Journal of the Ramanujan Mathematical Society

Volume 29, Issue 1, March 2014 pp. 75–92.

Partitions of graphs and Selmer groups of elliptic curves of Neumann-Setzer type

Authors: Tomasz Jedrzejak and Malgorzata Wieczorek
Author institution: University of Szczecin, Institute of Mathematics, Wielkopolska 15, 70-451 Szczecin, Poland

Summary: We consider the elliptic curves Eu:y2=x3 + ux2-16x and their quadratic twists Enu by a squarefree integer n, where u2 + 64 = p1 … pl, (pi are primes). When l ≤ 2, n ≡ 1 (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 Enu 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 Enu has rank zero for a positive proportion of squarefree integers n with a fixed number of prime divisors.

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