Selmer group
In arithmetic geometry, the Selmer group, named in honor of the work of Ernst Sejersted Selmer (1951) by John William Scott 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
- [math]\displaystyle{ \operatorname{Sel}^{(f)}(A/K)=\bigcap_v\ker(H^1(G_K,\ker(f))\rightarrow H^1(G_{K_v},A_v[f])/\operatorname{im}(\kappa_v)) }[/math]
where Av[f] denotes the f-torsion of Av and [math]\displaystyle{ \kappa_v }[/math] is the local Kummer map [math]\displaystyle{ B_v(K_v)/f(A_v(K_v))\rightarrow H^1(G_{K_v},A_v[f]) }[/math]. Note that [math]\displaystyle{ H^1(G_{K_v},A_v[f])/\operatorname{im}(\kappa_v) }[/math] is isomorphic to [math]\displaystyle{ H^1(G_{K_v},A_v)[f] }[/math]. Geometrically, the principal homogeneous spaces coming from elements of the Selmer group have Kv-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.
Ralph 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(GK,M) that have images inside certain given subgroups of H1(GKv,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
- Cassels, John William Scott (1991), Lectures on elliptic curves, London Mathematical Society Student Texts, 24, Cambridge University Press, doi:10.1017/CBO9781139172530, ISBN 978-0-521-41517-0, https://books.google.com/books?id=zgqUAuEJNJ4C
- 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
See also
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