``The subject of this lecture is rather a special one. I want to describe some computations undertaken by myself and Swinnerton-Dyer on EDSAC, by which we have calculated the zeta-functions of certain elliptic curves. As a result of these computations we have found an analogue for an elliptic curve of the Tamagawa number of an algebraic group; and conjectures have proliferated. [] I would like to stress that though the associated theory is both abstract and technically complicated, the objects about which I intend to talk are usually simply defined and often machine computable; experimentally we have detected certain relations between different invariants, but we have been unable to approach proofs of these relations, which must lie very deep.''
-Bryan Birch
Let
Recall from Section 1.5.2 that
the real period is the integral
To define the regulator let be a basis for modulo torsion and recall the Néron-Tate canonical height pairing from Section 1.2. The real number is the absolute value of the determinant of the matrix whose entry is . See [Cre97, §3.4] for a discussion of how to compute .
We defined the group in Section 2.2.4. In general it is not known to be finite, which led to Tate's famous assertion that the above conjecture ``relates the value of a function at a point at which it is not known to be defined to the order of a group that is not known to be finite.'' The paper [GJP+05] discusses methods for computing in practice, though no general algorithm for computing is known. In fact, in general even if we assume truth of the BSD rank conjecture (Conjecture 1.1) and assume that is finite, there is still no known way to compute , i.e., there is no analogue of Proposition 1.3. Given finiteness of we can compute the -part of for any prime , but we don't know when to stop considering new primes . (Note that when , Kolyvagin's work provides an explicit upper bound on , so in that case is computable.)
The Tamagawa numbers are for all primes
, where is the discriminant of (2.3.2).
When
, the number is a more refined measure
of the structure of the locally at . If is a prime
of additive reduction (see Section 1.3), then
one can prove that . The other alternatives are
that is a prime of split or nonsplit multiplicative reduction.
If is a nonsplit prime, then
For those that are very familiar with elliptic curves
over local fields,
For those with more geometric background, we offer the following conceptual definition of . Let be the Néron model of . This is the unique, up to unique isomorphism, smooth commutative (but not proper!) group scheme over that has generic fiber and satisfies the Néron mapping property:
for any smooth group scheme over the natural map
is an isomorphism.In particular, note that . For each prime , the reduction of the Néron model modulo is a smooth commutative group scheme over (smoothness is a property of morphisms that is closed under base extension). Let be the identity component of the group scheme , i.e., the connected component of that contains the section. The component group of at is the quotient group scheme
William 2007-05-25