Selmer stability for elliptic curves in Galois $\ell$-extensions

Abstract

We study the behavior of Selmer groups of an elliptic curve $E/\mathbb{Q}$ in finite Galois extensions with prescribed Galois group. Fix a prime $\ell \ge 5$, a finite group $G$ with $\#G={\ell }^n$, and an elliptic curve $E/\mathbb{Q}$ with $Se{l}_{\ell }(E/\mathbb{Q})=0$ and surjective mod-$\ell$ Galois representation. We show that there exist infinitely many Galois extensions $F/\mathbb{Q}$ with Galois group $Gal(F/\mathbb{Q})\simeq G$ for which the $\ell$-Selmer group $Se{l}_{\ell }(E/F)$ also vanishes. We obtain an asymptotic lower bound for the number $M(G,E;X)$ of such fields $F$ with absolute discriminant $|{\Delta }_F|\le X$, proving that there is an explicit constant $\delta >0$ such that $M(G,E;X)\gg X^{\frac{1}{{\ell }^{n-1}(\ell -1)}}(\log X{)}^{\delta -1}$. The asymptotic for $M(G,E;X)$ matches the conjectural count for all $G$-extensions $F/\mathbb{Q}$ for which $|{\Delta }_F|\le X$, up to a power of $\log X$. This demonstrates that Selmer stability is not a rare phenomenon.

Version 2: 24 pages, accepted for publication in Mathematische Nachrichten

References in corpus (3)

• Rank growth of elliptic curves over $N$-th root extensions
• Rank stability of elliptic curves in certain non-abelian extensions
• Rank growth of elliptic curves in $S_4$ and $A_4$ quartic extensions of the rationals