Building Bridges: 1st EU-US conference on Automorphic Forms and related topics

Building Bridges

Modular Symbols

Modular symbols were introduced by Manin several decades ago and since then
have been studied, refined, and reformulated by several authors. They provide an explicit
description of elliptic modular forms by a finite set of algebraic integers, and thus are the main
tool for computations of modular forms. They provide theoretical insights into the special
values of L-series of modular forms inside the critical strip. They are the main building
blocks for explicit formula for the Fourier coefficients of Jacobi forms, modular forms of half
integral and integral weight. They are the main data for a not well-understood reciprocity
law for solutions modulo primes of an elliptic curve defined over the integers. Though they
can be easily computed and described in a purely algebraic or combinatorial manner they
are still full of mysteries and yet not really understood.

In the simplest case, the complex vector space of modular symbols associated to a subgroup
G of finite index in the elliptic modular group consists of all homomorphisms of the group
of formal integral linear combinations of points of the projective line over the rationals with
sum of the coefficients equal to zero into the additive group of complex numbers which are
invariant under the group G. There is a natural isomorphism of the space of holomorphic
and antiholomorphic modular forms of weight 2 on the group G with the space of modular
symbols of G.

The mini-course will give definitions and proofs of the basic properties of modular symbols,
present algorithms to compute with modular symbols and to compute modular forms, and
derive explicit formulas for the Fourier coefficients of modular forms of integral weight and
their associated modular forms of half integral weight. Topics will include:

  1. The definition of spaces of modular symbols.
  2. How to use Manin symbols to do explicit computations, for example, computing
    elliptic curves.
  3. Manin symbols and labeled Schreier coset graphs.
  4. Modular forms of integral weight as theta series associated to integral quadratic forms
    of signature (2, 2).
  5. Modular forms of half integral weight as theta series associated to integral quadratic
    forms of signature (2, 1).
  6. Modular symbols and homology and cohomology of modular groups.