Tate curve

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

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

where

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

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

are power series with integer coefficients.[1]

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

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

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

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.

In the case of the curve over the complete field, ${\displaystyle k^{*}/q^{\mathbb {Z} }}$, the easiest case to visualize is ${\displaystyle \mathbb {C} ^{*}/q^{\mathbb {Z} }}$, where ${\displaystyle q^{\mathbb {Z} }}$ is the discrete subgroup generated by one multiplicative period ${\displaystyle e^{2\pi i\tau }}$, where the period ${\displaystyle \tau =\omega _{1}/\omega _{2}}$. Note that ${\displaystyle \mathbb {C} ^{*}}$ is isomorphic to ${\displaystyle {\mathbb {C} +}/\mathbb {Z} +}$, where ${\displaystyle \mathbb {C} +}$ 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, ${\displaystyle q}$ is already singly periodic; modding out by q's integral powers you are modding out ${\displaystyle \mathbb {C} }$ by ${\displaystyle \mathbb {Z} ^{2}}$, 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, ${\displaystyle \mathbb {C} }$, and gluing together the sides to make a cylinder ${\displaystyle \mathbb {C} /\mathbb {Z} }$, and then gluing together the edges of the cylinder to make a torus, ${\displaystyle \mathbb {C} /\mathbb {Z} ^{2}}$.

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).

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]

• Lang, Serge (1987), Elliptic functions, Graduate Texts in Mathematics, vol. 112 (2nd ed.), Berlin, New York: Springer-Verlag, doi:10.1007/978-1-4612-4752-4, 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, ISBN 9783525403013, 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, CiteSeerX 10.1.1.367.7205, ISBN 978-1-57146-026-4, MR 1363501