Z[[q]]
The Tate curve was introduced by 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 .
The Tate curve is the projective plane curve over the ring Z of formal power series with integer coefficients given (in an affine open subset of the projective plane) by the equation
y2+xy=x
3+a | |
4x+a |
6
-a4=5\sumn
n3qn | |
1-qn |
=5q+45q2+140q3+ …
-a6=\sumn
7n5+5n3 | x | |
12 |
qn | |
1-qn |
=q+23q2+154q3+ …
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)=\summ
(qmw)2 | |
(1-qmw)3 |
+\summ
qm | |
(1-qm)2 |
In the case of the curve over the complete field,
k*/qZ
C*/qZ
qZ
e2
\tau=\omega1/\omega2
C*
{(C,+)}/(Z,+)
(C,+)
To see why the Tate curve morally corresponds to a torus when the field is C with the usual norm,
q
C
Z2
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,
C
C/Z
C/Z2
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).
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]