Deuring’s mass formula of a Mumford family

Sheng, Mao; Zuo, Kang

doi:10.1090/S0002-9947-2015-06312-0

Deuring’s mass formula of a Mumford family
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by Mao Sheng and Kang Zuo PDF

Trans. Amer. Math. Soc. 368 (2016), 169-207 Request permission

Abstract:

We study the Newton polygon jumping locus of a Mumford family in char $p$. Our main result says that, under a mild assumption on $p$, the jumping locus consists of only supersingular points and its cardinality is equal to $(p^r-1)(g-1)$, where $r$ is the degree of the defining field of the base curve of a Mumford family in char $p$ and $g$ is the genus of the curve. The underlying technique is the $p$-adic Hodge theory.

References

Laurent Berger, Représentations $p$-adiques et équations différentielles, Invent. Math. 148 (2002), no. 2, 219–284 (French, with English summary). MR 1906150, DOI 10.1007/s002220100202
Don Blasius, A $p$-adic property of Hodge classes on abelian varieties, Motives (Seattle, WA, 1991) Proc. Sympos. Pure Math., vol. 55, Amer. Math. Soc., Providence, RI, 1994, pp. 293–308. MR 1265557
Christophe Breuil, Groupes $p$-divisibles, groupes finis et modules filtrés, Ann. of Math. (2) 152 (2000), no. 2, 489–549 (French, with French summary). MR 1804530, DOI 10.2307/2661391
Christophe Breuil, Integral $p$-adic Hodge theory, Algebraic geometry 2000, Azumino (Hotaka), Adv. Stud. Pure Math., vol. 36, Math. Soc. Japan, Tokyo, 2002, pp. 51–80. MR 1971512, DOI 10.2969/aspm/03610051
Henri Carayol, Sur la mauvaise réduction des courbes de Shimura, Compositio Math. 59 (1986), no. 2, 151–230 (French). MR 860139
Pierre Deligne, Travaux de Shimura, Séminaire Bourbaki, 23ème année (1970/71), Exp. No. 389, Lecture Notes in Math., Vol. 244, Springer, Berlin, 1971, pp. 123–165 (French). MR 0498581
Pierre Deligne, James S. Milne, Arthur Ogus, and Kuang-yen Shih, Hodge cycles, motives, and Shimura varieties, Lecture Notes in Mathematics, vol. 900, Springer-Verlag, Berlin-New York, 1982. MR 654325, DOI 10.1007/978-3-540-38955-2
Max Deuring, Die Typen der Multiplikatorenringe elliptischer Funktionenkörper, Abh. Math. Sem. Hansischen Univ. 14 (1941), 197–272 (German). MR 5125, DOI 10.1007/BF02940746
Jean-Marc Fontaine and Guy Laffaille, Construction de représentations $p$-adiques, Ann. Sci. École Norm. Sup. (4) 15 (1982), no. 4, 547–608 (1983) (French). MR 707328, DOI 10.24033/asens.1437
Gerd Faltings, Crystalline cohomology and $p$-adic Galois-representations, Algebraic analysis, geometry, and number theory (Baltimore, MD, 1988) Johns Hopkins Univ. Press, Baltimore, MD, 1989, pp. 25–80. MR 1463696
Gerd Faltings, Integral crystalline cohomology over very ramified valuation rings, J. Amer. Math. Soc. 12 (1999), no. 1, 117–144. MR 1618483, DOI 10.1090/S0894-0347-99-00273-8
William Fulton and Joe Harris, Representation theory, Graduate Texts in Mathematics, vol. 129, Springer-Verlag, New York, 1991. A first course; Readings in Mathematics. MR 1153249, DOI 10.1007/978-1-4612-0979-9
E. Z. Goren and F. Oort, Stratifications of Hilbert modular varieties, J. Algebraic Geom. 9 (2000), no. 1, 111–154. MR 1713522
Mark Kisin, Crystalline representations and $F$-crystals, Algebraic geometry and number theory, Progr. Math., vol. 253, Birkhäuser Boston, Boston, MA, 2006, pp. 459–496. MR 2263197, DOI 10.1007/978-0-8176-4532-8_{7}
Mark Kisin, Integral models for Shimura varieties of abelian type, J. Amer. Math. Soc. 23 (2010), no. 4, 967–1012. MR 2669706, DOI 10.1090/S0894-0347-10-00667-3
Stacy G. Langton, Valuative criteria for families of vector bundles on algebraic varieties, Ann. of Math. (2) 101 (1975), 88–110. MR 364255, DOI 10.2307/1970987
Giovanni Di Matteo, On admissible tensor products in $p$-adic Hodge theory, Compos. Math. 149 (2013), no. 3, 417–429. MR 3040745, DOI 10.1112/S0010437X1200070X
Ben Moonen, Models of Shimura varieties in mixed characteristics, Galois representations in arithmetic algebraic geometry (Durham, 1996) London Math. Soc. Lecture Note Ser., vol. 254, Cambridge Univ. Press, Cambridge, 1998, pp. 267–350. MR 1696489, DOI 10.1017/CBO9780511662010.008
D. Mumford, A note of Shimura’s paper “Discontinuous groups and abelian varieties”, Math. Ann. 181 (1969), 345–351. MR 248146, DOI 10.1007/BF01350672
Rutger Noot, Abelian varieties with $l$-adic Galois representation of Mumford’s type, J. Reine Angew. Math. 519 (2000), 155–169. MR 1739726, DOI 10.1515/crll.2000.010
Rutger Noot, On Mumford’s families of abelian varieties, J. Pure Appl. Algebra 157 (2001), no. 1, 87–106. MR 1809220, DOI 10.1016/S0022-4049(99)00174-7
Peter Norman, An algorithm for computing local moduli of abelian varieties, Ann. of Math. (2) 101 (1975), 499–509. MR 389928, DOI 10.2307/1970937
Peter Norman and Frans Oort, Moduli of abelian varieties, Ann. of Math. (2) 112 (1980), no. 3, 413–439. MR 595202, DOI 10.2307/1971152
Arthur Ogus, A $p$-adic analogue of the Chowla-Selberg formula, $p$-adic analysis (Trento, 1989) Lecture Notes in Math., vol. 1454, Springer, Berlin, 1990, pp. 319–341. MR 1094861, DOI 10.1007/BFb0091147
Shankar Sen, Lie algebras of Galois groups arising from Hodge-Tate modules, Ann. of Math. (2) 97 (1973), 160–170. MR 314853, DOI 10.2307/1970879
Mao Sheng, Jiajin Zhang, and Kang Zuo, Higgs bundles over the good reduction of a quaternionic Shimura curve, J. Reine Angew. Math. 671 (2012), 223–248. MR 2983201, DOI 10.1515/crelle.2011.158
J. Tits, Reductive groups over local fields, Automorphic forms, representations and $L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977) Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 29–69. MR 546588
L. Xiao, Tensor being crystalline implies each factor being crystalline up to twist, Appendix to On automorphy of certain Galois representations of $GO_4$-type by T. Liu and J.-K. Yu, 2011.
Eckart Viehweg and Kang Zuo, A characterization of certain Shimura curves in the moduli stack of abelian varieties, J. Differential Geom. 66 (2004), no. 2, 233–287. MR 2106125
Jean-Pierre Wintenberger, Motifs et ramification pour les points d’ordre fini des variétés abéliennes, Séminaire de Théorie des Nombres, Paris 1986–87, Progr. Math., vol. 75, Birkhäuser Boston, Boston, MA, 1988, pp. 453–471 (French). MR 990520