Tate curve

2020 Mathematics Subject Classification: Primary: 14H52 [MSN][ZBL]

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 [a4], Chapt. 15; [a8]. See also [a9], A.1.1, and [a10], Appendix C, Sect. 14. In [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 [a5], [a6] and by D. Mumford [a7]. This was generalized by G. Faltings and by C.-L. Chai to the case of stable reduction of Abelian varieties in [a3] and [a1], and it was used in the theory of compactifications of moduli schemes of Abelian varieties.

See also [a11], [a12], [b1].

References

[a1]	C.-L. Chai, "Compactifications of Siegel moduli schemes" , Cambridge Univ. Press (1985)
[a2]	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
[a3]	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
[a4]	S. Lang, "Elliptic functions" , Addison-Wesley (1973)
[a5]	H. Morikawa, "On theta functions and abelian varieties over valuation fields of rank one" Nagoya Math. J. , 20 (1962) pp. 1–27
[a6]	H. Morikawa, "On theta functions and abelian varieties over valuation fields of rank one" Nagoya Math. J. , 21 (1962) pp. 231–250
[a7]	D. Mumford, "An analytic construction of degenerating abelian varieties over complete rings" Compos. Math. , 24 (1972) pp. 129–174; 239–272
[a8]	P. Roquette, "Analytic theory of elliptic functions over local fields" , Vandenhoeck & Ruprecht (1970)
[a9]	J.-P. Serre, "Abelian $\ell$-adic representations and elliptic curves" , Benjamin (1986) (Translated from French)
[a10]	J.H. Silverman, "The arithmetic of elliptic curves" , Springer (1986)
[a11]	C.-L. Chai, G. Faltings, "Semiabelian degeneration and compactification" , Forthcoming
[a12]	S. Bosch, W. Lütkebohmert, M. Raynaud, "Néron models" , Springer (Forthcoming)
[b1]	Alain Robert, "Elliptic curves. Notes from postgraduate lectures given in Lausanne 1971/72", Lecture Notes in Mathematics 326 Springer (1973) Zbl 0256.14013