Realizations of the formal double Eisenstein space

Abstract: In this talk, we introduce the formal double Eisenstein space $\mathcal{E}_k$, which is a generalization of the formal double zeta space $\mathcal{D}_k$ of Gangl-Kaneko-Zagier. We show that $\mathbb{Q}$-linear from $\mathcal{E}_k$ to $A$, for some $\mathbb{Q}$-algebra $A$, can be constructed from formal Laurent series that satisfy the Fay identity. As the prototypical example, we define the Kronecker realization, which lifts Gangl-Kaneko-Zagier's Bernoulli realization, and whose image consists of quasimodular forms for the full modular group. As an application to the theory of modular forms, we obtain a purely combinatorial proof of Ramanujan's differential equations for classical Eisenstein series. This talk is based on a joint work with H. Bachmann and N. Matthes.

number theory

Audience: researchers in the topic

Japan Europe Number Theory Exchange Seminar

Series comments: The purpose of the Japan Europe Number Theory Exchange Seminar is to give a opportunity for researchers in Japan and Europe to exchange their research projects by giving short talks (30 min). The target audience are researchers of any level in the area of number theory.

Start for Fall 2021: 26th October