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In mathematics, the '''Tate curve''' is a curve defined over the ring of formal [[power series]] <math>Z[[q]]</math> 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 {{harvs|txt|first=John |last=Tate|authorlink=John Tate|year=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 {{harvtxt|Roquette|1970}}.

==Definition==

The Tate curve is the projective plane curve over the ring '''Z'''[<span/>[''q'']] of formal power series with integer coefficients given (in an affine open subset of the projective plane) by the equation
:<math> y^2+xy=x^3+a_4x+a_6</math>
where
:<math>-a_4=5\sum_n \frac{n^3q^n}{1-q^n} = 5q+45q^2+140q^3+\cdots</math>
:<math>-a_6=\sum_{n}\frac{7n^5+5n^3}{12}\times\frac{q^n}{1-q^n} = q+23q^2+154q^3+\cdots</math>
are power series with integer coefficients.<ref>Manin & Panchishkin (2007) p.220</ref>

==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''<sup>*</sup>/''q''<sup>'''Z'''</sup> to this elliptic curve, taking ''w'' to (''x''(''w''),''y''(''w'')) for ''w'' not a power of ''q'', where
:<math>x(w)= -y(w)-y(w^{-1})</math>
:<math>y(w) = \sum_{m\in Z}\frac{(t^mw)^2}{(1-t^mw)^3} + \sum_{m\ge 1} \frac{t^mw}{(1-t^mw)^2}</math>
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''.

==Properties==
The [[j-invariant]] of the Tate curve is given by a power series in ''q'' with leading term ''q''<sup>-1</sup>.<ref>Silverman (1994) p.423</ref>  Over a ''p''-adic [[local field]], therefore, ''j'' is non-integral and the Tate curve has [[Semistable abelian variety|semistable reduction]] of multiplicative type.  Conversely, every semistable elliptic curve over a local field is isomorphic to a Tate curve (up to [[quadratic twist]]).<ref> Manin & Panchiskin (2007) p.300</ref>

==References==
{{reflist}}
*{{Citation | last1=Lang | first1=Serge | author1-link=Serge Lang | title=Elliptic functions | publisher=[[Springer-Verlag]] | location=Berlin, New York | edition=2nd | series=Graduate Texts in Mathematics | isbn=978-0-387-96508-6 | id={{MR|890960}} | year=1987 | volume=112 | zbl=0615.14018 }}
*{{cite book | first1=Yu. I. | last1=Manin | authorlink1=Yuri I. Manin | first2=A. A. | last2=Panchishkin | title=Introduction to Modern Number Theory | series=Encyclopaedia of Mathematical Sciences | volume=49 | edition=Second | year=2007 | isbn=978-3-540-20364-3 | issn=0938-0396 | zbl=1079.11002 }}
*{{Citation | last1=Robert | first1=Alain | title=Elliptic curves | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Lecture Notes in Mathematics | isbn=978-3-540-06309-4 | id={{MR|0352107}} | year=1973 | volume=326 | doi=10.1007/978-3-540-46916-2 | zbl=0256.14013 }}
*{{Citation | last1=Roquette | first1=Peter | title=Analytic theory of elliptic functions over local fields | url=http://books.google.com/books?id=NHGmAAAAIAAJ | publisher=Vandenhoeck & Ruprecht | location=Göttingen | series=Hamburger Mathematische Einzelschriften (N.F.), Heft 1 | id={{MR|0260753}} | year=1970 | zbl=0194.52002 }}
*{{cite book | first=Joseph H. | last=Silverman | authorlink=Joseph H. Silverman | title=Advanced Topics in the Arithmetic of Elliptic Curves | series=[[Graduate Texts in Mathematics]] | publisher=[[Springer-Verlag]] | volume=151 | year=1994 | isbn=0-387-94328-5 | zbl=0911.14015 }}
*{{Citation | last1=Tate | first1=John | author1-link=John Tate | editor1-last=Coates | editor1-first=John | editor2-last=Yau | editor2-first=Shing-Tung | title=Elliptic curves, modular forms, & Fermat's last theorem (Hong Kong, 1993) | origyear=1959 | url=http://www.intlpress.com/books/Fermat-2.php | publisher=Int. Press, Cambridge, MA | series= Series in Number Theory | isbn=978-1-57146-026-4  | id={{MR|1363501}}  | year=1995 | volume=I | chapter=A review of non-Archimedean elliptic functions | chapterurl=http://www.ma.utexas.edu/users/voloch/Ellnotes/nonarch-ams.pdf | pages=162–184}}
[[Category:Elliptic curves]]