# Tate curve

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In mathematics, the Tate curve is a curve defined over the ring of formal power series ${\displaystyle \mathbb {Z} [[q]]}$ 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

${\displaystyle y^{2}+xy=x^{3}+a_{4}x+a_{6}}$

where

${\displaystyle -a_{4}=5\sum _{n}{\frac {n^{3}q^{n}}{1-q^{n}}}=5q+45q^{2}+140q^{3}+\cdots }$
${\displaystyle -a_{6}=\sum _{n}{\frac {7n^{5}+5n^{3}}{12}}\times {\frac {q^{n}}{1-q^{n}}}=q+23q^{2}+154q^{3}+\cdots }$

are power series with integer coefficients.[1]

## 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*/qZ to this elliptic curve, taking w to (x(w),y(w)) for w not a power of q, where

${\displaystyle x(w)=-y(w)-y(w^{-1})}$
${\displaystyle y(w)=\sum _{m\in Z}{\frac {(t^{m}w)^{2}}{(1-t^{m}w)^{3}}}+\sum _{m\geq 1}{\frac {t^{m}w}{(1-t^{m}w)^{2}}}}$

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, ${\displaystyle k^{*}/q^{\mathbb {Z} }}$, the easiest case to visualize is ${\displaystyle \mathbb {C} ^{*}/q^{\mathbb {Z} }}$, where ${\displaystyle q}$ is the discrete subgroup generated by one multiplicative period ${\displaystyle e^{2\pi i\tau }}$, where the period ${\displaystyle \tau =\omega _{1}/\omega _{2}}$. Note that ${\displaystyle \mathbb {C} ^{*}}$ is isomorphic to ${\displaystyle \mathbb {C} +/\mathbb {Z} +}$, where ${\displaystyle \mathbb {C} +}$ is the complex numbers under addition.

To see why the Tate Curve morally corresponds to a torus when the field is C with the usual norm, ${\displaystyle q}$ is already singly periodic; modding out by q's integral powers you are modding out ${\displaystyle \mathbb {C} }$ by ${\displaystyle \mathbb {Z} ^{2}}$, 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, ${\displaystyle \mathbb {C} }$, and gluing together the sides to make a cylinder ${\displaystyle \mathbb {C} /\mathbb {Z} }$, and then gluing together the edges of the cylinder to make a torus, ${\displaystyle \mathbb {C} /\mathbb {Z} ^{2}}$.

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's a family of curves depending on a formal parameter (when that formal parameter is zero it degenerates to a pinched torus, and when it's nonzero it's a torus).

## Properties

The j-invariant of the Tate curve is given by a power series in q with leading term q−1.[2] 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).[3]

## References

1. ^ Manin & Panchishkin (2007) p.220
2. ^ Silverman (1994) p.423
3. ^ Manin & Panchiskin (2007) p.300