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

  • Philipp Habegger (a1) and Fabien Pazuki (a2) (a3)


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.



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[Aut06] Autissier, P., Hauteur de Faltings et hauteur de Néron–Tate du diviseur thêta , Compos. Math. 142 (2006), 14511458.
[Bad10] Badzyan, A. I., The Euler–Kronecker constant , Mat. Zametki 87 (2010), 3547.
[BL04] Birkenhake, C. and Lange, H., Complex abelian varieties, Grundlehren Math. Wiss., vol. 302 (Springer, Berlin, 2004).
[BG06] Bombieri, E. and Gubler, W., Heights in Diophantine geometry, New Mathematical Monographs, vol. 4 (Cambridge University Press, Cambridge, 2006).
[BLR90] Bosch, S., Lütkebohmert, W. and Raynaud, M., Néron models, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 21 (Springer, Berlin, 1990).
[BMM90] 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), 69105; Séminaire sur les pinceaux de courbes elliptiques (Paris, 1988).
[CU05] Clozel, L. and Ullmo, E., Équidistribution de mesures algébriques , Compos. Math. 141 (2005), 12551309.
[Coh05] Cohen, P. B., Hyperbolic equidistribution problems on Siegel 3-folds and Hilbert modular varieties , Duke Math. J. 129 (2005), 87127.
[Coh07] Cohen, H., Number theory volume II: analytic and modern tools, Graduate Texts in Mathematics, vol. 240 (Springer, New York, 2007).
[Col93] Colmez, P., Périodes des variétés abéliennes à multiplication complexe , Ann. of Math. (2) 138 (1993), 625683.
[Col98] Colmez, P., Sur la hauteur de Faltings des variétés abéliennes à multiplication complexe , Compos. Math. 111 (1998), 359368.
[DM69] Deligne, P. and Mumford, D., The irreducibility of the space of curves of given genus , Inst. Hautes Études Sci. Publ. Math. 36 (1969), 75109.
[Fal83] Faltings, G., Endlichkeitssätze für abelsche Varietäten über Zahlkörpern , Invent. Math. 73 (1983), 349366.
[Fon85] Fontaine, J.-M., Il n’y a pas de variété abélienne sur ℤ , Invent. Math. 81 (1985), 515538.
[Gor97] Goren, E. Z., On certain reduction problems concerning abelian surfaces , Manuscripta Math. 94 (1997), 3343.
[GL06] Goren, E. Z. and Lauter, K. E., Evil primes and superspecial moduli , Int. Math. Res. Not. IMRN 2006 (2006), Art. ID 53864.
[GL07] Goren, E. Z. and Lauter, K. E., Class invariants for quartic CM fields , Ann. Inst. Fourier (Grenoble) 57 (2007), 457480.
[GH78] Griffiths, P. and Harris, J., Principles of algebraic geometry (Wiley-Interscience, New York, 1978).
[Hab15] Habegger, P., Singular moduli that are algebraic units , Algebra Number Theory 9 (2015), 15151524.
[IKO86] Ibukiyama, T., Katsura, T. and Oort, F., Supersingular curves of genus two and class numbers , Compos. Math. 57 (1986), 127152.
[Igu60] Igusa, J., Arithmetic variety of moduli for genus two , Ann. of Math. (2) 72 (1960), 612649.
[ID02] Iwaniec, H., Duke, W. and Friedlander, J. B., The subconvexity problem for Artin L-functions , Invent. Math. 149 (2002), 489577.
[IK04] Iwaniec, H. and Kowalski, E., Analytic number theory, American Mathematical Society Colloquium Publications, vol. 53 (American Mathematical Society, Providence, RI, 2004).
[dJN91] 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), 177192.
[Kli90] Klingen, H., Introductory lectures on Siegel modular forms, Cambridge Studies in Advanced Mathematics, vol. 20 (Cambridge University Press, Cambridge, 1990).
[Liu93] Liu, Q., Courbes stables de genre 2 et leur schéma de modules , Math. Ann. 295 (1993), 201222.
[Liu94] Liu, Q., Conducteur et discriminant minimal de courbes de genre 2 , Compos. Math. 94 (1994), 5179.
[Liu02] 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.
[Loc94] Lockhart, P., On the discriminant of a hyperelliptic curve , Trans. Amer. Math. Soc. 342 (1994), 729752.
[MV10] Michel, P. and Venkatesh, A., The subconvexity problem for GL2 , Publ. Math. Inst. Hautes Études Sci. 111 (2010), 171271.
[MB01] Moret-Bailly, L., Problèmes de Skolem sur les champs algébriques , Compos. Math. 125 (2001), 130.
[Mum84] Mumford, D., Tata lectures on theta. II, Progress in Mathematics, vol. 43 (Birkhäuser, Boston, MA, 1984).
[NT91] Nakkajima, Y. and Taguchi, Y., A generalization of the Chowla–Selberg formula , J. Reine Angew. Math. 419 (1991), 119124.
[NU73] Namikawa, Y. and Ueno, K., The complete classification of fibres in pencils of curves of genus two , Manuscripta Math. 9 (1973), 143186.
[Neu99] Neukirch, J., Algebraic number theory, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 322 (Springer, Berlin, 1999).
[Obu13] Obus, A., On Colmez’s product formula for periods of CM-abelian varieties , Math. Ann. 356 (2013), 401418.
[Paz12a] Pazuki, F., Theta height and Faltings height , Bull. Soc. Math. France 140 (2012), 1949.
[Paz12b] Pazuki, F., Décompositions en hauteurs locales, Preprint, 2012, arXiv:1205.4525.
[Paz13] Pazuki, F., Minoration de la hauteur de Néron–Tate sur les surfaces abéliennes , Manuscripta Math. 142 (2013), 6199.
[PT13] Pila, J. and Tsimerman, J., The André–Oort conjecture for the moduli space of abelian surfaces , Compos. Math. 149 (2013), 204216.
[PT14] Pila, J. and Tsimerman, J., Ax–Lindemann for A g , Ann. of Math. (2) 179 (2014), 659681.
[Sai88] Saito, T., Conductor, discriminant, and the Noether formula of arithmetic surfaces , Duke Math. J. 57 (1988), 151173.
[Sai89] Saito, T., The discriminants of curves of genus 2 , Compos. Math. 69 (1989), 229240.
[Sch03] Schoof, R., Abelian varieties over cyclotomic fields with good reduction everywhere , Math. Ann. 325 (2003), 413448.
[ST68] Serre, J.-P. and Tate, J. T., Good reduction of abelian varieties , Ann. of Math. (2) 88 (1968), 492517.
[Shi97] Shimura, G., Abelian varieties with complex multiplication and modular functions, Princeton Mathematical Series, vol. 46 (Princeton University Press, Princeton, NJ, 1997).
[Szp85] Szpiro, L. (ed.), Séminaire sur les pinceaux arithmétiques: la conjecture de Mordell, Astérisque, vol. 127 (Société Mathématique de France, Paris, 1985).
[Uen88] Ueno, K., Discriminants of curves of genus 2 and arithmetic surfaces , in Algebraic geometry and commutative algebra, Vol. II (Kinokuniya, Tokyo, 1988), 749770.
[vdG88] 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).
[vWa99a] van Wamelen, P., Examples of genus two CM curves defined over the rationals , Math. Comp. 68 (1999), 307320.
[vWa99b] van Wamelen, P., Proving that a genus 2 curve has complex multiplication , Math. Comp. 68 (1999), 16631677.
[Voj99] Vojta, P., Integral points on subvarieties of semiabelian varieties. II , Amer. J. Math. 121 (1999), 283313.
[Was82] Washington, L. C., Introduction to cyclotomic fields, Graduate Texts in Mathematics, vol. 83 (Springer, New York, 1982).
[Yan10] Yang, T., The Chowla–Selberg formula and the Colmez conjecture , Canad. J. Math. 62 (2010), 456472.
[Zha05] Zhang, S., Equidistribution of CM-points on quaternion Shimura varieties , Int. Math. Res. Not. IMRN 2005 (2005), 36573689.
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Bad reduction of genus $2$ curves with CM jacobian varieties

  • Philipp Habegger (a1) and Fabien Pazuki (a2) (a3)


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