Selmer stability for elliptic curves in Galois $\ell$-extensions
Siddhi Pathak, Anwesh Ray
TL;DR
This work proves that Selmer stability for an elliptic curve $E/\mathbb{Q}$ persists along many finite Galois extensions with prescribed $\ell$-group G, under the assumptions $Sel_{\ell}(E/\mathbb{Q})=0$ and surjectivity of the mod-$\ell$ Galois representation. The authors combine embedding-problem techniques (via Reichardt–Scholz), Galois cohomology, and Chebotarev density with analytic methods (Wiles’ formula and Delange Tauberian theorems) to construct towers where $Sel_{\ell}(E/\cdot)$ vanishes and to count such fields by discriminant. They obtain a sharp asymptotic lower bound $M(G,E;X) \gg X^{1/(\ell^{n-1}(\ell-1))}(\log X)^{\delta-1}$ with $\delta=(\ell^2-\ell-1)/(\ell^{n-1}(\ell^2-1))$, showing the phenomenon aligns with Malle-type counts up to a logarithmic factor. In the case $\ell=5$, the results are compatible with positive-density families of curves for which $Sel_{\ell}(E/\mathbb{Q})=0$ and surjectivity holds, underscoring the generality and significance of Selmer stability in arithmetic statistics.
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 $\ell \geq 5$, a finite group $G$ with $\#G = \ell^n$, and an elliptic curve $E/\mathbb{Q}$ with $Sel_\ell(E/\mathbb{Q}) = 0$ 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 $Sel_\ell(E/F)$ also vanishes. We obtain an asymptotic lower bound for the number $M(G, E; X)$ of such fields $F$ with absolute discriminant $|Δ_F|\leq X$, proving that there is an explicit constant $δ>0$ such that $M(G, E; X) \gg X^{\frac{1}{\ell^{n-1}(\ell - 1)}} (\log X)^{δ- 1}$. The asymptotic for $M(G, E; X)$ matches the conjectural count for all $G$-extensions $F/\mathbb{Q}$ for which $|Δ_F|\leq X$, up to a power of $\log X$. This demonstrates that Selmer stability is not a rare phenomenon.