Partitions of Graphs and Selmer Groups of Elliptic Curves of Neumann-Setzer Type

Abstract

We consider the elliptic curves Eu : y 2 = x 3 + ux2 - 16x and their quadratic twists E~ by a square free integer n, where u 2 + 64 = Pl ... pz, (p; are primes). When l :::, 2, n = l(mod 4) and all prime divisors of n are congruent to 3 modulo 4 we give a complete description of sizes of Selmer groups of E~ in terms of number of even partitions of some graphs. If n is even or l > 2, we give some conditions for twists of rank zero. We deduce also that E~ has rank zero for a positive proportion of squarefreed integers n with a fixed number of prime divisors.