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A Tate curve is a uniformization of an elliptic curve having stable bad reduction with the help of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092260/t0922601.png" />-parametrization.
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A Tate curve is a uniformization of an [[elliptic curve]] having stable bad reduction with the help of a $q$-parametrization.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092260/t0922602.png" /> be a [[Local field|local field]] (e.g., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092260/t0922603.png" /> or a finite extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092260/t0922604.png" />). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092260/t0922605.png" /> be an [[Elliptic curve|elliptic curve]] over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092260/t0922606.png" /> such that it has stable reduction. Then it can have good reduction (i.e. with integral <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092260/t0922607.png" />-invariant) or bad reduction (i.e. with non-integral <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092260/t0922608.png" />-invariant). In the case of stable bad reduction one can construct an elliptic curve <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092260/t0922609.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092260/t09226010.png" />, which analytically is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092260/t09226011.png" /> (where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092260/t09226012.png" /> is the subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092260/t09226013.png" /> generated by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092260/t09226014.png" />), such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092260/t09226015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092260/t09226016.png" /> are isomorphic over a finite extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092260/t09226017.png" />. One of the marvels of this theorem is the fact that the construction of the period <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092260/t09226018.png" /> starting from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092260/t09226019.png" />, and the computation of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092260/t09226020.png" />-value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092260/t09226021.png" />, can be done without denominators (hence can be done in every characteristic): the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092260/t09226022.png" />-value of the Tate curve with period <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092260/t09226023.png" /> is a power series in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092260/t09226024.png" /> with coefficients in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092260/t09226025.png" />:
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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 $\mathbb{Z}$:
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$$
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j = q^{-1} + 744 + 196884 q + 21493760 q^2  + \cdots \ .
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$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092260/t09226026.png" /></td> </tr></table>
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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.
  
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092260/t09226027.png" /> 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.
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See also [[#References|[a11]]], [[#References|[a12]]], [[#References|[b1]]].
 
 
See also [[#References|[a11]]], [[#References|[a12]]].
 
  
 
====References====
 
====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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/t/t092/t092260/t09226028.png" />-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></table>
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<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>

Revision as of 21:47, 21 December 2014

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
How to Cite This Entry:
Tate curve. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Tate_curve&oldid=17928
This article was adapted from an original article by F. Oort (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article