Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 14 (2018), 090, 32 pages      arXiv:1803.06072      https://doi.org/10.3842/SIGMA.2018.090
Contribution to the Special Issue on Modular Forms and String Theory in honor of Noriko Yui

Computing Special $L$-Values of Certain Modular Forms with Complex Multiplication

Wen-Ching Winnie Li a, Ling Long b and Fang-Ting Tu b
a) Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA
b) Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, USA

Received April 03, 2018, in final form August 18, 2018; Published online August 29, 2018

Abstract
In this expository paper, we illustrate two explicit methods which lead to special $L$-values of certain modular forms admitting complex multiplication (CM), motivated in part by properties of $L$-functions obtained from Calabi-Yau manifolds defined over $\mathbb Q$.

Key words: $L$-values; modular forms; complex multiplications; hypergeometric functions; Eisenstein series.

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