(Quasi-)magnetic modular forms

Abstract: Given a positive even integer $k$, let $E_k(\tau)$ stands for the normalised Eisenstein series of weight $k$; denote $$\Delta(\tau)=q\prod_{m=1}^\infty(1-q^m)^{24}=(E_4^3-E_6^2)/1728$$ with $q=e^{2\pi i\tau}$, and $\delta=\frac{1}{2\pi i}\frac{d}{\d\tau}=q\frac{d}{dq}$. About ten years ago Honda and Kaneko observed surprising arithmetic properties of the meromorphic modular function $\Delta^{5/6}/E_4^2$ of weight 2, while the recent work of Li and Neururer (inspired by an observation of Broadhurst and this speaker) brought to life an even stronger arithmetic for the modular function $\Delta/E_4^2$ of weight 4. To convince the attendee about it, you are invited to verify that the anti-derivatives $\delta^{-1}(\Delta/E_4^2)$ and $\delta^{-1}(E_4\Delta/E_6^2)$ have integer coefficients in their $q$-expansions. At the same time, these series are transcendental over the field of quasi-modular functions. I will discuss this phenomenon (in a greater generality!) and ideas behind its proof in my talk. The talk is based on my joint work with Vicenţiu Paşol.

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