# Monthly Archives: May 2016

## Our friend the Tate elliptic curve

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 ${\bf C}_p$ be the completion of the algebraic closure of the p-adic numbers ${\bf Q}_p$. The difficulty in defining analytic spaces over ${\bf C}_p$, by analogy with complex analytic spaces, is that ${\bf C}_p$ 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 $X$ over ${\bf Q}_p$ could be viewed as a quotient of the affine line minus the origin as an analytic space: ${\bf Q}_p^*/\langle q^{\bf Z}\rangle \cong X({\bf Q}_p).$

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 $\{ z: |z| \leq 1\}$ acts as though it is connected, because its covering by the two disjoint open subsets $\{ z: |z| < 1\}$ and $\{ z: |z| = 1\}$ is not an admissible covering. (“Affinoids,” playing the role of compact open sets, include closed balls such as $|z|\leq a$ for any real number $a$, but not the open ball $|z|<1$. An admissible covering of an affinoid such as $\{ z: |z| \leq 1\}$ is required to have a refinement by finitely many affinoids.)

Tate’s formulas for the p-adic analytic map $G_m \rightarrow X$, modeled on similar formulas for the Weierstrass $p$-function, are as follows.

Theorem. Let $K$ be a complete field with respect to a non-archimedean absolute value, and let $q \in K^*$ have $0<|q|<1$. Then the following power series define a isomorphism of abelian groups $K^*/q^{\bf Z}\cong X(K)$, for the elliptic curve $X$ below:

$x(w)=\sum_{m\in {\bf Z}}\frac{q^m w}{(1-q^mw)^2} -2s_1$

$y(w)=\sum_{m\in {\bf Z}}\frac{q^{2m} w}{(1-q^mw)^2} +s_1,$

where $s_l=\sum_{m\geq 1}\frac{m^lq^m}{1-q^m}$ for positive integers $l$. The corresponding elliptic curve $X$ in ${\bf P}^2$ is defined in affine coordinates by $y^2+xy=x^3+Bx+C,$ where $B=-5s_3$ and $C=(5s_3+7s_5)/12$. Its $j$-invariant is $j(q)=1/q+744+196884q+\cdots.$ For every element $j\in K$ with $|j|>1$ (corresponding to an elliptic curve over $K$ that does not have potentially good reduction), there is a unique $q\in K$ with $j(q)=j$.

It is worth contemplating why the formulas for $x(w)$ and $y(w)$ make sense, for $w\in K^*$. The series both have poles when $w$ is an integer power of ${q}$, 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 $x(qw)=x(w)$ and $y(qw)=y(w)$, but the series do not obviously converge; the terms are small for $m \rightarrow \infty$, but they are large for $m\rightarrow -\infty$.

To make sense of the formulas, one has to use the identity of rational functions $\frac{w}{(1-w)^2} = \frac{w^{-1}}{(1-w^{-1})^2}.$ As a result, the series for $x(w)$ (for example) can be written as

$x(w)=\frac{w}{(1-w)^2}+\sum_{m\geq 1}\big(\frac{q^mw}{(1-q^mw)^2}+\frac{q^mw^{-1}}{(1-q^mw^{-1})^2} -2\frac{q^m}{(1-q^m)^2}\big),$

which manifestly converges. One checks from this description that the series $x(w)$ satisfies $x(qw)=x(w)$, as we want.

References:

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).