Tate curve

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

 y^2+xy=x^3+a_4x+a_6

where

-a_4=5\sum_n \frac{n^3q^n}{1-q^n} = 5q+45q^2+140q^3+\cdots
-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[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

x(w)= -y(w)-y(w^{-1})
y(w) = \sum_{m\in Z}\frac{(t^mw)^2}{(1-t^mw)^3} + \sum_{m\ge 1} \frac{t^mw}{(1-t^mw)^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.

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