arithmetic geometry In mathematics, arithmetic geometry is roughly the application of techniques from algebraic geometry to problems in number theory. Arithmetic geometry is centered around Diophantine geometry, the study of rational points of algebraic variety, alg ...

, 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 underlyi ...

abelian varieties In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a Algebraic variety#Projective variety, projective algebraic variety that is also an algebraic group, i.e., has a group law th ...

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 nat ...

as :

/\operatorname(\kappa_v))

where ''A''_v 'f''denotes the ''f''-

torsion Torsion may refer to: Science * Torsion (mechanics), the twisting of an object due to an applied torque * Torsion of spacetime, the field used in Einstein–Cartan theory and ** Alternatives to general relativity * Torsion angle, in chemistry Bi ...

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 An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next. Definition In the context o ...

: 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

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 representation In mathematics, a Galois module is a ''G''-module, with ''G'' being the Galois group of some extension of fields. The term Galois representation is frequently used when the ''G''-module is a vector space over a field or a free module over a ring ...

s and to ''p''-adic variations of

motive Motive(s) or The Motive(s) may refer to: * Motive (law) Film and television * ''Motives'' (film), a 2004 thriller * ''The Motive'' (film), 2017 * ''Motive'' (TV series), a 2013 Canadian TV series * ''The Motive'' (TV series), a 2020 Israeli T ...

s 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