Integrable Lagrangians and Picard modular forms

Evgeny Ferapontov, Loughborough University

We consider first-order Lagrangians such that the corresponding Euler-Lagrange equations belong to the class of 3D dispersionless integrable systems. It is demonstrated that the generic integrable Lagrangian density is a Picard modular form  of its arguments.

Explicit parametrisation of such densities by generalised hypergeometric functions of Appell type is obtained. Alternative representations and degenerations are also discussed.

The talk is based on joint work with F. Clery, A. Odesskii, and notes of D. Zagier.