Quadratic families of elliptic curves and unirationality of degree 1 conic bundles

TL;DR

This paper studies elliptic curves with quadratic polynomial coefficients, proving that infinitely many have positive rank, and shows the associated algebraic surface is birational to a conic bundle with unirational properties, even over small finite fields.

Contribution

It establishes the unirationality of certain conic bundles related to quadratic families of elliptic curves, extending results to small finite fields.

Findings

Infinitely many quadratic-parameterized elliptic curves have rank at least 1.

The algebraic surface formed by these curves is birational to a conic bundle with 7 singular fibers.

Unirationality of the conic bundles is proven, including over small finite fields.

Abstract

We consider elliptic curves whose coefficients are degree 2 polynomials in a variable t. We prove that for infinitely many values of t the resulting elliptic curve has rank at least 1. All such curves together form an algebraic surface which is birational to a conic bundle with 7 singular fibers. The main step of the proof is to show that such conic bundles are unirational. V.2: Main theorem is corrected for small finite fields.