The trace of modular forms and its application to number theory

TL;DR

This paper generalizes a method for calculating the trace of elliptic modular forms and applies it to establish relationships between algebraic number fields associated with modular forms and elliptic curves.

Contribution

It introduces a generalized algebraic linear combination for the trace of modular forms and connects it to algebraic number fields via elliptic curves.

Findings

Derived a new formula linking different algebraic number fields

Extended the algebraic linear combination for modular form traces

Established a connection between modular forms and elliptic curves

Abstract

We provide a generalization of an algebraic linear combination for the trace of certain elliptic modular forms, and through specializing the expression at a suitable pair consisting of an elliptic curve over algebraic number fields and its a certain cyclic subgroup with finite order, show a formula between distinct algebraic number fields, the one related to modular forms and the other related to elliptic curves.