Selmer Groups as Flat Cohomology Groups

Given a prime number p, Bloch and Kato showed how the p8-Selmer group of an abelian variety A over a number field K is determined by the p-adic Tate module. In general, the p-Selmer group Selpm A need not be determined by the mod p Galois representation Arps; 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 Arps for all p R Σ. In the course of the argument we describe the flat cohomology group H fppfpOK ,Arpsq of the ring of integers of K with coefficients in the p-torsion Arps of the Néron model of A by local conditions for p R Σ, compare them with the local conditions defining Selpm A, and prove that Arps itself is determined by Arps for such p. Our method sharpens the relationship between Selpm A and H fppfpOK ,Arpsq which was observed by Mazur 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. Standard glueing techniques developed in the course of the proofs have applications to finite flat group schemes over global bases, permitting us to transfer many of the known local results to the global setting.