Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

Request Permissions   Purchase Content 
 
 

 

Paramodular cusp forms


Authors: Cris Poor and David S. Yuen
Journal: Math. Comp. 84 (2015), 1401-1438
MSC (2010): Primary 11F46; Secondary 11F50
DOI: https://doi.org/10.1090/S0025-5718-2014-02870-6
Published electronically: August 20, 2014
MathSciNet review: 3315514
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We classify Siegel modular cusp forms of weight two for the paramodular group $ K(p)$ for primes $ p< 600$. We find evidence that rational weight two Hecke eigenforms beyond the Gritsenko lifts correspond to certain abelian surfaces defined over $ \mathbb{Q}$ of conductor $ p$. The arithmetic classification is in the companion article by A. Brumer and K. Kramer, Paramodular abelian varieties of odd conductor. The Paramodular Conjecture, supported by these computations and consistent with the Langlands philosophy and the work of H. Yoshida, Siegel's modular forms and the arithmetic of quadratic forms, is a partial extension to degree two of the Shimura-Taniyama Conjecture. These nonlift Hecke eigenforms share Euler factors with the corresponding abelian variety $ A$ and satisfy congruences modulo $ \ell $ with Gritsenko lifts, whenever $ A$ has rational $ \ell $-torsion.


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

  • [1] A. N. Andrianov, Quadratic Forms and Hecke Operators 254, Springer Verlag, Berlin, 1980.
  • [2] Avner Ash, Paul E. Gunnells, and Mark McConnell, Cohomology of congruence subgroups of $ {\rm SL}(4,\mathbb{Z})$. II, J. Number Theory 128 (2008), no. 8, 2263-2274. MR 2394820 (2009d:11084), https://doi.org/10.1016/j.jnt.2007.09.002
  • [3] Avner Ash, Paul E. Gunnells, and Mark McConnell, Cohomology of congruence subgroups of $ {\rm SL}_4(\mathbb{Z})$. III, Math. Comp. 79 (2010), no. 271, 1811-1831. MR 2630015 (2011e:11095), https://doi.org/10.1090/S0025-5718-10-02331-8
  • [4] 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
  • [5] A. Brumer and K. Kramer, Paramodular abelian varieties of odd conductor, Trans. Amer. Math. Soc. 366 (2014), 2463-2516.
  • [6] Dohoon Choi, YoungJu Choie, and Olav K. Richter, Congruences for Siegel modular forms, Ann. Inst. Fourier (Grenoble) 61 (2011), no. 4, 1455-1466 (2012) (English, with English and French summaries). MR 2951499, https://doi.org/10.5802/aif.2646
  • [7] D. Choi, Y. Choie and T. Kikuta, Bounds for Siegel modular forms of genus 2 modulo $ p$, arXiv:1103.0821v1.
  • [8] Tobias Dern, Paramodular forms of degree 2 and level 3, Comment. Math. Univ. St. Paul. 51 (2002), no. 2, 157-194. MR 1955170 (2003m:11072)
  • [9] Michael Dewar and Olav K. Richter, Ramanujan congruences for Siegel modular forms, Int. J. Number Theory 6 (2010), no. 7, 1677-1687. MR 2740728 (2011m:11099), https://doi.org/10.1142/S179304211000371X
  • [10] Martin Eichler and Don Zagier, The Theory of Jacobi Forms, Progress in Mathematics, vol. 55, Birkhäuser Boston Inc., Boston, MA, 1985. MR 781735 (86j:11043)
  • [11] 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
  • [12] V. Gritsenko and K. Hulek, Commutator coverings of Siegel threefolds, Duke Math. J. 94 (1998), no. 3, 509-542. MR 1639531 (99e:11075), https://doi.org/10.1215/S0012-7094-98-09421-2
  • [13] V. Gritsenko, N.-P. Skoruppa, D. Zagier, Theta Blocks, (in preparation).
  • [14] Ki-ichiro Hashimoto and Tomoyoshi Ibukiyama, On relations of dimensions of automorphic forms of $ {\rm Sp}(2, {\bf R})$ and its compact twist $ {\rm Sp}(2)$. II, Automorphic forms and number theory (Sendai, 1983) Adv. Stud. Pure Math., vol. 7, North-Holland, Amsterdam, 1985, pp. 31-102. MR 876101 (88d:11042)
  • [15] 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)
  • [16] Tomoyoshi Ibukiyama, On relations of dimensions of automorphic forms of $ {\rm Sp}(2,{\bf R})$ and its compact twist $ {\rm Sp}(2)$. I, Automorphic forms and number theory (Sendai, 1983) Adv. Stud. Pure Math., vol. 7, North-Holland, Amsterdam, 1985, pp. 7-30. MR 876100 (88d:11041)
  • [17] T. Ibukiyama and F. Onodera, On the graded ring of modular forms of the Siegel paramodular group of level $ 2$, Abh. Math. Sem. Univ. Hamburg 67 (1997), 297-305. MR 1481544 (98j:11035), https://doi.org/10.1007/BF02940837
  • [18] T. Ibukiyama, email, Private communication to C. Poor, February 24, 2006.
  • [19] T. Ibukiyama, Dimension formulas of Siegel modular forms of weight $ 3$ and supersingular abelian surfaces, Siegel Modular Forms and Abelian Varieties, Proceedings of the 4-th Spring Conference on Modular Forms and Related Topics, 2007, pp. 39-60.
  • [20] Jun-ichi Igusa, On the ring of modular forms of degree two over $ {\bf Z}$, Amer. J. Math. 101 (1979), no. 1, 149-183. MR 527830 (80d:10039), https://doi.org/10.2307/2373943
  • [21] Henryk Iwaniec, Topics in Classical Automorphic Forms, Graduate Studies in Mathematics, vol. 17, American Mathematical Society, Providence, RI, 1997. MR 1474964 (98e:11051)
  • [22] A. Marschner, Paramodular cusp forms of degree two with parcticular emphasis on the level t=5, Thesis (2005), 1-144.
  • [23] Gordon L. Nipp, Quaternary quadratic forms, Springer-Verlag, New York, 1991. Computer generated tables; With a $ 3.5''$ IBM PC floppy disk. MR 1118842 (92h:11030)
  • [24] O. T. O'Meara, Introduction to Quadratic Forms, Springer-Verlag, New York.
  • [25] Cris Poor and David S. Yuen, Linear dependence among Siegel modular forms, Math. Ann. 318 (2000), no. 2, 205-234. MR 1795560 (2001j:11024), https://doi.org/10.1007/s002080000083
  • [26] C. Poor and D. S. Yuen, The extreme core, Abh. Math. Sem. Univ. Hamburg 75 (2005), 51-75. MR 2187578 (2006k:11086), https://doi.org/10.1007/BF02942035
  • [27] Cris Poor and David S. Yuen, Computations of spaces of Siegel modular cusp forms, J. Math. Soc. Japan 59 (2007), no. 1, 185-222. MR 2302669 (2008a:11055)
  • [28] C. Poor and D. S. Yuen, Dimensions of cusp forms for $ \Gamma _0(p)$ in degree two and small weights, Abh. Math. Sem. Univ. Hamburg 77 (2007), 59-80. MR 2379329 (2009b:11087), https://doi.org/10.1007/BF03173489
  • [29] C. Poor and D. Yuen, Authors' website: math.lfc.edu/$ \sim $yuen/paramodular.
  • [30] Nathan C. Ryan and Gonzalo Tornaría, A Böcherer-type conjecture for paramodular forms, Int. J. Number Theory 7 (2011), no. 5, 1395-1411. MR 2825979 (2012j:11109), https://doi.org/10.1142/S1793042111004629
  • [31] Goro Shimura, On the Fourier coefficients of modular forms of several variables, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II 17 (1975), 261-268. MR 0485706 (58 #5528)
  • [32] N.-P. Skoruppa, Über den Zusammenhang zwischen Jacobi-Formen und Modulformen halbganzen Gewichts, Dissertation, Universität Bonn, (1984).
  • [33] Jacob Sturm, On the congruence of modular forms, Number theory (New York, 1984-1985) Lecture Notes in Math., vol. 1240, Springer, Berlin, 1987, pp. 275-280. MR 894516 (88h:11031), https://doi.org/10.1007/BFb0072985
  • [34] J. Tate, A nonbanal cusp form of weight one and level 133, Letter of June 18 to A. Atkin (1974).
  • [35] Ernst Witt, Eine Identität zwischen Modulformen zweiten Grades, Abh. Math. Sem. Hansischen Univ. 14 (1941), 323-337 (German). MR 0005508 (3,163d)
  • [36] Hiroyuki Yoshida, Siegel's modular forms and the arithmetic of quadratic forms, Invent. Math. 60 (1980), no. 3, 193-248. MR 586427 (81m:10051), https://doi.org/10.1007/BF01390016
  • [37] H. Yoshida, On generalization of the Shimura-Taniyama conjecture $ I$ and $ II$, Siegel Modular Forms and Abelian Varieties, Proceedings of the 4-th Spring Conference on Modular Forms and Related Topics, 2007, pp. 1-26.

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2010): 11F46, 11F50

Retrieve articles in all journals with MSC (2010): 11F46, 11F50


Additional Information

Cris Poor
Affiliation: Department of Mathematics, Fordham University, Bronx, New York 10458
Email: poor@fordham.edu

David S. Yuen
Affiliation: Mathematics and Computer Science Department, Lake Forest College, 555 N. Sheridan Road, Lake Forest, Illinois 60045
Email: yuen@lakeforest.edu

DOI: https://doi.org/10.1090/S0025-5718-2014-02870-6
Keywords: Paramodular, Hecke eigenform
Received by editor(s): April 19, 2012
Received by editor(s) in revised form: June 14, 2013, and August 6, 2013
Published electronically: August 20, 2014
Article copyright: © Copyright 2014 American Mathematical Society

American Mathematical Society