In arithmetic geometry, the Selmer group, named in honor of the work of by , is a group constructed from an

isogeny In mathematics, in particular, in algebraic geometry, an isogeny is a morphism of algebraic groups (also known as group varieties) that is surjective and has a finite kernel. If the groups are abelian varieties, then any morphism of the underlyin ...

of abelian varieties.

The Selmer group of an isogeny

The Selmer group of an abelian variety ''A'' with respect to an

''f'' : ''A'' → ''B'' of abelian varieties can be defined in terms of

Galois cohomology In mathematics, Galois cohomology is the study of the group cohomology of Galois modules, that is, the application of homological algebra to modules for Galois groups. A Galois group ''G'' associated to a field extension ''L''/''K'' acts in a natur ...

as :

/\operatorname(\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_,A_v

. Note that

/\operatorname(\kappa_v)

is isomorphic to

H^1(G_,A_v) /math>. 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 In arithmetic geometry, the Tate–Shafarevich group of an abelian variety (or more generally a group scheme) defined over a number field consists of the elements of the Weil–Châtelet group that become trivial in all of the completions of ...

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 In mathematics, the Mordell–Weil theorem states that for an abelian variety A over a number field K, the group A(K) of k-rational point, ''K''-rational points of A is a finitely-generated abelian group, called the Mordell–Weil group. The case ...

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

is in fact finite, in which case any prime ''p'' would work. However, if (as seems unlikely) the

has an infinite ''p''-component for every prime ''p'', then the procedure may never terminate. 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 In number theory, Iwasawa theory is the study of objects of arithmetic interest over infinite towers of number fields. It began as a Galois module theory of ideal class groups, initiated by (), as part of the theory of cyclotomic fields. In th ...

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 ''H''¹(''G''_''K'',''M'') that have images inside certain given subgroups of ''H''¹(''G''_{''K''_''v''},''M'').

References

* * * *

The Selmer group of an isogeny

The Selmer group of a finite Galois module

References

See also