Elliptic Curves with Large Rank over Function Fields

Abstract

We produce explicit elliptic curves over 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. * This paper is based upon work supported by the National Science Foundation under Grant No. DMS-0070839.2. In fact we prove the conjecture of Birch and Swinnerton-Dyer for E over F q (t) for q any power of p, and we show that Rank E(F p 2n (t)) = Rank E(F p (t)) = p n if 6 | d and p n − 2 if 6|d.3. In Section 10 we explain that the curves in Theorem 1.5 asymptotically have maximal ranks for their conductor and we make a conjecture about ranks of elliptic curves over number fields.4. The displayed Weierstrass equation also defines an elliptic curve over Q(t). It turns out that this curve has rank which is bounded independently of d, even over Q(t).1.7. The proof of the theorem involves an appealing mix of geometry and arithmetic. We begin with the geometry: First, we construct an elliptic surface E → P 1 over F p whose generic fiber is E/K. The rank of the Mordell-Weil

Ranks of quadratic twists of elliptic curves

“…The principal rationale here is that (perhaps from L-function considerations) the rank might be roughly as large as √ log d, with the speculation then being phrased in a simple form (though see Footnote 32, and compare [20]). However, alternative schools of thought (e.g., descent bounds or function field analogues [56], or rank bounds under a Riemann hypothesis [7,36]) suggest the rank might be as large as log d log log d . The numerics are conflated by the fact that the smallest d of rank 7 is rather small (perhaps abnormally so), for the second smallest d is more than 2500 times as large.…”

“…Honda [26, §4, p. 98] conjectures the rank is bounded 1 in any such family, basing this on an analogy [26, between Mordell-Weil groups of abelian varieties and Dirichlet's unit theorem. Schneiders and Zimmer [53] presented some preliminary evidence for Honda's conjecture back in 1989, though it seems that modern thought has preferred to intuit that ranks are unbounded in quadratic twist families, partially relying on function field analogues such as in [55] and [56]. 2 In this paper, we give some experimental data regarding quadratic twists of rank 6-8 for the congruent number curve.…”

L-functions with large analytic rank and abelian varieties with large algebraic rank over function fields

“…In [Ulm02] we showed that the main term of this arithmetic bound, as well as the geometric bound, are sharp for L-functions of elliptic curves. The towers Theorem 4.7 gives a large supply of other examples related to this question.…”

“…The main tools are a beautiful observation of Shioda [Shi86], already exploited in [Ulm02], that surfaces defined by four monomials are often dominated by Fermat surfaces and so satisfy the Tate conjecture, and well-known connections between the conjectures of Tate and of Birch and Swinnerton-Dyer.…”

Torsion des modules de Drinfeld de rang 2 et formes modulaires de Drinfeld

“…Cette condition Ꮿ n'est certainement pas vérifiée par tout idéal premier p. En effet, d'après Ulmer [2002] il existe des courbes elliptiques non-isotriviales sur ‫ކ‬ q (T ), pour q premier, de rang arithmétique arbitrairement grand. Par le théo-rème de modularité pour les courbes elliptiques sur K (corollaire des travaux de Grothendieck, Deligne, Jacquet-Langlands et Drinfeld, voir [Gekeler et Reversat 1996]) et en supposant la conjecture de Birch et Swinnerton-Dyer pour ces courbes, on voit que la condition Ꮿ n'est pas satisfaite pour tout p. Cependant, on s'attend à ce qu'elle le soit assez fréquemment car une philosophie courante prédit que les courbes elliptiques de rangs élevés (≥ 2) sont rares.…”