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 ${\mathrm{rk}}_p(E/F)$ of the $p$-power Selmer group of $E$ over $F$. We obtain lower bounds for ${\mathrm{rk}}_p(E/F)$, generalizing the results in [MR], which applied to dihedral extensions.

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

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 $F_p^{\times }$ 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 ${\mathrm{rk}}_p(E/Q(A[p^n]))\ge B{p}^{2n}$ for every $n$