Abstract
We show how to obtain the difference function of the Weierstrass zeta function very directly, by choosing an appropriate order of summation in the series defining this function. As a byproduct, we show how to obtain the quasi-modularity of the weight 2 Eisenstein series immediately from the fact that it appears in this difference function and the homogeneity properties of the latter.
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References
Diamond, F., Shurman, J.: A First Course in Modular Forms, Graduate Texts in Mathematics, vol. 228. Springer, New York (2005)
Lang, S.: Elliptic Functions, 2nd edn. Graduate Texts in Mathematics, vol. 112. Springer, New York (1987)
Acknowledgments
I wish to thank J. Shurman, with whom I had a long and enlightening correspondence while I was studying modular forms from [1]. Thanks are also due to J. Bruinier and E. Freitag, who read this paper and offered useful advice and clarifications.
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The initial stage of this research has been carried out as part of S. Zemel’s Ph.D. thesis work at the Hebrew University of Jerusalem, Israel. The final stage of this work was supported by the Minerva Fellowship (Max-Planck-Gesellschaft).
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Zemel, S. A direct evaluation of the periods of the Weierstrass zeta function. Ann Univ Ferrara 60, 495–505 (2014). https://doi.org/10.1007/s11565-013-0186-8
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DOI: https://doi.org/10.1007/s11565-013-0186-8