Modular Parametrization as Polyakov Path Integral: Cases with CM Elliptic Curves as Target Spaces

TL;DR

This paper explores the connection between CM elliptic curves, modular parametrizations, and string theory, showing that certain string theory correlation functions produce Hecke theta functions related to these curves.

Contribution

It demonstrates that chiral correlation functions in Type II string theory with CM elliptic curves as target spaces reproduce Hecke theta functions, linking string theory and modular forms.

Findings

String theory correlation functions yield Hecke theta functions.

Modular parametrization relates elliptic curves to modular curves.

Kähler parameters correspond to indices in modular curve limits.

Abstract

For an elliptic curve E over an abelian extension k/K with CM by K of Shimura type, the L-functions of its [k:K] Galois representations are Mellin transforms of Hecke theta functions; a modular parametrization (surjective map) from a modular curve to E pulls back the 1-forms on E to give the Hecke theta functions. This article refines the study of our earlier work and shows that certain class of chiral correlation functions in Type II string theory with [E]_C (E as real analytic manifold) as a target space yield the same Hecke theta functions as objects on the modular curve. The Kahler parameter of the target space [E]_C in string theory plays the role of the index (partially ordered) set in defining the projective/direct limit of modular curves.