View source for Tate curve

A Tate curve is a uniformization of an [[elliptic curve]] having stable bad reduction with the help of a $q$-parametrization.

Let $K$ be a [[local field]] (e.g., $\mathbb{C}((t))$ or a finite extension of $\mathbb{Q}_p$). Let $E$ be an elliptic curve over $K$ such that it has stable reduction. Then it can have good reduction (i.e. with integral $j$-invariant) or bad reduction (i.e. with non-integral $j$-invariant). In the case of stable bad reduction one can construct an elliptic curve $E_q$ over $K$, which analytically is $K^*/q^{\mathbb{Z}}$ (where $q^{\mathbb{Z}}$ is the subgroup of $K^*$ generated by $q \in K^*$), such that $E$ and $E_q$ are isomorphic over a finite extension of $K$. One of the marvels of this theorem is the fact that the construction of the period $q$ starting from $E$, and the computation of the $j$-value of $E_q$, can be done without denominators (hence can be done in every characteristic): the $j$-value of the Tate curve with period $q$ is a power series in $q$ with coefficients in $\mathbb{Z}$:
$$
j = q^{-1} + 744 + 196884 q + 21493760 q^2  + \cdots \ .
$$

Such formulas can be found in [[#References|[a4]]], Chapt. 15; [[#References|[a8]]]. See also [[#References|[a9]]], A.1.1, and [[#References|[a10]]], Appendix C, Sect. 14. In [[#References|[a2]]], VII, constructions over $\mathbb{Z}$ are given, with applications to compactifications of moduli schemes of elliptic curves. A generalization to higher-dimensional Abelian varieties over local fields with totally bad, stable reduction was given by H. Morikawa [[#References|[a5]]], [[#References|[a6]]] and by D. Mumford [[#References|[a7]]]. This was generalized by G. Faltings and by C.-L. Chai to the case of stable reduction of Abelian varieties in [[#References|[a3]]] and [[#References|[a1]]], and it was used in the theory of compactifications of moduli schemes of Abelian varieties.

See also [[#References|[a11]]], [[#References|[a12]]],  [[#References|[b1]]].

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  C.-L. Chai,   "Compactifications of Siegel moduli schemes" , Cambridge Univ. Press  (1985)</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  P. Deligne,   M. Rapoport,   "Les schémas de modules de courbes elliptiques"  P. Deligne (ed.)  W. Kuyk (ed.) , ''Modular Functions II'' , ''Lect. notes in math.'' , '''349''' , Springer  (1973)  pp. 143–316</TD></TR>
<TR><TD valign="top">[a3]</TD> <TD valign="top">  G. Faltings,   "Arithmetische Kompaktifizierung des Modulraums der abelschen Varietäten"  F. Hirzebruch (ed.)  J. Schwermer (ed.)  S. Suter (ed.) , ''Arbeitstagung Bonn 1984'' , ''Lect. notes in math.'' , '''1111''' , Springer  (1985)  pp. 321–383</TD></TR>
<TR><TD valign="top">[a4]</TD> <TD valign="top">  S. Lang,   "Elliptic functions" , Addison-Wesley  (1973)</TD></TR>
<TR><TD valign="top">[a5]</TD> <TD valign="top">  H. Morikawa,   "On theta functions and abelian varieties over valuation fields of rank one"  ''Nagoya Math. J.'' , '''20'''  (1962)  pp. 1–27</TD></TR>
<TR><TD valign="top">[a6]</TD> <TD valign="top">  H. Morikawa,   "On theta functions and abelian varieties over valuation fields of rank one"  ''Nagoya Math. J.'' , '''21'''  (1962)  pp. 231–250</TD></TR>
<TR><TD valign="top">[a7]</TD> <TD valign="top">  D. Mumford,   "An analytic construction of degenerating abelian varieties over complete rings"  ''Compos. Math.'' , '''24'''  (1972)  pp. 129–174; 239–272</TD></TR>
<TR><TD valign="top">[a8]</TD> <TD valign="top">  P. Roquette,   "Analytic theory of elliptic functions over local fields" , Vandenhoeck &amp; Ruprecht  (1970)</TD></TR>
<TR><TD valign="top">[a9]</TD> <TD valign="top">  J.-P. Serre,   "Abelian $\ell$-adic representations and elliptic curves" , Benjamin  (1986)  (Translated from French)</TD></TR>
<TR><TD valign="top">[a10]</TD> <TD valign="top">  J.H. Silverman,   "The arithmetic of elliptic curves" , Springer  (1986)</TD></TR>
<TR><TD valign="top">[a11]</TD> <TD valign="top">  C.-L. Chai,   G. Faltings,   "Semiabelian degeneration and compactification" , Forthcoming</TD></TR>
<TR><TD valign="top">[a12]</TD> <TD valign="top">  S. Bosch,   W. Lütkebohmert,   M. Raynaud,   "Néron models" , Springer  (Forthcoming)</TD></TR>
<TR><TD valign="top">[b1]</TD> <TD valign="top">  Alain Robert, "Elliptic curves. Notes from postgraduate lectures given in Lausanne 1971/72", Lecture Notes in Mathematics '''326''' Springer (1973) {{ZBL|0256.14013}}</TD></TR>
</table>