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

We study the elliptic curve E given by y 2 = x(x + 1)(x + t) over the rational function field k(t) and its extensions K d = k(µ d , t 1/d ). When k is finite of characteristic p and d = p f + 1, we write down explicit points on E and show by elementary arguments that they generate a subgroup V d of rank d − 2 and of finite index in E(K d ). Using more sophisticated methods, we then show that the Birch and Swinnerton-Dyer conjecture holds for E over K d , and we relate the index of V d in E(K d ) to the order of the Tate-Shafarevich group X(E/K d ). When k has characteristic 0, we show that E has rank 0 over K d for all d.We note that in [Ca09], Conceição gives an equally simple construction of polynomial points on certain isotrivial elliptic curves over F p (t) and uses them to show that the rank is large.

We study the arithmetic of abelian varieties over K = k(t) where k is an arbitrary field. The main result relates Mordell-Weil groups of certain Jacobians over K to homomorphisms of other Jacobians over k. Our methods also yield completely explicit points on elliptic curves with unbounded rank over Fp(t) and a new construction of elliptic curves with moderately high rank over C(t).

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Background on curves and function fields This "Lecture 0" covers definitions and notations that are probably familiar to many readers and that were reviewed very quickly during the PCMI lectures. Readers are invited to skip it and refer back as necessary.