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Texas A&M Number Theory Seminar

Department of Mathematics
Milner 317
Wednesdays, 12:30-1:30 PM

Ahmad El-Guindy

Texas A&M University

Wednesday, September 12

Milner 317, 12:30PM

Title:Fourier Expansions with Modular Form Coefficients

Abstract:For non-negative integers $ n$, let $ j_n$ be the unique modular function that is holomorphic on the upper half-plane and with Fourier expansion $ q^{-n}+O(q)$. Asai, Kaneko, and Ninomiya showed that

$\displaystyle \sum_{n=0}^\infty j_n(\tau)q^n=E_{14}(z)\Delta(z)^{-1}/(j(z)-j(\tau)), $

where as usual $ E_k$ is the weight $ k$ Eisenstein series and $ \Delta$ is the unique normalized weight $ 12$ cusp form. In this talk we show how to obtain similar formulas starting from any meromorphic modular form $ F$ of weight $ k$. (In this context the above formula corresponds to $ F=1, k=0$). We also discuss generalizations to forms on groups of higher level for which the underlying modular curve is hyperelliptic.