Elliptic curves with large rank over function fields

By Douglas Ulmer. 2001

127 citations recorded by the source corpus.

We produce explicit elliptic curves over Fp(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

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/03d91197f0bd4f543c6935e38bacfbbb99682abd

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In Algebra and Number Theory

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Three lectures on elliptic surfaces and curves of high rank (2007, background): “Indeed, when F has characteristic p > 0 the rank is unbounded for “isotrivial” curves with constant and supersingular j-invariant [Šafarevič–Tate 1967], and also for non-isotrivial ones [Ulmer 2002].”
Some structure theorems for algebraic groups (2015, background): “(Such a pair (E,L) exists unless k is algebraic over a finite field, as follows from [55]; see also [59]).”