Tate curve

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

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

$y^{2}+xy=x^{3}+a_{4}x+a_{6}$

where

$-a_{4}=5\sum _{n}{\frac {n^{3}q^{n}}{1-q^{n}}}=5q+45q^{2}+140q^{3}+\cdots$

$-a_{6}=\sum _{n}{\frac {7n^{5}+5n^{3}}{12}}\times {\frac {q^{n}}{1-q^{n}}}=q+23q^{2}+154q^{3}+\cdots$

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|<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^*/q^Z to this elliptic curve, taking w to (x(w),y(w)) for w not a power of q, where

$x(w)=-y(w)-y(w^{-1})$

$y(w)=\sum _{m\in Z}{\frac {(t^{m}w)^{2}}{(1-t^{m}w)^{3}}}+\sum _{m\geq 1}{\frac {t^{m}w}{(1-t^{m}w)^{2}}}$

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, $k^{*}/q^{Z}$, the easiest case to visualize is $C^{*}/q$, where $q$ is the discrete subgroup generated by one multiplicative period Failed to parse (unknown function "\math"): {\displaystyle e^{2 \pi i \tau} <\math>, where the period <math> \tau = \omega_1/\omega_2} .

To see why the Tate Curve corresponds to a torus when the elliptic field is C with the usual norm, visualize a piece of paper, we can cut out the origin; this gives us a complex plane with a hole in it, i.e., $C^{*}$. Now, cut along the edge of the piece of paper until it is circular, we now have a disk with a hole in it. Glue the boundary of the disk, i.e., a circle traced out by $e^{it}$ as $t$ ranges throughout the real numbers, to the edges of the hole in the center of our disk. In other words, we have an annulus, and we glue inner and outer edges.

This is an alternative way to form a torus, which 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.

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

• Lang, Serge (1987), Elliptic functions, Graduate Texts in Mathematics, vol. 112 (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-96508-6, MR 0890960, Zbl 0615.14018
• Manin, Yu. I.; Panchishkin, A. A. (2007). Introduction to Modern Number Theory. Encyclopaedia of Mathematical Sciences. Vol. 49 (Second ed.). ISBN 978-3-540-20364-3. ISSN 0938-0396. Zbl 1079.11002.
• Robert, Alain (1973), Elliptic curves, Lecture Notes in Mathematics, vol. 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. Vol. 151. Springer-Verlag. ISBN 0-387-94328-5. Zbl 0911.14015.
• Tate, John (1995) [1959], "A review of non-Archimedean elliptic functions", in Coates, John; Yau, Shing-Tung (eds.), Elliptic curves, modular forms, & Fermat's last theorem (Hong Kong, 1993), Series in Number Theory, vol. I, Int. Press, Cambridge, MA, pp. 162–184, ISBN 978-1-57146-026-4, MR 1363501 {{citation}}: External link in |chapterurl= (help); Unknown parameter |chapterurl= ignored (|chapter-url= suggested) (help)