Periods of modular forms on $\Gamma_0(N)$ and products of Jacobi theta functions

YoungJu Choie; Yoon Kyung Park; Don B. Zagier

doi:10.4171/jems/864

JournalsjemsVol. 21, No. 5pp. 1379–1410

Periods of modular forms on $\Gamma_0(N)$ and products of Jacobi theta functions

YoungJu Choie
Pohang University of Science and Technology, Pohang City, Republic of Korea
Yoon Kyung Park
Gongju National University of Education, Gongju, Republic of Korea
Don B. Zagier
Max Planck Institute for Mathematics, Bonn, Germany

$Periods of modular forms on $\Gamma_0(N)$ and products of Jacobi theta functions cover$

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Abstract

Generalizing a result of [15] for modular forms of level one, we give a closed formula for the sum of all Hecke eigenforms on $\Gamma_0(N)$ , multiplied by their odd period polynomials in two variables, as a single product of Jacobi theta series for any squarefree level $N$ . We also show that for $N = 2, 3$ and $5$ this formula completely determines the Fourier expansions all Hecke eigenforms of all weights on $\Gamma_0(N)$ .

Cite this article

YoungJu Choie, Yoon Kyung Park, Don B. Zagier, Periods of modular forms on $\Gamma_0(N)$ and products of Jacobi theta functions. J. Eur. Math. Soc. 21 (2019), no. 5, pp. 1379–1410

DOI 10.4171/JEMS/864