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).
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
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|<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.
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).
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).
- Manin & Panchishkin (2007) p.220
- Silverman (1994) p.423
- Manin & Panchiskin (2007) p.300
- Lang, Serge (1987), Elliptic functions, Graduate Texts in Mathematics, 112 (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-96508-6, MR 890960, Zbl 0615.14018
- Manin, Yu. I.; Panchishkin, A. A. (2007). Introduction to Modern Number Theory. Encyclopaedia of Mathematical Sciences. 49 (Second ed.). ISBN 978-3-540-20364-3. ISSN 0938-0396. Zbl 1079.11002.
- Robert, Alain (1973), Elliptic curves, Lecture Notes in Mathematics, 326, Berlin, New York: Springer-Verlag, doi:10.1007/978-3-540-46916-2, ISBN 978-3-540-06309-4, MR 0352107, Zbl 0256.14013
- Roquette, Peter (1970), Analytic theory of elliptic functions over local fields, Hamburger Mathematische Einzelschriften (N.F.), Heft 1, Göttingen: Vandenhoeck & Ruprecht, MR 0260753, Zbl 0194.52002
- Silverman, Joseph H. (1994). Advanced Topics in the Arithmetic of Elliptic Curves. Graduate Texts in Mathematics. 151. Springer-Verlag. ISBN 0-387-94328-5. Zbl 0911.14015.
- Tate, John (1995) , "A review of non-Archimedean elliptic functions" (PDF), in Coates, John; Yau, Shing-Tung, Elliptic curves, modular forms, & Fermat's last theorem (Hong Kong, 1993), Series in Number Theory, I, Int. Press, Cambridge, MA, pp. 162–184, ISBN 978-1-57146-026-4, MR 1363501