Selmer group

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

${\displaystyle \operatorname {Sel} ^{(f)}(A/K)=\bigcap _{v}\ker(H^{1}(G_{K},\ker(f))\rightarrow H^{1}(G_{K_{v}},A_{v}[f])/\operatorname {im} (\kappa _{v}))}$

where A_v[f] denotes the f-torsion of A_v and ${\displaystyle \kappa _{v}}$ is the local Kummer map ${\displaystyle B_{v}(K_{v})/f(A_{v}(K_{v}))\rightarrow H^{1}(G_{K_{v}},A_{v}[f])}$. Note that ${\displaystyle H^{1}(G_{K_{v}},A_{v}[f])/\operatorname {im} (\kappa _{v})}$ is isomorphic to ${\displaystyle H^{1}(G_{K_{v}},A_{v})[f]}$. 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.

In some important cases, A = B = E, an elliptic curve (or A = E and B = E', two elliptic curves), and the map f = [m], is the "multiply by m ∈ ℤ+" map which is an isogeny (the other isogenies are complex multiplication). The image of the map mE covers E but it isn't quite the same thing, so E/mE isn't 0. There is a SES and a corresponding LES via coker and cohomology &c. We can extract an SES from the LES, this SES can be denoted (in general) by 0->U->V->W->0, then U injects into V and U is the ker(V->W), a general fact. This is why the Selmer group is expressed as a kernel. In this case V and W are cohomology groups from the LES. There are other details best explained via Homogeneous Spaces and Cohomology, see Silverman [1]

This matters because the Hasse principle is decisive for Quadratic Forms (2nd order), but the Hasse principle often fails for Elliptic Curves (3rd order). The Selmer group represents the hypothesis of the Hasse principle and the Shaw group Ш represents the failure of the conclusion of the Hasse principle. The Hasse principle is based on the fact that a point in ℚ will, necessarily be a point in ℝ and in ℚ_v. (Here ℚ_v, or ℚ_p, is the p-adic completion of ℚ. The general case is for a valuation v and a number field K.) In the case of quadratic forms the Hasse principle is also sufficient, but (again) the Hasse principle can often fail for Elliptic Curves. The Hasse principle is also known as the Hasse Local/Global principle.

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.

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).

• 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, 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