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
editThe 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
![{\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}))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3e4329c680d5e243b954d7e847077c28ce030c80)
where Av[f] denotes the f-torsion of Av and is the local Kummer map . Note that is isomorphic to . 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
![{\displaystyle B_{v}(K_{v})/f(A_{v}(K_{v}))\rightarrow H^{1}(G_{K_{v}},A_{v}[f])}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3b583e5fef6574b1c261377964668b66489ccc45)
![{\displaystyle H^{1}(G_{K_{v}},A_{v}[f])/\operatorname {im} (\kappa _{v})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/85a2c184f892c5f192c819201f88225896d0ee0c)
![{\displaystyle H^{1}(G_{K_{v}},A_{v})[f]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/11ccb9654ebd3a9934eeda733e6bf7cb23b44171)
- 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
editMore 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
edit- 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, vol. 24, Cambridge University Press, doi:10.1017/CBO9781139172530, 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. (eds.), Motives, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-1637-0, MR 1265554
- 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