Constructing elliptic curves from Galois representations

TL;DR

This paper proves that any rank-two lisse sheaf with specific properties over an arithmetic surface originates from an elliptic curve, establishing a correspondence between sheaves and elliptic curves.

Contribution

It demonstrates that certain rank-two lisse sheaves with particular properties are precisely those arising from elliptic curves, providing a new characterization.

Findings

Any such lisse sheaf comes from an elliptic curve.

Properties like cyclotomic determinant and finite ramification characterize these sheaves.

The result bridges sheaf theory and elliptic curve construction.

Abstract

Given a non-isotrivial elliptic curve over an arithmetic surface, one obtains a lisse $\ell$-adic sheaf of rank two over the surface. This lisse sheaf has a number of straightforward properties: cyclotomic determinant, finite ramification, rational traces of Frobenius, and somewhere not potentially good reduction. We prove that any lisse sheaf of rank two possessing these properties comes from an elliptic curve.