Growth of Selmer Rank in Nonabelian Extensions of Number Fields

Abstract

Let \(p\) be an odd prime number, let E be an elliptic curve over a number field \(k\), and let \(F/k\) be a Galois extension of degree twice a power of p. We study the \(Z_p\)-corank \(rk_p(E/F)\) of the \(p\)-power Selmer group of \(E\) over \(F\). We obtain lower bounds for \(rk_p(E/F)\), generalizing the results in [MR], which applied to dihedral extensions.

If \(K\) is the (unique) quadratic extension of \(k\) in \(F\), if \(G = Gal(F/K)\), if \(G+\) is the subgroup of elements of \(G\) commuting with a choice of involution of \(F\) over \(k\), and if \(rk_p(E/K)\) is odd, then we show that (under mild hypotheses) \(rkp(E/F)\ge[G:G+]\).

As a very specific example of this, suppose that \(A\) is an elliptic curve over \(Q\) with a rational torsion point of order \(p\) and without complex multiplication. If \(E\) is an elliptic curve over \(Q\) with good ordinary reduction at \(p\) such that every prime where both \(E\) and \(A\) have bad reduction has odd order in \(F\frac{x}{p}\) and such that the negative of the conductor of \(E\) is not a square modulo \(p\), then there is a positive constant \(B\) depending on \(A\) but not on \(E\) or \(n\) such that \(rk_p(E/Q(A[p^n]))/geBp^{2n}\) for every \(n\).