L-functions of twisted Legendre curves

Let K be a global field of char p and let F-q be the algebraic closure of F-P in K. For an elliptic curve E/K with nonconstant j-invariant, the L-function L(T, E/K) is a polynomial in 1 + T . Z[T]. For any N > 1 invertible in K and finite subgroup J subset of E(K) of order N, we compute the mod N reduction of L(T, E/K) and determine an upper-bound for the order of vanishing at 1/q, the so-called analytic rank of E/K. We construct infinite families of curves of rank zero when q is an odd prime power such that q equivalent to 1 mod l for some odd prime l. Our construction depends upon a construction of infinitely many twin-prime pairs (Lambda, Lambda - 1) in Fq [Lambda] x F-q [Lambda]. We also construct infinitely many quadratic twists with minimal analytic rank, half of which have rank zero and half have (analytic) rank one. In both cases we bound the analytic rank by letting J congruent to Z/2 circle plus Z/2 and studying the mod-4 reduction of L (T, El K). (c) 2005 Elsevier Inc. All rights reserved.