In mathematics, the Tate curve is a curve defined over the ring of formal power series
Contents
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
where
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*/qZ 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.
Intuitive example
In the case of the curve over the complete field,
To see why the Tate Curve morally corresponds to a torus when the field is C with the usual norm,
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,
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. 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).