Modular elliptic directions with complex multiplication (with an application to Gross's elliptic curves)

TL;DR

This paper investigates modular parametrizations of elliptic curves with complex multiplication, focusing on Gross's elliptic curves, and explores their applications within the context of modular forms and abelian varieties.

Contribution

It provides new insights into the modular parametrizations of CM elliptic curves, especially Gross's elliptic curves, linking them to modular forms and abelian varieties.

Findings

Characterization of modular parametrizations for CM elliptic curves

Application of results to Gross's elliptic curves A(p)

Enhanced understanding of the structure of abelian varieties associated with CM forms

Abstract

For every normalized newform f in S_2(Gamma_1(N)) with complex multiplication, we study the modular parametrizations of elliptic curves C from the abelian variety A_f. We apply the results obtained when C is Gross's elliptic curve A(p).