## Section: New Results

### Modular forms and $L$-functions

Participant : Henri Cohen.

Members of the team have taken part in an international autumn school on computational number theory at the Izmir Institute of Technology (IZTECH) in 2017. Henri Cohen has transformed his two lectures in book chapters. The text on modular forms [23] presents the (of course extremely condensed) view of the book [6] he has coauthored. The chapter on $L$-functions [24] is closely related to new developments in PARI/GP.

In [25] the same author explains how to compute Fourier expansions at all cusps of any modular form of integral or half-integral weight thanks to a theorem of Borisov–Gunnells and explicit expansions of Eisenstein series at all cusps. Using this, he gives a number of methods for computing arbitrary Petersson products. Implementations in our PARI/GP software are also described.

A complementary approach using modular symbols is used in [14] by Karim Belabas, Dominique Bernardi and Bernadette Perrin-Riou to compute Manin's constant and the modular degree of elliptic curves defined over $\mathbb{Q}$.