Rigid analytic spaces are all the rage these days, thanks to the work of Peter Scholze and his collaborators on perfectoid spaces. In this post, I want to briefly describe the example that inspired the whole subject of rigid analytic spaces: the Tate elliptic curve. Tate’s original 1959 notes were not published until 1995. (My thanks to Martin Gallauer for his explanations of the theory.)
Let be the completion of the algebraic closure of the p-adic numbers . The difficulty in defining analytic spaces over , by analogy with complex analytic spaces, is that is totally disconnected, and so there are too many locally analytic (or even locally constant) functions. Tate became convinced that it should be possible to get around this problem by his discovery of the Tate elliptic curve. Namely, by explicit power series, he argued that some elliptic curves over could be viewed as a quotient of the affine line minus the origin as an analytic space:
Trying to make sense of the formulas led Tate to his definition of rigid analytic spaces. In short, one has to view a rigid analytic space not just as a topological space, but as a space with a Grothendieck topology — that is, a space with a specified class of admissible coverings. So, for example, the closed unit disc acts as though it is connected, because its covering by the two disjoint open subsets and is not an admissible covering. (“Affinoids,” playing the role of compact open sets, include closed balls such as for any real number , but not the open ball . An admissible covering of an affinoid such as is required to have a refinement by finitely many affinoids.)
Tate’s formulas for the p-adic analytic map , modeled on similar formulas for the Weierstrass -function, are as follows.
Theorem. Let be a complete field with respect to a non-archimedean absolute value, and let have . Then the following power series define a isomorphism of abelian groups , for the elliptic curve below:
where for positive integers . The corresponding elliptic curve in is defined in affine coordinates by where and . Its -invariant is For every element with (corresponding to an elliptic curve over that does not have potentially good reduction), there is a unique with .
It is worth contemplating why the formulas for and make sense, for . The series both have poles when is an integer power of , just because these points map to the origin of the elliptic curve, which is at infinity in affine coordinates. More important, these formulas make it formally clear that and , but the series do not obviously converge; the terms are small for , but they are large for .
To make sense of the formulas, one has to use the identity of rational functions As a result, the series for (for example) can be written as
which manifestly converges. One checks from this description that the series satisfies , as we want.
S. Bosch, U. Güntzer, R. Remmert. Non-Archimedean Analysis. Springer (1984).
B. Conrad. Several approaches to non-Archimedean geometry. P-adic Geometry, 9–63, Amer. Math. Soc. (2008).
W. Lütkebohmert. From Tate’s elliptic curve to abeloid varieties. Pure and Applied Mathematics Quarterly 5 (2009), 1385–1427.
J. Tate. A review of non-Archimedean elliptic functions. Elliptic Curves, Modular Forms, & Fermat’s Last Theorem (Hong Kong, 1993), 162–184. Int. Press (1995).