Relations de Manin d'ordre 2 v\'erifi\'ees par le polyn\^ome des bi-p\'eriodes d'un couple de formes modulaires

TL;DR

This paper explores relations satisfied by bi-period polynomials of modular forms, introduces a finite set of such relations, constructs an Eisenstein component, and proves irrationality of certain period ratios.

Contribution

It introduces a finite set of relations for bi-period polynomials and constructs an Eisenstein component, advancing understanding of period relations in modular forms.

Findings

Finite set of relations checked by bi-period polynomials.

Construction of an Eisenstein part of depth 2.

Proof of irrationality of certain period ratios.

Abstract

Manin's relations characterize period polynomials of a modular form. In this paper, we propose a finite set of relations checked by bi-period polynomials of a couple of forms. Then we construct an Eisenstein part of depth 2 which essentially completed the set of bi-periods polynomials among polynomials canceled by the relations. By giving a computational description of that part, we prove a result of irrationality of ratio of the periods for some weights.