Selmer groups of elliptic curves that can be arbitrarily large

Abstract

In this article, it is shown that certain kinds of Selmer groups of elliptic curves can be arbitrarily large. The main result is that if p is a prime at least 5, then p-Selmer groups of elliptic curves can be arbitrarily large if one ranges over number fields of degree at most g+1 over the rationals, where g is the genus of X 0( p). As a corollary, one sees that p-Selmer groups of elliptic curves over the rationals can be arbitrarily large for p=5,7 and 13 (the cases p⩽7 were already known). It is also shown that the number of elements of order N in the N-Selmer group of an elliptic curve over the rationals can be arbitrarily large for N=9,10,12,16 and 25.