Selmer group

Given an elliptic curveMathworldPlanetmath E we can define two very interesting and important groups, the Selmer groupMathworldPlanetmath and the Tate-Shafarevich group, which together provide a measure of the failure of the Hasse principleMathworldPlanetmath for elliptic curves, by measuring whether the curve is everywhere locally soluble. Here we present the construction of these groups.

Let E,E be elliptic curves defined over and let ¯ be an algebraic closureMathworldPlanetmath of . Let ϕ:EE be an non-constant isogenyMathworldPlanetmath (for example, we can let E=E and think of ϕ as being the “multiplication by n” map, [n]:EE). The following standard result asserts that ϕ is surjective over ¯:

Theorem 1.

Let C1,C2 be curves defined over an algebraically closed field K and let


be a morphismMathworldPlanetmath (or algebraic map) of curves. Then ψ is either constant or surjective.


See [4], Chapter II.6.8. ∎

Since ϕ:E(¯)E(¯) is non-constant, it must be surjective and we obtain the following exact sequencePlanetmathPlanetmathPlanetmath:

0E(¯)[ϕ]E(¯)E(¯)0  (1)

where E(¯)[ϕ]=Kerϕ. Let G=Gal(¯/), the absolute Galois group of , and consider the ith-cohomology groupPlanetmathPlanetmath Hi(G,E(¯)) (we abbreviate by Hi(G,E)). Using equation (1) we obtain the following long exact sequence (see Proposition 1 in group cohomologyMathworldPlanetmathPlanetmath):

0H0(G,E(¯)[ϕ])H0(G,E)H0(G,E)H1(G,E(¯)[ϕ])H1(G,E)H1(G,E)  (2)

Note that


and similarly


From (2) we can obtain an exact sequence:


We could repeat the same procedure but this time for E,E defined over p,for some prime numberMathworldPlanetmath p, and obtain a similar exact sequence but with coefficients in p which relates to the original in the following commutative diagramMathworldPlanetmath (here Gp=Gal(p¯/p)):

0E()/ϕ(E()) H1(G,E(¯)[ϕ]) H1(G,E)[ϕ]0
0E(p)/ϕ(E(p)) H1(Gp,E(p¯)[ϕ]) H1(Gp,E)[ϕ]0

The goal here is to find a finite group containing E()/ϕ(E()). Unfortunately H1(G,E(¯)[ϕ]) is not necessarily finite. With this purpose in mind, we define the ϕ-Selmer group:


Equivalently, the ϕ-Selmer group is the set of elements γ of H1(G,E(¯)[ϕ]) whose image γp in H1(Gp,E(𝕡¯)[ϕ]) comes from some element in E(p).

Finally, by imitation of the definition of the Selmer group, we define the Tate-Shafarevich group:


The Tate-Shafarevich group is precisely the group that measures the Hasse principle in the elliptic curve E. It is unknown if this group is finite.


Title Selmer group
Canonical name SelmerGroup
Date of creation 2013-03-22 13:50:55
Last modified on 2013-03-22 13:50:55
Owner alozano (2414)
Last modified by alozano (2414)
Numerical id 6
Author alozano (2414)
Entry type Definition
Classification msc 14H52
Related topic GroupCohomology
Related topic RankOfAnEllipticCurve
Related topic ArithmeticOfEllipticCurves
Defines Selmer group
Defines Tate-Shafarevich group