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Bad reduction of genus $2$ curves with CM jacobian varieties

Part of: Arithmetic algebraic geometry Abelian varieties and schemes Curves Arithmetic problems. Diophantine geometry

Published online by Cambridge University Press:  11 September 2017

Philipp Habegger  and
Fabien Pazuki
Philipp Habegger
Affiliation:
Departement Mathematik und Informatik, Spiegelgasse 1, 4051 Basel, Switzerland email [email protected]
Fabien Pazuki
Affiliation:
Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen, Denmark email [email protected] IMB, Université de Bordeaux, 351, cours de la Libération, 33405 Talence, France
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Abstract

We show that a genus $2$ curve over a number field whose jacobian has complex multiplication will usually have stable bad reduction at some prime. We prove this by computing the Faltings height of the jacobian in two different ways. First, we use a known case of the Colmez conjecture, due to Colmez and Obus, that is valid when the CM field is an abelian extension of the rationals. It links the height and the logarithmic derivatives of an $L$ -function. The second formula involves a decomposition of the height into local terms based on a hyperelliptic model. We use the reduction theory of genus $2$ curves as developed by Igusa, Liu, Saito, and Ueno to relate the contribution at the finite places with the stable bad reduction of the curve. The subconvexity bounds by Michel and Venkatesh together with an equidistribution result of Zhang are used to bound the infinite places.

Keywords

curveshyperelliptic jacobianscomplex multiplication

MSC classification

Primary: 14K22: Complex multiplication
Secondary: 11G18: Arithmetic aspects of modular and Shimura varieties 11G30: Curves of arbitrary genus or genus $ne 1$ over global fields 11G50: Heights 14H40: Jacobians, Prym varieties 14G40: Arithmetic varieties and schemes; Arakelov theory; heights
Type
Research Article
Information
Compositio Mathematica , Volume 153 , Issue 12 , December 2017 , pp. 2534 - 2576
Copyright
© The Authors 2017 

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References

Autissier, P., Hauteur de Faltings et hauteur de Néron–Tate du diviseur thêta , Compos. Math. 142 (2006), 1451–1458.Google Scholar
Badzyan, A. I., The Euler–Kronecker constant , Mat. Zametki 87 (2010), 35–47.Google Scholar
Birkenhake, C. and Lange, H., Complex abelian varieties, Grundlehren Math. Wiss., vol. 302 (Springer, Berlin, 2004).Google Scholar
Bombieri, E. and Gubler, W., Heights in Diophantine geometry, New Mathematical Monographs, vol. 4 (Cambridge University Press, Cambridge, 2006).Google Scholar
Bosch, S., Lütkebohmert, W. and Raynaud, M., Néron models, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 21 (Springer, Berlin, 1990).CrossRefGoogle Scholar
Bost, J.-B., Mestre, J.-F. and Moret-Bailly, L., Sur le calcul explicite des ‘classes de Chern’ des surfaces arithmétiques de genre 2 , Astérisque 183 (1990), 69–105; Séminaire sur les pinceaux de courbes elliptiques (Paris, 1988).Google Scholar
Clozel, L. and Ullmo, E., Équidistribution de mesures algébriques , Compos. Math. 141 (2005), 1255–1309.Google Scholar
Cohen, P. B., Hyperbolic equidistribution problems on Siegel 3-folds and Hilbert modular varieties , Duke Math. J. 129 (2005), 87–127.CrossRefGoogle Scholar
Cohen, H., Number theory volume II: analytic and modern tools, Graduate Texts in Mathematics, vol. 240 (Springer, New York, 2007).Google Scholar
Colmez, P., Périodes des variétés abéliennes à multiplication complexe , Ann. of Math. (2) 138 (1993), 625–683.CrossRefGoogle Scholar
Colmez, P., Sur la hauteur de Faltings des variétés abéliennes à multiplication complexe , Compos. Math. 111 (1998), 359–368.Google Scholar
Deligne, P. and Mumford, D., The irreducibility of the space of curves of given genus , Inst. Hautes Études Sci. Publ. Math. 36 (1969), 75–109.Google Scholar
Faltings, G., Endlichkeitssätze für abelsche Varietäten über Zahlkörpern , Invent. Math. 73 (1983), 349–366.Google Scholar
Fontaine, J.-M., Il n’y a pas de variété abélienne sur ℤ , Invent. Math. 81 (1985), 515–538.CrossRefGoogle Scholar
Goren, E. Z., On certain reduction problems concerning abelian surfaces , Manuscripta Math. 94 (1997), 33–43.CrossRefGoogle Scholar
Goren, E. Z. and Lauter, K. E., Evil primes and superspecial moduli , Int. Math. Res. Not. IMRN 2006 (2006), Art. ID 53864.Google Scholar
Goren, E. Z. and Lauter, K. E., Class invariants for quartic CM fields , Ann. Inst. Fourier (Grenoble) 57 (2007), 457–480.Google Scholar
Griffiths, P. and Harris, J., Principles of algebraic geometry (Wiley-Interscience, New York, 1978).Google Scholar
Habegger, P., Singular moduli that are algebraic units , Algebra Number Theory 9 (2015), 1515–1524.CrossRefGoogle Scholar
Ibukiyama, T., Katsura, T. and Oort, F., Supersingular curves of genus two and class numbers , Compos. Math. 57 (1986), 127–152.Google Scholar
Igusa, J., Arithmetic variety of moduli for genus two , Ann. of Math. (2) 72 (1960), 612–649.Google Scholar
Iwaniec, H., Duke, W. and Friedlander, J. B., The subconvexity problem for Artin L-functions , Invent. Math. 149 (2002), 489–577.Google Scholar
Iwaniec, H. and Kowalski, E., Analytic number theory, American Mathematical Society Colloquium Publications, vol. 53 (American Mathematical Society, Providence, RI, 2004).Google Scholar
de Jong, J. and Noot, R., Jacobians with complex multiplication , in Arithmetic algebraic geometry (Texel, 1989), Progress in Mathematics, vol. 89 (Birkhäuser, Boston, MA, 1991), 177–192.Google Scholar
Klingen, H., Introductory lectures on Siegel modular forms, Cambridge Studies in Advanced Mathematics, vol. 20 (Cambridge University Press, Cambridge, 1990).Google Scholar
Liu, Q., Courbes stables de genre 2 et leur schéma de modules , Math. Ann. 295 (1993), 201–222.Google Scholar
Liu, Q., Conducteur et discriminant minimal de courbes de genre 2 , Compos. Math. 94 (1994), 51–79.Google Scholar
Liu, Q., Algebraic geometry and arithmetic curves, Oxford Graduate Texts in Mathematics, vol. 6 (Oxford University Press, Oxford, 2002); translated from the French by Reinie Erné, Oxford Science Publications.CrossRefGoogle Scholar
Lockhart, P., On the discriminant of a hyperelliptic curve , Trans. Amer. Math. Soc. 342 (1994), 729–752.Google Scholar
Michel, P. and Venkatesh, A., The subconvexity problem for GL2 , Publ. Math. Inst. Hautes Études Sci. 111 (2010), 171–271.Google Scholar
Moret-Bailly, L., Problèmes de Skolem sur les champs algébriques , Compos. Math. 125 (2001), 1–30.Google Scholar
Mumford, D., Tata lectures on theta. II, Progress in Mathematics, vol. 43 (Birkhäuser, Boston, MA, 1984).Google Scholar
Nakkajima, Y. and Taguchi, Y., A generalization of the Chowla–Selberg formula , J. Reine Angew. Math. 419 (1991), 119–124.Google Scholar
Namikawa, Y. and Ueno, K., The complete classification of fibres in pencils of curves of genus two , Manuscripta Math. 9 (1973), 143–186.Google Scholar
Neukirch, J., Algebraic number theory, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 322 (Springer, Berlin, 1999).Google Scholar
Obus, A., On Colmez’s product formula for periods of CM-abelian varieties , Math. Ann. 356 (2013), 401–418.CrossRefGoogle Scholar
Pazuki, F., Theta height and Faltings height , Bull. Soc. Math. France 140 (2012), 19–49.Google Scholar
Pazuki, F., Décompositions en hauteurs locales, Preprint, 2012, arXiv:1205.4525.Google Scholar
Pazuki, F., Minoration de la hauteur de Néron–Tate sur les surfaces abéliennes , Manuscripta Math. 142 (2013), 61–99.Google Scholar
Pila, J. and Tsimerman, J., The André–Oort conjecture for the moduli space of abelian surfaces , Compos. Math. 149 (2013), 204–216.Google Scholar
Pila, J. and Tsimerman, J., Ax–Lindemann for A g , Ann. of Math. (2) 179 (2014), 659–681.Google Scholar
Saito, T., Conductor, discriminant, and the Noether formula of arithmetic surfaces , Duke Math. J. 57 (1988), 151–173.Google Scholar
Saito, T., The discriminants of curves of genus 2 , Compos. Math. 69 (1989), 229–240.Google Scholar
Schoof, R., Abelian varieties over cyclotomic fields with good reduction everywhere , Math. Ann. 325 (2003), 413–448.Google Scholar
Serre, J.-P. and Tate, J. T., Good reduction of abelian varieties , Ann. of Math. (2) 88 (1968), 492–517.CrossRefGoogle Scholar
Shimura, G., Abelian varieties with complex multiplication and modular functions, Princeton Mathematical Series, vol. 46 (Princeton University Press, Princeton, NJ, 1997).Google Scholar
Szpiro, L. (ed.), Séminaire sur les pinceaux arithmétiques: la conjecture de Mordell, Astérisque, vol. 127 (Société Mathématique de France, Paris, 1985).Google Scholar
Ueno, K., Discriminants of curves of genus 2 and arithmetic surfaces , in Algebraic geometry and commutative algebra, Vol. II (Kinokuniya, Tokyo, 1988), 749–770.Google Scholar
van der Geer, G., Hilbert modular surfaces, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 16 (Springer, Berlin, 1988).Google Scholar
van Wamelen, P., Examples of genus two CM curves defined over the rationals , Math. Comp. 68 (1999), 307–320.Google Scholar
van Wamelen, P., Proving that a genus 2 curve has complex multiplication , Math. Comp. 68 (1999), 1663–1677.Google Scholar
Vojta, P., Integral points on subvarieties of semiabelian varieties. II , Amer. J. Math. 121 (1999), 283–313.Google Scholar
Washington, L. C., Introduction to cyclotomic fields, Graduate Texts in Mathematics, vol. 83 (Springer, New York, 1982).Google Scholar
Yang, T., The Chowla–Selberg formula and the Colmez conjecture , Canad. J. Math. 62 (2010), 456–472.Google Scholar
Zhang, S., Equidistribution of CM-points on quaternion Shimura varieties , Int. Math. Res. Not. IMRN 2005 (2005), 3657–3689.Google Scholar