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 $\overline{\mathbb{Q}}$ be an algebraic closure^{} of $\mathbb{Q}$. Let $\varphi :E\to {E}^{\prime}$ be an non-constant isogeny^{} (for example, we can let $E={E}^{\prime}$ and think of $\varphi $ as being the “multiplication by $n$” map, $[n]:E\to E$). The following standard result asserts that $\varphi $ is surjective over $\overline{\mathbb{Q}}$:
Theorem 1.
Let ${C}_{\mathrm{1}}\mathrm{,}{C}_{\mathrm{2}}$ be curves defined over an algebraically closed field $K$ and let
$$\psi :{C}_{1}\to {C}_{2}$$ |
be a morphism^{} (or algebraic map) of curves. Then $\psi $ is either constant or surjective.
Proof.
See [4], Chapter II.6.8. ∎
Since $\varphi :E(\overline{\mathbb{Q}})\to {E}^{\prime}(\overline{\mathbb{Q}})$ is non-constant, it must be surjective and we obtain the following exact sequence^{}:
$$0\to E(\overline{\mathbb{Q}})[\varphi ]\to E(\overline{\mathbb{Q}})\to {E}^{\prime}(\overline{\mathbb{Q}})\to 0\mathit{\hspace{1em}\hspace{1em}}(1)$$ |
where $E(\overline{\mathbb{Q}})[\varphi ]=\mathrm{Ker}\varphi $. Let $G=\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$, the absolute Galois group of $\mathbb{Q}$, and consider the ${i}^{th}$-cohomology group^{} ${H}^{i}(G,E(\overline{\mathbb{Q}}))$ (we abbreviate by ${H}^{i}(G,E)$). Using equation $(1)$ we obtain the following long exact sequence (see Proposition 1 in group cohomology^{}):
$$0\to {H}^{0}(G,E(\overline{\mathbb{Q}})[\varphi ])\to {H}^{0}(G,E)\to {H}^{0}(G,{E}^{\prime})\to {H}^{1}(G,E(\overline{\mathbb{Q}})[\varphi ])\to {H}^{1}(G,E)\to {H}^{1}(G,{E}^{\prime})\mathit{\hspace{1em}\hspace{1em}}(2)$$ |
Note that
$${H}^{0}(G,E(\overline{\mathbb{Q}})[\varphi ])={(E(\overline{\mathbb{Q}})[\varphi ])}^{G}=E(\mathbb{Q})[\varphi ]$$ |
and similarly
$${H}^{0}(G,E)=E(\mathbb{Q}),{H}^{0}(G,{E}^{\prime})={E}^{\prime}(\mathbb{Q})$$ |
From $(2)$ we can obtain an exact sequence:
$$0\to {E}^{\prime}(\mathbb{Q})/\varphi (E(\mathbb{Q}))\to {H}^{1}(G,E(\overline{\mathbb{Q}})[\varphi ])\to {H}^{1}(G,E)[\varphi ]\to 0$$ |
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 ${G}_{p}=\mathrm{Gal}(\overline{{\mathbb{Q}}_{p}}/{\mathbb{Q}}_{p})$):
$0\to {E}^{\prime}(\mathbb{Q})/\varphi (E(\mathbb{Q}))\to $ | ${H}^{1}(G,E(\overline{\mathbb{Q}})[\varphi ])$ | $\to {H}^{1}(G,E)[\varphi ]\to 0$ | ||
$\downarrow $ | $\downarrow $ | $\mathrm{\hspace{1em}}\mathit{\hspace{1em}\hspace{1em}}\downarrow $ | ||
$0\to {E}^{\prime}({\mathbb{Q}}_{p})/\varphi (E({\mathbb{Q}}_{p}))\to $ | ${H}^{1}({G}_{p},E(\overline{{\mathbb{Q}}_{p}})[\varphi ])$ | $\to {H}^{1}({G}_{p},E)[\varphi ]\to 0$ |
The goal here is to find a finite group containing ${E}^{\prime}(\mathbb{Q})/\varphi (E(\mathbb{Q}))$. Unfortunately ${H}^{1}(G,E(\overline{\mathbb{Q}})[\varphi ])$ is not necessarily finite. With this purpose in mind, we define the $\varphi $-Selmer group:
$${S}^{\varphi}(E/\mathbb{Q})=\mathrm{Ker}\left({H}^{1}(G,E(\overline{\mathbb{Q}})[\varphi ])\to \prod _{p}{H}^{1}({G}_{p},E)\right)$$ |
Equivalently, the $\varphi $-Selmer group is the set of elements $\gamma $ of ${H}^{1}(G,E(\overline{\mathbb{Q}})[\varphi ])$ whose image ${\gamma}_{p}$ in ${H}^{1}({G}_{p},E(\overline{{\mathbb{Q}}_{\mathbb{p}}})[\varphi ])$ 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:
$$TS(E/\mathbb{Q})=\mathrm{Ker}\left({H}^{1}(G,E)\to \prod _{p}{H}^{1}({G}_{p},E)\right)$$ |
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, The Arithmetic of Elliptic Curves. Springer-Verlag, New York, 1986.
- 4 R. Hartshorne, Algebraic Geometry^{}, Springer-Verlag, 1977.
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 |