Elliptic curves with large rank over function fields

2001/09/30 by Douglas Ulmer
Mathematics · #math.NT #math.AG #msc:11G05 #msc:11G40 #msc:14G10

paper · pdf

published as Ann. of Math. (2), Vol. 155 (2002), no. 1, 295--315 · 21 pages, published version

arxiv created 2004/05/20 · arxiv updated 2009/11/30

Abstract

We produce explicit elliptic curves over \Bbb F_p(t) whose Mordell-Weil groups have arbitrarily large rank. Our method is to prove the conjecture of Birch and Swinnerton-Dyer for these curves (or rather the Tate conjecture for related elliptic surfaces) and then use zeta functions to determine the rank. In contrast to earlier examples of Shafarevitch and Tate, our curves are not isotrivial. Asymptotically these curves have maximal rank for their conductor. Motivated by this fact, we make a conjecture about the growth of ranks of elliptic curves over number fields.

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• Mordell-Weil groups and the rank of elliptic curves over large fields
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