Elliptic curves with large rank over function fields

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51 Scopus citations

Abstract

We produce explicit elliptic curves over (double-struck 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 Tare 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.

Original languageEnglish (US)
Pages (from-to)295-315
Number of pages21
JournalAnnals of Mathematics
Volume155
Issue number1
DOIs
StatePublished - 2002

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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