Modular parametrizations of certain elliptic curves

Abstract

AbstractKaneko and Sakai [11] recently observed that certain elliptic curves whose associatednewforms (by the modularity theorem) are given by the eta-quotients can be character-ized by a particular differential equation involving modular forms and Ramanujan-Serredifferential operator.In this paper, we study certain properties of modular parametrization associated tothe elliptic curves over Q, and as a consequence we generalize and explain some of theirfindings.1. IntroductionBy the modularity theorem [4, 8], an elliptic curve Eover Qadmits a modular parametrizationΦ E : X 0 (N) → Efor some integer N. If Nis the smallest such integer, then it is equal to theconductor of Eand the pullback of the N´eron differential of Eunder Φ E is a rational multipleof 2πif E (τ), where f E (τ) ∈ S 2 (Γ 0 (N)) is a newform with rational Fourier coefficients. The factthat the L-function of f E (τ) coincides with the Hasse-Weil zeta function of E(which followsfrom Eichler-Shimura theory) is central to the proof of Fermat’s last theorem, and is related tothe Birch and Swinnerton-Dyer conjecture. In addition to this, modular parametrization is usedfor constructing rational points on elliptic curves, and appears in the Gross-Zagier formula [9].In this paper, we study some general properties of Φ