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 $\mathbf{Z}_p$-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 $\mathbf{Q}$ with a rational torsion point of order $p$ and without complex multiplication. If $E$ is an elliptic curve over $\mathbf{Q}$ with good ordinary reduction at $p$ such that every prime where both $E$ and $A$ have bad reduction has odd order in $\mathbf{F}_p^\times$ 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 $\mathrm{rk}_p(E/\mathbf{Q}(A[p^n])) \ge B p^{2n}$ for every $n$