2013/01/21 by Kęstutis Česnavičius, Cesnavicius, Kestutis · 3 citations
Mathematics · #11G10 (Primary) #14F20 #14K02 #14L15 (Secondary) #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric and Algebraic Topology #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.1301.4724
openalex publication_date 2013/01/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Given a prime number p, Bloch and Kato showed how the p^∞-Selmer group of an abelian variety A over a number field K is determined by the p-adic Tate module. In general, the pm-Selmer group Selpm A need not be determined by the mod pm Galois representation A[pm]; we show, however, that this is the case if p is large enough. More precisely, we exhibit a finite explicit set of rational primes Σ depending on K and A, such that Selpm A is determined by A[pm] for all p \not ∈ Σ. In the course of the argument we describe the flat cohomology group H1fppf(OK, A[pm]) of the ring of integers of K with coefficients in the pm-torsion A[pm] of the Néron model of A by local conditions for p\not∈ Σ, compare them with the local conditions defining Selpm A, and prove that A[pm] itself is determined by A[pm] for such p. Our method sharpens the known relationship between Selpm A and H1fppf(OK, A[pm]) and continues to work for other isogenies ϕ between abelian varieties over global fields provided that deg ϕ is constrained appropriately. To illustrate it, we exhibit resulting explicit rank predictions for the elliptic curve 11A1 over certain families of number fields.