Tate curve

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).

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

where

are power series with integer coefficients.

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

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.

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.

...
Wikipedia