In this post, we will discuss (without any proofs) Tate’s parameterization of elliptic curves over -adic fields. Aside from intrinsic importance, his work marked the beginning of , which I’ll likely discuss in future posts.
Recall that an elliptic curve over a field is a smooth, irreducible projective curve of genus with a choice of distinguished point . An elliptic curve has a group law (in fact, ), for which the point is the identity; over , this is the same as a Riemann surface of topological genus equipped with a choice of point. Weierstrass was able to elliptic curves over : by starting with their description as a complex torus , where is a lattice of rank two, he used the -function (which expresses a complex torus as a branched, two-sheeted cover of the Riemann sphere at the group of four -torsion points ) to realize as a cubic in via the map . This renders more explicit the shape of the -torsion subgroups, the group law, and the possible endomorphism rings.
Tate wanted to mimic this construction for elliptic curves over , but immediately ran into a problem: the -adic numbers have no nontrivial lattices! To see this, let be a nonzero subgroup. Take , not zero, and note that for all . Moreover, , so accumulates at , contradicting discreteness.
To mitigate this, we can apply the exponential in the complex setting to obtain an alternative description of elliptic curves as those of the form . This topologically checks out, although it is slightly harder to see – the space is biholomorphic to the cylinder , and then modding out by powers of under the homomorphism effects quotienting by another rank-one lattice. Thus, these successive quotients recover the description of elliptic curves as modulo a rank-two lattice. Formally mimicking this construction bodes much better for , since the group has plenty of discrete subgroups; for example, any with defines the discrete subgroup . We will use this description to furnish a -adic analytic isomorphism between with a -adic elliptic curve , and subsequently obtain a similar parameterization to that in the complex case.
All of this is summarized in the following theorem: (Tate) Let be a finite extension of . Let satisfy , and define
The series converge in , enabling us to define the by the equation
The Tate curve is an elliptic curve over with discriminant and -invariant given by
The series converge for all . They define a surjective morphism sending for , and if .
The morphism is compatible with the action of the absolute Galois group , namely, for all .
The arithmetic contribution from the last part of the theorem, absent in the theory of elliptic curves over , tells us that for any algebraic extension , we have an isomorphism
Our programme is as follows: we will flesh out the story of elliptic curves over to indicate where the formulas in Tate’s result come from, and then prove each part of the -adic version. After that, we will indicate some interpretations, as well as an application to the theory of complex multiplication.
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