Selmer stability for elliptic curves in Galois ℓ-extensions

2025/04/22 by Pathak, Siddhi, Ray, Anwesh
#11G05 #11R45 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2504.15945

Abstract

We study the behavior of Selmer groups of an elliptic curve E/ℚ in finite Galois extensions with prescribed Galois group. Fix a prime ℓ ≥ 5, a finite group G with #G = ℓⁿ, and an elliptic curve E/ℚ with Sel_ℓ(E/ℚ) = 0 and surjective mod-ℓ Galois representation. We show that there exist infinitely many Galois extensions F/ℚ with Galois group Gal(F/ℚ) ≃ G for which the ℓ-Selmer group Sel_ℓ(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|≤ X, proving that there is an explicit constant δ>0 such that M(G, E; X) ≫ X^\frac1ℓ^n-1(ℓ - 1) (log X)^{δ- 1}. The asymptotic for M(G, E; X) matches the conjectural count for all G-extensions F/ℚ for which |Δ_F|≤ X, up to a power of log X. This demonstrates that Selmer stability is not a rare phenomenon.

Citations

• Rank growth of elliptic curves over S3 extensions with fixed quadratic resolvents
• Vanishing of Quartic and Sextic Twists of L-functions
• Rank growth of elliptic curves in nonabelian extensions
• Diophantine stability
• The average size of the 5-Selmer group of elliptic curves is 6, and the average rank is less than 1
• A Markov model for Selmer ranks in families of twists
• Ranks of twists of elliptic curves and Hilbert’s tenth problem
• Counting nilpotent Galois extensions
• Counting nilpotent Galois extensions
• Cohomology of Number Fields

Related

• The distribution of ℓ^∞-Selmer groups in degree ℓ twist families I
• Selmer groups of elliptic curves in degree p extensions
• Selmer groups for elliptic curves in Z_l^d-extensions of function fields of characteristic p
• Remarks on Greenberg's conjecture for Galois representations associated to elliptic curves
• Growth of Selmer rank in nonabelian extensions of number fields