Tate curve

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In mathematics, the Tate curve is a curve defined over the ring of formal power series 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[edit]

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.[1]

The Tate curve over a complete field[edit]

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[edit]

In the case of the curve over the complete field, , the easiest case to visualize is , where is the discrete subgroup generated by one multiplicative period , where the period . Note that is isomorphic to , where 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, is already singly periodic; modding out by q's integral powers you are modding out by , 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, , and gluing together the sides to make a cylinder , and then gluing together the edges of the cylinder to make a torus, .

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[edit]

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[edit]

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