## Abelian Groups Attached to Elliptic Curves

If is an elliptic curve over , then we give the set of all -rational points on  the structure of abelian group with identity element . If we embed in the projective plane, then this group is determined by the condition that three points sum to the zero element if and only if they lie on a common line.

For example on the curve , we have . This is because , , and are on a common line (so sum to zero):

and , , and (the point at infinity on the curve) are also on a common line, so . See the illustration below:

Iterating the group operation often leads quickly to very complicated points:

That the above condition--three points on a line sum to zero--defines an abelian group structure on is not obvious. Depending on your perspective, the trickiest part is seeing that the operation satisfies the associative axiom. The best way to understand the group operation on is to view as being related to a class group. As a first observation, note that the ring

is a Dedekind domain, so is defined, and every nonzero fractional ideal can be written uniquely in terms of prime ideals. When is a perfect field, the prime ideals correspond to the Galois orbits of affine points of .

Let be the free abelian group on the Galois orbits of points of  , which as explained above is analogous to the group of fractional ideals of a number field (here we do include the point at infinity). We call the elements of divisors. Let be the quotient of by the principal divisors, i.e., the divisors associated to rational functions via

Note that the principal divisor associated to is analogous to the principal fractional ideal associated to a nonzero element of a number field. The definition of is analogous to the power of  that divides the principal ideal generated by ''. Define the class group to be the quotient of the divisors by the principal divisors, so we have an exact sequence:

A key difference between elliptic curves and algebraic number fields is that the principal divisors in the context of elliptic curves all have degree 0, i.e., the sum of the coefficients of the divisor  is always 0. This might be a familiar fact to you: the number of zeros of a nonzero rational function on a projective curve equals the number of poles, counted with multiplicity. If we let denote the subgroup of divisors of degree 0, then we have an exact sequence

To connect this with the group law on , note that there is a natural map

Using the Riemann-Roch theorem, one can prove that this map is a bijection, which is moreover an isomorphism of abelian groups. Thus really when we discuss the group of -rational points on an , we are talking about the class group .

Recall that we proved (Theorem 7.1.2) that the class group of a number field is finite. The group of an elliptic curve can be either finite (e.g., for ) or infinite (e.g., for ), and determining which is the case for any particular curve is one of the central unsolved problems in number theory.

The Mordell-Weil theorem (see Chapter 12) asserts that if is an elliptic curve over a number field , then there is a nonnegative integer such that

 (10.2)

where is a finite group. This is similar to Dirichlet's unit theorem, which gives the structure of the unit group of the ring of integers of a number field. The main difference is that need not be cyclic, and computing appears to be much more difficult than just finding the number of real and complex roots of a polynomial!

Also, if is an arbitrary extension of fields, and is an elliptic curve over , then there is a natural inclusion homomorphism . Thus instead of just obtaining one group attached to an elliptic curve, we obtain a whole collection, one for each extension of . Even more generally, if is an arbitrary scheme, then is a group, and the association defines a functor from the category of schemes to the category of groups. Thus each elliptic curve gives rise to map:

William Stein 2012-09-24