Modular graph functions and odd cuspidal functions. Fourier and Poincare series

Modular graph functions are SL(2, Z)-invariant functions associated with Feynman graphs of a two-dimensional conformal field theory on a torus of modulus . For one-loop graphs they reduce to real analytic Eisenstein series. We obtain the Fourier series, including the constant and non-constant Fourier modes, of all two-loop modular graph functions, as well as their Poincare series with respect to \PSL(2, Z). The Fourier and Poincare series provide the tools to compute the Petersson inner product of two-loop modular graph functions using Rankin-Selberg-Zagier methods. Modular graph functions which are odd under are cuspidal functions, with exponential decay near the cusp, and exist starting at two loops. Holomorphic subgraph reduction and the sieve algorithm, developed in earlier work, are used to give a lower bound on the dimension of the space A(w) of odd two-loop modular graph functions of weight w. For w 11 the bound is saturated and we exhibit a basis for (w).