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

TL;DR

This paper presents two explicit methods for computing special $L$-values of modular forms with complex multiplication, inspired by properties of $L$-functions from Calabi-Yau manifolds over $ ext{Q}$.

Contribution

It introduces novel explicit techniques for calculating special $L$-values of CM modular forms, connecting number theory and algebraic geometry.

Findings

Demonstrates two explicit computational methods

Links $L$-values to properties of Calabi-Yau manifolds

Provides illustrative examples of the methods

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 $\mathbb Q$.