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arXiv · 1412.3673

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

János Kollár·Massimiliano Mella
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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.

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János Kollár, Massimiliano Mella. 2014-12-11. Quadratic families of elliptic curves and unirationality of degree 1 conic bundles. https://arxiv.org/abs/1412.3673

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