Selmer group

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

Let $E,E^{\prime}$ be elliptic curves defined over $\mathbb{Q}$ and let $\bar{\mathbb{Q}}$ be an algebraic closure of $\mathbb{Q}$. Let $\phi\colon E\to E^{\prime}$ be an non-constant isogeny (for example, we can let $E=E^{\prime}$ and think of $\phi$ as being the “multiplication by $n$” map, $[n]\colon E\to E$). The following standard result asserts that $\phi$ is surjective over $\bar{\mathbb{Q}}$:

Theorem 1.

Let $C_{1},C_{2}$ be curves defined over an algebraically closed field $K$ and let

 $\psi\colon C_{1}\to C_{2}$

be a morphism (or algebraic map) of curves. Then $\psi$ is either constant or surjective.

Proof.

See [4], Chapter II.6.8. ∎

Since $\phi\colon E(\bar{\mathbb{Q}})\to E^{\prime}(\bar{\mathbb{Q}})$ is non-constant, it must be surjective and we obtain the following exact sequence:

 $0\to E(\bar{\mathbb{Q}})[\phi]\to E(\bar{\mathbb{Q}})\to E^{\prime}(\bar{% \mathbb{Q}})\to 0\quad\quad(1)$

where $E(\bar{\mathbb{Q}})[\phi]=\operatorname{Ker}\phi$. Let $G=\operatorname{Gal}({\bar{\mathbb{Q}}/\mathbb{Q}})$, the absolute Galois group of $\mathbb{Q}$, and consider the $i^{th}$-cohomology group $H^{i}(G,E(\bar{\mathbb{Q}}))$ (we abbreviate by $H^{i}(G,E)$). Using equation $(1)$ we obtain the following long exact sequence (see Proposition 1 in group cohomology):

 $0\to H^{0}(G,E(\bar{\mathbb{Q}})[\phi])\to H^{0}(G,E)\to H^{0}(G,E^{\prime})% \to H^{1}(G,E(\bar{\mathbb{Q}})[\phi])\to H^{1}(G,E)\to H^{1}(G,E^{\prime})% \quad\quad(2)$

Note that

 $H^{0}(G,E(\bar{\mathbb{Q}})[\phi])={(E(\bar{\mathbb{Q}})[\phi])}^{G}=E(\mathbb% {Q})[\phi]$

and similarly

 $H^{0}(G,E)=E(\mathbb{Q}),\quad H^{0}(G,E^{\prime})=E^{\prime}(\mathbb{Q})$

From $(2)$ we can obtain an exact sequence:

 $0\to E^{\prime}(\mathbb{Q})/\phi(E(\mathbb{Q}))\to H^{1}(G,E(\bar{\mathbb{Q}})% [\phi])\to H^{1}(G,E)[\phi]\to 0$

We could repeat the same procedure but this time for $E,E^{\prime}$ defined over $\mathbb{Q}_{p}$,for some prime number $p$, and obtain a similar exact sequence but with coefficients in $\mathbb{Q}_{p}$ which relates to the original in the following commutative diagram (here $G_{p}=\operatorname{Gal}({\bar{\mathbb{Q}_{p}}/\mathbb{Q}_{p}})$):

 $\displaystyle 0\to E^{\prime}(\mathbb{Q})/\phi(E(\mathbb{Q}))\to$ $\displaystyle H^{1}(G,E(\bar{\mathbb{Q}})[\phi])$ $\displaystyle\to H^{1}(G,E)[\phi]\to 0$ $\displaystyle\downarrow$ $\displaystyle\downarrow$ $\displaystyle\quad\quad\quad\downarrow$ $\displaystyle 0\to E^{\prime}(\mathbb{Q}_{p})/\phi(E(\mathbb{Q}_{p}))\to$ $\displaystyle H^{1}(G_{p},E(\bar{\mathbb{Q}_{p}})[\phi])$ $\displaystyle\to H^{1}(G_{p},E)[\phi]\to 0$

The goal here is to find a finite group containing $E^{\prime}(\mathbb{Q})/\phi(E(\mathbb{Q}))$. Unfortunately $H^{1}(G,E(\bar{\mathbb{Q}})[\phi])$ is not necessarily finite. With this purpose in mind, we define the $\phi$-Selmer group:

 $S^{\phi}(E/\mathbb{Q})=\operatorname{Ker}\left(H^{1}(G,E(\bar{\mathbb{Q}})[% \phi])\to\prod_{p}H^{1}(G_{p},E)\right)$

Equivalently, the $\phi$-Selmer group is the set of elements $\gamma$ of $H^{1}(G,E(\bar{\mathbb{Q}})[\phi])$ whose image $\gamma_{p}$ in $H^{1}(G_{p},E(\bar{\mathbb{Q_{p}}})[\phi])$ comes from some element in $E(\mathbb{Q}_{p})$.

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

 $TS(E/\mathbb{Q})=\operatorname{Ker}\left(H^{1}(G,E)\to\prod_{p}H^{1}(G_{p},E)\right)$

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.

References

• 1 J.P. Serre, Galois Cohomology, Springer-Verlag, New York.
• 2 James Milne, Elliptic Curves, http://www.jmilne.org/math/CourseNotes/math679.htmlonline course notes.
• 3 Joseph H. Silverman, . Springer-Verlag, New York, 1986.
• 4 R. Hartshorne, , Springer-Verlag, 1977.
Title Selmer group SelmerGroup 2013-03-22 13:50:55 2013-03-22 13:50:55 alozano (2414) alozano (2414) 6 alozano (2414) Definition msc 14H52 GroupCohomology RankOfAnEllipticCurve ArithmeticOfEllipticCurves Selmer group Tate-Shafarevich group