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.
![{\displaystyle \mathbb {Z} [[q]]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c1aa0c08bbbeba12df7dd38cef74ee7aee965025)
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).