$-series resp. local L-factors and $\varepsilon$-factors defined for both sides so that the underlying correspondences are defined by a comparison of these factors. Through deep results of M.Harris and R.Taylor locally this has been fully established for the linear groups $G=Gl(n)$. In particular, for local fields $K$ and $G=Gl(n)$ it is well known how to define $L$- and $\varepsilon$-factors attached to irreducible admissible representations $\pi$ of $G(K)$. For other reductive groups $G$, however, not much is known beyond the case of the symplectic group of similitudes $GSp(4)$. There, similar $L$-factors and $\varepsilon$-factors have been defined, first by A.Andrianov and then later, using Bessel models of $\pi$, by I.Piateskii-Shapiro. Recent calculations of these local $L$-factors indicate certain new phenomena, and this gives rise to interesting questions. A detailed discussion of the case $GSp(4)$ will be presented in the lectures.

Periods of an algebraic variety satisfy a special class of linear differential equations, known as Picard--Fuchs equations. The differential equations themselves encode some important arithmetic information; in particular, they provide a natural link to "counting" on the variety and to modular (automorphic) forms. In the minicourse, we will walk through examples of such differential equations of low order and their connection to several arithmetic phenomena, including integrality of related "mirror" expansions, modular parametrizations and formulas for $1/\pi$.

Local coordinators: