Let be an odd prime number, and an elliptic curve over a global field that has good ordinary reduction at . Let be any (infinite degree) Galois extension with a continuous injective homomorphism of its Galois group to . To the data , one associates6.1 a canonical (bilinear, symmetric) ( -adic) height pairing
Such pairings are of great interest for the arithmetic of over , and they arise specifically in -adic analogues of the Birch and Swinnerton-Dyer conjecture.6.2
The goal of this project is to investigate some computational questions regarding -adic height pairings. The main stumbling block to computing them efficiently is in calculating, for each of the completions at the places of dividing , the value of the -adic modular form associated to the elliptic curve with a chosen Weierstrass form of good reduction over .
The paper [MST06] contains an algorithm for computing these quantities (for ), i.e., for computing the value of of an elliptic curve (that builds on the works of Katz and Kedlaya listed in our bibliography).
The paper [MST06] also discusses the -adic convergence rate of canonical expansions of the -adic modular form on the Hasse domain, where for we view as an infinite sum of classical modular forms divided by powers of the (classical) modular form , while for we view it as a sum of classical modular forms divided by powers of .
We were led to our fast method of computing by our realization that the more naive methods, of computing it by integrality or by approximations to it as function on the Hasse domain, were not practical, because the convergence is ``logarithmic'' in the sense that the th convergent gives only an accuracy of .
The reason why this constant enters the calculation is because it is needed for the computation of the -adic sigma function [MT91], which in turn is the critical element in the formulas for height pairings.
For example, let us consider the cyclotomic -adic height pairing in the special case where and .
If is the Galois group of an algebraic closure of over , we have the natural surjective continuous homomorphism pinned down by the standard formula where and is any -power root of unity. The -adic logarithm is the unique group homomorphism with that extends the homomorphism defined by the usual power series of about . Explicitly, if , then
where is the unit part of , and the usual series for converges at .
The composition is a cyclotomic linear functional which, in the body of our text, will be dealt with (thanks to class field theory) as the idele class functional that we denote .
Let denote the Néron model of over . Let be a non-torsion point that reduces to and to the connected component of at all primes of bad reduction for . Because is a unique factorization domain, any nonzero point can be written uniquely in the form , where , , and . The function assigns to this square root of the denominator of .
Here is the formula for the cyclotomic -adic height of , i.e., the value of
where is the height pairing attached to , the cyclotomic linear functional described above:
Here is the -adic sigma function of [MT91] associated to the pair . The -function depends only on and not on a choice of Weierstrass equation, and behaves like a modular form of weight , that is . It is ``quadratic'' the sense that for any and point in the formal group , we have
The height function of (6.1.1) extends uniquely to a function on the full Mordell-Weil group that satisfies for all integers and . For , setting
we obtain a pairing on . The -adic regulator of is the discriminant of the induced pairing on (well defined up to sign), and we have the following standard conjecture about this height pairing.
The -adic function is the most mysterious quantity in (6.1.1). There are many ways to define , e.g., [MT91] contains different characterizations of ! We now describe a characterization that leads directly to a (slow!) algorithm to compute . Let
In (6.1.1), by we mean , where . We have thus given a complete definition of for any point and a prime of good ordinary reduction for .