Selmer group

In arithmetic geometry, the Selmer group, named in honor of the work of Selmer (1951) by Cassels (1962), is a group constructed from an isogeny of abelian varieties.

The Selmer group of an isogeny

The Selmer group of an abelian variety A with respect to an isogeny f : A → B of abelian varieties can be defined in terms of Galois cohomology as

$\mathrm{Sel}^{(f)}(A/K)=\bigcap_v\mathrm{ker}(H^1(G_K,\mathrm{ker}(f)))\rightarrow H^1(G_{K_v},A_v[f])/\mathrm{im}(\kappa_v))$

where A_v[f] denotes the f-torsion of A_v and $\kappa_v$ is the local Kummer map $B_v(K_v)/f(A_v(K_v))\rightarrow H^1(G_{K_v},A_v[f])$. Note that $H^1(G_{K_v},A_v[f])/\mathrm{im}(\kappa_v)$ is isomorphic to $H^1(G_{K_v},A_v)[f]$. Geometrically, the principal homogeneous spaces coming from elements of the Selmer group have K_v-rational points for all places v of K. The Selmer group is finite. This implies that the part of the Tate–Shafarevich group killed by f is finite due to the following exact sequence

0 → B(K)/f(A(K)) → Sel^(f)(A/K) → Ш(A/K)[f] → 0.

The Selmer group in the middle of this exact sequence is finite and effectively computable. This implies the weak Mordell–Weil theorem that its subgroup B(K)/f(A(K)) is finite. There is a notorious problem about whether this subgroup can be effectively computed: there is a procedure for computing it that will terminate with the correct answer if there is some prime p such that the p-component of the Tate–Shafarevich group is finite. It is conjectured that the Tate–Shafarevich group is in fact finite, in which case any prime p would work. However, if (as seems unlikely) the Tate–Shafarevich group has an infinite p-component for every prime p, then the procedure may never terminate.

Greenberg (1994) has generalized the notion of Selmer group to more general p-adic Galois representations and to p-adic variations of motives in the context of Iwasawa theory.

The Selmer group of a finite Galois module

More generally one can define the Selmer group of a finite Galois module M (such as the kernel of an isogeny) as the elements of H¹(G_K,M) that have images inside certain given subgroups of H¹(G_Kv,M).

References

• Cassels, John William Scott (1962), "Arithmetic on curves of genus 1. III. The Tate–Šafarevič and Selmer groups", Proceedings of the London Mathematical Society. Third Series 12: 259–296, doi:10.1112/plms/s3-12.1.259, ISSN 0024-6115, MR 0163913
• Cassels, John William Scott (1991), Lectures on elliptic curves, London Mathematical Society Student Texts 24, Cambridge University Press, ISBN 978-0-521-41517-0, MR 1144763
• Greenberg, Ralph (1994), "Iwasawa Theory and p-adic Deformation of Motives", in Serre, Jean-Pierre; Jannsen, Uwe; Kleiman, Steven L., Motives, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-1637-0
• Selmer, Ernst S. (1951), "The Diophantine equation ax³ + by³ + cz³  = 0", Acta Mathematica 85: 203–362, doi:10.1007/BF02395746, ISSN 0001-5962, MR 0041871