Modular graph functions and odd cuspidal functions -- Fourier and Poincar\'e series

Understand

Modular graph functions are $SL(2,{\mathbb Z})$-invariant functions associated with Feynman graphs of a two-dimensional conformal field theory on a torus of modulus $\tau$.

• 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 Poincar\'e series with respect to $\Gamma_\infty \backslash PSL(2,{\mathbb Z})$.
• The Fourier and Poincar\'e series provide the tools to compute the Petersson inner product of two-loop modular graph functions using Rankin-Selberg-Zagier methods.