Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Paramodular abelian varieties of odd conductor


Authors: Armand Brumer and Kenneth Kramer
Journal: Trans. Amer. Math. Soc. 366 (2014), 2463-2516
MSC (2010): Primary 11G10; Secondary 14K15, 11F46
DOI: https://doi.org/10.1090/S0002-9947-2013-05909-0
Published electronically: November 21, 2013
MathSciNet review: 3165645
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A precise and testable modularity conjecture for rational abelian surfaces $ A$ with trivial endomorphisms, $ \mathrm {End}_{\mathbb{Q}} A = \mathbb{Z}$, is presented. It is consistent with our examples, our non-existence results and recent work of C. Poor and D. S. Yuen on weight 2 Siegel paramodular forms. We obtain fairly precise information on $ \ell $-division fields of semistable abelian varieties, mainly when $ A[\ell ]$ is reducible, by considering extension problems for group schemes of small rank.


References [Enhancements On Off] (What's this?)

  • [1] V. A. Abrashkin, Galois modules of group schemes of period $ p$ over the ring of Witt vectors, Izv. Akad. Nauk SSSR Ser. Mat. 51 (1987), no. 4, 691-736, 910 (Russian); English transl., Math. USSR-Izv. 31 (1988), no. 1, 1-46. MR 914857 (89a:14062)
  • [2] Anatolij N. Andrianov, Quadratic forms and Hecke operators, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 286, Springer-Verlag, Berlin, 1987. MR 884891 (88g:11028)
  • [3] Christina Birkenhake and Herbert Lange, Complex abelian varieties, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 302, Springer-Verlag, Berlin, 2004. MR 2062673 (2005c:14001)
  • [4] Michael Aschbacher, Finite group theory, Cambridge Studies in Advanced Mathematics, vol. 10, Cambridge University Press, Cambridge, 1986. MR 895134 (89b:20001)
  • [5] Wieb Bosma, John Cannon, and Catherine Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput. 24 (1997), no. 3-4, 235-265. Computational algebra and number theory (London, 1993). MR 1484478, https://doi.org/10.1006/jsco.1996.0125
  • [6] Armand Brumer and Kenneth Kramer, Non-existence of certain semistable abelian varieties, Manuscripta Math. 106 (2001), no. 3, 291-304. MR 1869222 (2003b:14054), https://doi.org/10.1007/PL00005885
  • [7] Armand Brumer and Kenneth Kramer, Semistable abelian varieties with small division fields, Galois theory and modular forms, Dev. Math., vol. 11, Kluwer Acad. Publ., Boston, MA, 2004, pp. 13-37. MR 2059756 (2005m:11106), https://doi.org/10.1007/978-1-4613-0249-0_2
  • [8] Armand Brumer and Kenneth Kramer, Arithmetic of division fields, Proc. Amer. Math. Soc. 140 (2012), no. 9, 2981-2995. MR 2917071, https://doi.org/10.1090/S0002-9939-2012-11500-X
  • [9] A. Brumer and K. Kramer, Pryms of Bielliptic Quartics (in preparation).
  • [10] H. Cohen et al: Tables of number fields. http://pari.math.u-bordeaux1.fr/pub/numberfields
  • [11] B. Conrad, Wild ramification and deformation rings, Unpublished Notes (1999).
  • [12] C. W. Curtis and I. Reiner, Representations of finite groups and associative algebras, John Wiley, 1962.
  • [13] Gerd Faltings, Gisbert Wüstholz, Fritz Grunewald, Norbert Schappacher, and Ulrich Stuhler, Rational points, 3rd ed., Aspects of Mathematics, E6, Friedr. Vieweg & Sohn, Braunschweig, 1992. Papers from the seminar held at the Max-Planck-Institut für Mathematik, Bonn/Wuppertal, 1983/1984; With an appendix by Wüstholz. MR 1175627 (93k:11060)
  • [14] Gerd Faltings and Ching-Li Chai, Degeneration of abelian varieties, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 22, Springer-Verlag, Berlin, 1990. With an appendix by David Mumford. MR 1083353 (92d:14036)
  • [15] Jean-Marc Fontaine, Il n'y a pas de variété abélienne sur $ {\bf Z}$, Invent. Math. 81 (1985), no. 3, 515-538 (French). MR 807070 (87g:11073), https://doi.org/10.1007/BF01388584
  • [16] Valeri Gritsenko, Arithmetical lifting and its applications, Number theory (Paris, 1992-1993) London Math. Soc. Lecture Note Ser., vol. 215, Cambridge Univ. Press, Cambridge, 1995, pp. 103-126. MR 1345176 (96d:11049), https://doi.org/10.1017/CBO9780511661990.008
  • [17] Valeri Gritsenko, Irrationality of the moduli spaces of polarized abelian surfaces, Abelian varieties (Egloffstein, 1993) de Gruyter, Berlin, 1995, pp. 63-84. With an appendix by the author and K. Hulek. MR 1336601 (96e:14022)
  • [18] Valeri Gritsenko and Klaus Hulek, Minimal Siegel modular threefolds, Math. Proc. Cambridge Philos. Soc. 123 (1998), no. 3, 461-485. MR 1607981 (99c:14048), https://doi.org/10.1017/S0305004197002259
  • [19] B. Gross, Letter to J.-P. Serre, June 22, 2010. Private communication.
  • [20] A. Grothendieck, Modèles de Néron et monodromie. Sém. de Géom. 7, Exposé IX, Lecture Notes in Math. 288, New York: Springer-Verlag 1973.
  • [21] Everett W. Howe, Isogeny classes of abelian varieties with no principal polarizations, Moduli of abelian varieties (Texel Island, 1999) Progr. Math., vol. 195, Birkhäuser, Basel, 2001, pp. 203-216. MR 1827021 (2002g:11079)
  • [22] Bertram Huppert and Norman Blackburn, Finite groups. II, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 242, Springer-Verlag, Berlin, 1982. AMD, 44. MR 650245 (84i:20001a)
  • [23] Tomoyoshi Ibukiyama, On symplectic Euler factors of genus two, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 30 (1984), no. 3, 587-614. MR 731520 (85j:11053)
  • [24] Christian U. Jensen, Arne Ledet, and Noriko Yui, Generic polynomials, Mathematical Sciences Research Institute Publications, vol. 45, Cambridge University Press, Cambridge, 2002. Constructive aspects of the inverse Galois problem. MR 1969648 (2004d:12007)
  • [25] Jennifer Johnson-Leung and Brooks Roberts, Siegel modular forms of degree two attached to Hilbert modular forms, J. Number Theory 132 (2012), no. 4, 543-564. MR 2887605, https://doi.org/10.1016/j.jnt.2011.08.004
  • [26] John W. Jones and David P. Roberts, A database of local fields, J. Symbolic Comput. 41 (2006), no. 1, 80-97. MR 2194887 (2006k:11230), https://doi.org/10.1016/j.jsc.2005.09.003
  • [27] C. Khare and J.-P. Wintenberger, Serre's modularity conjecture (I), (II), Inv. Math. 178(3) (2009), 485-504, 505-586.
  • [28] Qing Liu, Modèles entiers des courbes hyperelliptiques sur un corps de valuation discrète, Trans. Amer. Math. Soc. 348 (1996), no. 11, 4577-4610 (French, with English summary). MR 1363944 (97h:11062), https://doi.org/10.1090/S0002-9947-96-01684-4
  • [29] B. Mazur, Modular curves and the Eisenstein ideal, Inst. Hautes Études Sci. Publ. Math. 47 (1977), 33-186 (1978). MR 488287 (80c:14015)
  • [30] J. S. Milne, Abelian varieties, Arithmetic geometry (Storrs, Conn., 1984) Springer, New York, 1986, pp. 103-150. MR 861974
  • [31] J. S. Milne, On the arithmetic of abelian varieties, Invent. Math. 17 (1972), 177-190. MR 0330174 (48 #8512)
  • [32] Laurent Moret-Bailly, Pinceaux de variétés abéliennes, Astérisque 129 (1985), 266 (French, with English summary). MR 797982 (87j:14069)
  • [33] V. V. Nikulin, Integral symmetric bilinear forms and some of their applications, Trans. Moscow Math. Soc. 38 (1980) 71-135.
  • [34] Tadao Oda, The first de Rham cohomology group and Dieudonné modules, Ann. Sci. École Norm. Sup. (4) 2 (1969), 63-135. MR 0241435 (39 #2775)
  • [35] A. M. Odlyzko, Lower bounds for discriminants of number fields. II, Tôhoku Math. J. 29 (1977), no. 2, 209-216. MR 0441918 (56 #309)
  • [36] Takeo Okazaki, Proof of R. Salvati Manni and J. Top's conjectures on Siegel modular forms and abelian surfaces, Amer. J. Math. 128 (2006), no. 1, 139-165. MR 2197070 (2007a:11062)
  • [37] Vincent Pilloni, Modularité, formes de Siegel et surfaces abéliennes, J. Reine Angew. Math. 666 (2012), 35-82 (French, with French summary). MR 2920881, https://doi.org/10.1515/CRELLE.2011.123
  • [38] Harriet Pollatsek, First cohomology groups of some linear groups over fields of characteristic two, Illinois J. Math. 15 (1971), 393-417. MR 0280596 (43 #6316)
  • [39] C. Poor and D. S. Yuen, Paramodular Cusp Forms, arXiv:0912.0049v1 (2009).
  • [40] C. Poor and D. S. Yuen, in preparation. See http://math.lfc.edu/$ \sim $yuen/paramodular.
  • [41] Michel Raynaud, Schémas en groupes de type $ (p,\dots , p)$, Bull. Soc. Math. France 102 (1974), 241-280 (French). MR 0419467 (54 #7488)
  • [42] L. Rédei and H. Reichardt, Die Anzahl der durch 4 teilbaren Invarianten der Klassengruppe eines beliebigen quadratisschen Zahlkörpers, J. reine ang. Math. 170 (1933), 69-74.
  • [43] Kenneth A. Ribet, Endomorphisms of semi-stable abelian varieties over number fields, Ann. Math. (2) 101 (1975), 555-562. MR 0371903 (51 #8120)
  • [44] Kenneth A. Ribet, Galois action on division points of Abelian varieties with real multiplications, Amer. J. Math. 98 (1976), no. 3, 751-804. MR 0457455 (56 #15660)
  • [45] Kenneth A. Ribet, Images of semistable Galois representations, Pacific J. Math. Special Issue (1997), 277-297. Olga Taussky-Todd: in memoriam. MR 1610883 (99a:11065), https://doi.org/10.2140/pjm.1997.181.277
  • [46] Kenneth A. Ribet, Abelian varieties over $ \bf Q$ and modular forms, Modular curves and abelian varieties, Progr. Math., vol. 224, Birkhäuser, Basel, 2004, pp. 241-261. MR 2058653 (2005k:11120), https://doi.org/10.1007/978-3-0348-7919-4_15
  • [47] Brooks Roberts and Ralf Schmidt, On modular forms for the paramodular groups, Automorphic forms and zeta functions, World Sci. Publ., Hackensack, NJ, 2006, pp. 334-364. MR 2208781 (2006m:11065), https://doi.org/10.1142/9789812774415_0015
  • [48] Brooks Roberts and Ralf Schmidt, Local newforms for GSp(4), Lecture Notes in Mathematics, vol. 1918, Springer, Berlin, 2007. MR 2344630 (2008g:11080)
  • [49] David P. Roberts, Twin sextic algebras, Rocky Mountain J. Math. 28 (1998), no. 1, 341-368. MR 1639810 (99j:12001), https://doi.org/10.1216/rmjm/1181071838
  • [50] Riccardo Salvati Manni and Jaap Top, Cusp forms of weight $ 2$ for the group $ \Gamma _2(4,8)$, Amer. J. Math. 115 (1993), no. 2, 455-486. MR 1216438 (94e:11050), https://doi.org/10.2307/2374865
  • [51] René Schoof, Abelian varieties over cyclotomic fields with good reduction everywhere, Math. Ann. 325 (2003), no. 3, 413-448. MR 1968602 (2005b:11076), https://doi.org/10.1007/s00208-002-0368-7
  • [52] René Schoof, Abelian varieties over $ \mathbb{Q}$ with bad reduction in one prime only, Compos. Math. 141 (2005), no. 4, 847-868. MR 2148199 (2006c:11068), https://doi.org/10.1112/S0010437X05001107
  • [53] René Schoof, Semistable abelian varieties with good reduction outside 15, Manuscripta Math. 139 (2012), no. 1-2, 49-70. MR 2959670, https://doi.org/10.1007/s00229-011-0509-y
  • [54] R. Schoof, On the modular curve $ X_0(23)$ in Faber, C. et al., Geom. and Arith., EMS Publishing House, Zürich (to appear).
  • [55] Jean-Pierre Serre, Local fields, Graduate Texts in Mathematics, vol. 67, Springer-Verlag, New York, 1979. Translated from the French by Marvin Jay Greenberg. MR 554237 (82e:12016)
  • [56] J.-P. Serre, Facteurs locaux des fonctions zêta des variétés algébriques. Séminaire Delange-Pisot-Poitou. Théorie des nombres 11, 1969-1970.
  • [57] Jean-Pierre Serre, Sur les représentations modulaires de degré $ 2$ de $ {\rm Gal}(\overline {\bf Q}/{\bf Q})$, Duke Math. J. 54 (1987), no. 1, 179-230 (French). MR 885783 (88g:11022), https://doi.org/10.1215/S0012-7094-87-05413-5
  • [58] Goro Shimura, On analytic families of polarized abelian varieties and automorphic functions, Ann. of Math. (2) 78 (1963), 149-192. MR 0156001 (27 #5934)
  • [59] N. P. Smart, $ S$-unit equations, binary forms and curves of genus $ 2$, Proc. London Math. Soc. (3) 75 (1997), no. 2, 271-307. MR 1455857 (98d:11072), https://doi.org/10.1112/S002461159700035X
  • [60] Michio Suzuki, Group theory. I, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 247, Springer-Verlag, Berlin, 1982. Translated from the Japanese by the author. MR 648772 (82k:20001c)
  • [61] John Tate, Finite flat group schemes, Modular forms and Fermat's last theorem (Boston, MA, 1995) Springer, New York, 1997, pp. 121-154. MR 1638478
  • [62] Richard Taylor, Galois representations associated to Siegel modular forms of low weight, Duke Math. J. 63 (1991), no. 2, 281-332. MR 1115109 (92j:11044), https://doi.org/10.1215/S0012-7094-91-06312-X
  • [63] Richard Taylor, $ l$-adic representations associated to modular forms over imaginary quadratic fields. II, Invent. Math. 116 (1994), no. 1-3, 619-643. MR 1253207 (95h:11050a), https://doi.org/10.1007/BF01231575
  • [64] J. Tilouine, Nearly ordinary rank four Galois representations and $ p$-adic Siegel modular forms, Compos. Math. 142 (2006), no. 5, 1122-1156. With an appendix by Don Blasius. MR 2264659 (2007j:11071), https://doi.org/10.1112/S0010437X06002119
  • [65] J. Tilouine, Siegel varieties and $ p$-adic Siegel modular forms, Doc. Math. Extra Vol. (2006), 781-817. MR 2290605 (2007k:11067)
  • [66] J. Top, Hecke L-series related with algebraic cycles or with Siegel modular forms, thesis, Utrecht, 1989.
  • [67] Lawrence C. Washington, Introduction to cyclotomic fields, Graduate Texts in Mathematics, vol. 83, Springer-Verlag, New York, 1982. MR 718674 (85g:11001)
  • [68] John Wilson, Degrees of polarizations on an abelian surface with real multiplication, Bull. London Math. Soc. 33 (2001), no. 3, 257-264. MR 1817763 (2002c:11062), https://doi.org/10.1017/S0024609301007974
  • [69] André Weil, Zum Beweis des Torellischen Satzes, Nachr. Akad. Wiss. Göttingen. Math.-Phys. Kl. IIa. 1957 (1957), 33-53 (German). MR 0089483 (19,683e)
  • [70] Hiroyuki Yoshida, On Siegel modular forms obtained from theta series, J. Reine Angew. Math. 352 (1984), 184-219. MR 758701 (86e:11036), https://doi.org/10.1515/crll.1984.352.184

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 11G10, 14K15, 11F46

Retrieve articles in all journals with MSC (2010): 11G10, 14K15, 11F46


Additional Information

Armand Brumer
Affiliation: Department of Mathematics, Fordham University, Bronx, New York 10458
Email: brumer@fordham.edu

Kenneth Kramer
Affiliation: Department of Mathematics, Queens College, Flushing, New York 11367 – and – The Graduate Center CUNY, New York, New York 10016
Email: kkramer@gc.cuny.edu

DOI: https://doi.org/10.1090/S0002-9947-2013-05909-0
Keywords: Abelian variety, finite flat group scheme, polarization, division field, paramodular group
Received by editor(s): May 24, 2010
Received by editor(s) in revised form: July 4, 2012
Published electronically: November 21, 2013
Additional Notes: The research of the second author was partially supported by NSF grant DMS 0739346
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society