In mathematics, the Tate curve is a curve defined over the ring of formal power series
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.
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.
is the discrete subgroup generated by one multiplicative period
To see why the Tate curve morally corresponds to a torus when the field is C with the usual norm,
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.
This is slightly different from the usual method beginning with a flat sheet of paper,
, and gluing together the sides to make a cylinder
, and then gluing together the edges of the cylinder to make a torus,
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).