Tate curve

Definition

The Tate curve is the projective plane curve over the ring \mathbb{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.

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|. 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^{\mathbb{Z}} 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 \mathbb{Z}}\frac{(q^mw)^2}{(1-q^mw)^3} + \sum_{m \ge 1} \frac{q^m}{(1-q^m)^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.

Intuitive example

In the case of the curve over the complete field, k^/q^\mathbb{Z}, the easiest case to visualize is \mathbb{C}^/q^\mathbb{Z}, where q^\mathbb{Z} is the discrete subgroup generated by one multiplicative period e^{2 \pi i \tau} , where the period \tau = \omega_1/\omega_2. Note that \mathbb{C}^* is isomorphic to {(\mathbb{C},+)}/ (\mathbb{Z},+), where (\mathbb{C},+) is the complex numbers under addition.

To see why the Tate curve morally corresponds to a torus when the field is \mathbb{C} with the usual norm, q is already singly periodic; modding out by q's integral powers you are modding out \mathbb{C} by \mathbb{Z}^2, 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, \mathbb{C}, and gluing together the sides to make a cylinder \mathbb{C}/\mathbb{Z}, and then gluing together the edges of the cylinder to make a torus, \mathbb{C}/\mathbb{Z}^2.

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 is a family of curves depending on a formal parameter. When that formal parameter is zero it degenerates to a pinched torus, and when it is nonzero it is 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).

References

References

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