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 $E, E'$ be elliptic curves defined over $\mathbb{Q}$ and let $\bar{\mathbb{Q}}$ be an algebraic closure of $\mathbb{Q}$ . Let $\phi\colon E \to E'$ be an non-constant isogeny (for example, we can let $E=E'$ 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 $\bar{\mathbb{Q}}$ :

Proof. See $\cite{hart}$ , Chapter II.6.8.

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

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

Note that $$H^0(G,E(\bar{\mathbb{Q}})[\phi])={(E(\bar{\mathbb{Q}})[\phi])}^G=E(\mathbb{Q})[\phi]$$ and similarly $$H^0(G,E)=E(\mathbb{Q}),\quad H^0(G,E')=E'(\mathbb{Q})$$

From $(2)$ we can obtain an exact sequence: $$0\to E'(\mathbb{Q})/\phi(E(\mathbb{Q})) \to H^1(G,E(\bar{\mathbb{Q}})[\phi]) \to H^1(G,E)[\phi]\to 0$$

We could repeat the same procedure but this time for $E,E'$ 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 $G_p=\operatorname{Gal}({\bar{\mathbb{Q}_p}/\mathbb{Q}_p})$ ): \begin{eqnarray*} 0\to E'(\mathbb{Q})/\phi(E(\mathbb{Q})) \to &H^1(G,E(\bar{\mathbb{Q}})[\phi])& \to H^1(G,E)[\phi]\to 0\\ \downarrow \quad\quad\quad\quad &\downarrow& \quad\quad\quad \downarrow \\ 0\to E'(\mathbb{Q}_p)/\phi(E(\mathbb{Q}_p)) \to &H^1(G_p,E(\bar{\mathbb{Q}_p})[\phi])& \to H^1(G_p,E)[\phi]\to 0 \end{eqnarray*}The goal here is to find a finite group containing $E'(\mathbb{Q})/\phi(E(\mathbb{Q}))$ . 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.

Bibliography

1

J.P. Serre, Galois Cohomology, Springer-Verlag, New York.

2

James Milne, Elliptic Curves, online course notes.

3

Joseph H. Silverman, The Arithmetic of Elliptic Curves. Springer-Verlag, New York, 1986.

4

R. Hartshorne, Algebraic Geometry, Springer-Verlag, 1977.