# Tate curve

In mathematics, the Tate curve is a curve defined over the ring of formal power series $\mathbb {Z} [[q]]$ with integer coefficients. Over the open subscheme where q is invertible, the Tate curve is an elliptic curve. The Tate curve can also be defined for q as an element of a complete field of norm less than 1, in which case the formal power series converge.
The Tate curve was introduced by John Tate (1995) in a 1959 manuscript originally titled "Rational Points on Elliptic Curves Over Complete Fields"; he did not publish his results until many years later, and his work first appeared in Roquette (1970).

## Definition

The Tate curve is the projective plane curve over the ring Z[[q]] of formal power series with integer coefficients given (in an affine open subset of the projective plane) by the equation
    $y^{2}+xy=x^{3}+a_{4}x+a_{6}$
where
    $-a_{4}=5\sum _{n}{\frac {n^{3}q^{n}}{1-q^{n}}}=5q+45q^{2}+140q^{3}+\cdots$
    $-a_{6}=\sum _{n}{\frac {7n^{5}+5n^{3}}{12}}\times {\frac {q^{n}}{1-q^{n}}}=q+23q^{2}+154q^{3}+\cdots$
are power series with integer coefficients.

## The Tate curve over a complete field

Suppose that the field k is complete with respect to some absolute value | |, and q is a non-zero element of the field k with |q|<1. Then the series above all converge, and define an elliptic curve over k. If in addition q is non-zero then there is an isomorphism of groups from k$^*$/q$^Z$ to this elliptic curve, taking w to (x(w),y(w)) for w not a power of q, where
    $x(w)=-y(w)-y(w^{-1})$
    $y(w)=\sum _{m\in \mathbf {Z} }{\frac {(q^{m}w)^{2}}{(1-q^{m}w)^{3}}}+\sum _{m\geq 1}{\frac {q^{m}}{(1-q^{m})^{2}}}$
and taking powers of q to the point at infinity of the elliptic curve. The series x(w) and y(w) are not formal power series in w.

## Intuitive example

In the case of the curve over the complete field, $k^{*}/q^{\mathbb {Z} }$, the easiest case to visualize is $\mathbb {C} ^{*}/q^{\mathbb {Z} }$, where $q^{\mathbb {Z} }$ is the discrete subgroup generated by one multiplicative period $e^{2\pi i\tau }$, where the period $\tau =\omega _{1}/\omega _{2}$. Note that $\mathbb {C} ^{*}$ is isomorphic to ${(\mathbb {C} ,+)}/(\mathbb {Z} ,+)$, where $(\mathbb {C} ,+)$ is the complex numbers under addition.
To see why the Tate curve morally corresponds to a torus when the field is $\mathbb {C}$ with the usual norm, $q$ is already singly periodic; modding out by $q$'s integral powers you are modding out $\mathbb {C}$ by $\mathbb {Z} ^{2}$, which is a torus. In other words, we have an annulus, and we glue inner and outer edges.
But the annulus does not correspond to the circle minus a point: the annulus is the set of complex numbers between two consecutive powers of $q$; say all complex numbers with magnitude between 1 and $q$. That gives us two circles, i.e., the inner and outer edges of an annulus.
The image of the torus given here is a bunch of inlaid circles getting narrower and narrower as they approach the origin.
This is slightly different from the usual method beginning with a flat sheet of paper, $\mathbb {C}$, and gluing together the sides to make a cylinder $\mathbb {C} /\mathbb {Z}$, and then gluing together the edges of the cylinder to make a torus, $\mathbb {C} /\mathbb {Z} ^{2}$.
This is slightly oversimplified. The Tate curve is really a curve over a formal power series ring rather than a curve over C. Intuitively, it is a family of curves depending on a formal parameter. When that formal parameter is zero it degenerates to a pinched torus, and when it is nonzero it is a torus).

## Properties

The j-invariant of the Tate curve is given by a power series in q with leading term q$^{−1}$. Over a p-adic local field, therefore, j is non-integral and the Tate curve has semistable reduction of multiplicative type. Conversely, every semistable elliptic curve over a local field is isomorphic to a Tate curve (up to quadratic twist).