Automorphic forms and cubic twists of elliptic curves

TL;DR

This survey explores the relationship between elliptic curves defined by x^3 + y^3 = D and metaplectic forms on the cubic cover of GL(3), revealing how their Fourier coefficients relate to L-series and implications for rational solutions.

Contribution

It connects elliptic curves with metaplectic forms, providing new insights into their behavior and rational solutions without requiring prior advanced background.

Findings

Existence of infinitely many D with no rational solutions in certain congruence classes

Connection between Fourier coefficients of metaplectic forms and L-series of elliptic curves

Survey provides a self-contained overview without proofs

Abstract

This paper surveys the connection between the elliptic curve E_D: x^3 + y^3 = D and a certain metaplectic form on the cubic cover of GL(3) which has the property that its m,n^{th} Whittaker--Fourier coefficient is essentially the L--series of the curve E_{m^2n}. One may obtain information about the collective behavior the curves E_D by exploiting this connection; for example, one can prove: Theorem: Fix any prime p \ne 3, and any congruence class c mod p. Then there are infinitely many D congruent to c mod p such that the curve E_D has no rational solutions. This paper is fairly self-contained; no prior knowledge of algebraic number theory, analytic number theory or metaplectic forms is assumed. On the other hand, this paper is a survey, no proofs are included.