Elliptic Curves over Q - Elliptic Curves: Number Theory and Cryptography - page 251

The group G p can be regarded as a subgroup of G . Recall that cocycles in
Z ( G, E [2]) are maps from G to E [2] with certain properties. Such maps may
be restricted to G p to obtain elements of Z ( G p ,E [2]). A curve C a,b,c yields an
element of H 1 ( G, E [ 2] ). This yields an element of H 1 ( G p ,E [2]) that becomes
trivial in H 1 ( G p ,E ( Q p )) if and only if C a,b,c has a p -adic point.
In Section 8.7, we defined S 2 to be those triples ( a, b, c ) such that C a,b,c
has a p -adic point for all p ≤∞ . This means that S 2 is the set of triples
( a, b, c ) such that th e c orresponding cohomology class in H 1 ( G, E [2]) becomes
trivial in H 1 ( G p ,E ( Q p )) for all p

.Moreover, 2 is S 2 modulo those
triples coming from points in E ( Q ). All of this can be expressed in terms
of cohomology. We can also replace 2 by an arbitrary n

≤∞

≥ 1. Define the

Shafarevich-Tate group to be

⎛ ⎞

⎝ H 1 ( G, E ( Q )) → ⎠

H 1 ( G p ,E ( Q p ))

=Ker

p ≤∞

and define the n -Selmer group to be

⎛ ⎞

⎝ H 1 ( G p ,E [ n ]) ⎠ .

H 1 ( G p ,E ( Q p )) S n =Ker →

p ≤∞

The Shafarevich-Tate group can be thought of as consisting of equivalence
classes of pairs ( C, φ ) such that C has a p -adic point for all p ≤∞ .This
group is nontrivial if there exists such a C that has no rational points. In
Section 8.8, we gave an example of such a curve. The n -Selmer group S n can
be regarded as the generalization to n -descents of the curves C a,b,c that arise
in 2-descents. It is straightforward to use the definitions to deduce the basic
descent sequence

0 → E ( Q ) /nE ( Q ) → S n → [ n ] → 0 ,

where [ n ]isthe n -torsion in . When one is doing descent, the goal is to
obtain information about E ( Q ) /nE ( Q ). However, the calculations take place
in S n . The group [ n ] is the obstruction to transferring information back to

E ( Q ) /nE ( Q ).
The group S n depends on n . It is finite (we proved this in the case where n =
2and E [2] E ( Q )). The group is independent of n .Its n -torsion [ n ]is

finite since it is the quotient of the finite group S n . It was conjectured by Tate
and Shafarevich in the early 1960s that is finite; this is still unproved in
general. The first examples where was proved finite were given by Rubin
in 1986 (for all CM curves over Q with analytic rank 0; see Section 14.2) and
by Kolyvagin in 1987 (for all elliptic curves over Q with analytic rank 0 or 1).
No other examples over Q are known.

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