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COMPUTATIONS OF SPACES OF PARAMODULAR FORMS OF GENERAL LEVEL
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 Title & Authors
COMPUTATIONS OF SPACES OF PARAMODULAR FORMS OF GENERAL LEVEL
Breeding, Jeffery II; Poor, Cris; Yuen, David S.;
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 Abstract
This article gives upper bounds on the number of Fourier-Jacobi coefficients that determine a paramodular cusp form in degree two. The level N of the paramodular group is completely general throughout. Additionally, spaces of Jacobi cusp forms are spanned by using the theory of theta blocks due to Gritsenko, Skoruppa and Zagier. We combine these two techniques to rigorously compute spaces of paramodular cusp forms and to verify the Paramodular Conjecture of Brumer and Kramer in many cases of low level. The proofs rely on a detailed description of the zero dimensional cusps for the subgroup of integral elements in each paramodular group.
 Keywords
paramodular;theta block;Fourier-Jacobi;
 Language
English
 Cited by
 References
1.
H. Aoki, Estimating Siegel modular forms of genus 2 using Jacobi forms, J. Math. Kyoto Univ. 40 (2000), no. 3, 581-588.

2.
S. Bocherer and T. Ibukiyama, Surjectivity of Siegel ${\phi}$-operator for square free level and small weight, Ann. Inst. Fourier (Grenoble) 62 (2012), no. 1, 121-144. crossref(new window)

3.
J. Breeding, C. Poor, and D. S. Yuen, Author's website, http://jeffbreeding.com/para_jacobi.

4.
A. Brumer and K. Kramer, Paramodular abelian varieties of odd conductor, Trans. Amer. Math. Soc. 366 (2014), no. 5, 2463-2516.

5.
J. Brunier, Vector valued formal Fourier-Jacobi expansions, arXiv:1303.3699.2013.

6.
M. Eichler, Uber die Anzahl der linear unabhangigen Siegelschen Modulformen von gegebenem Gewicht, Math. Ann. 312 (1975), 281-291.

7.
M. Eichler and D. Zagier, The Theory of Jacobi Forms, Progress in Mathematics 55, Birkhauser Verlag, Berlin 1985.

8.
V. Gritsenko, Irrationality of the moduli spaces of polarized abelian surfaces, Internat. Math. Res. Notices 6 (1994), 9 pages.

9.
V. Gritsenko, Arithmetical lifting and its applications, Number theory (Paris 1992-1993), 103-126, London Math. Soc. Lecture Note Ser., 215, Cambridge Univ. Press, Cambridge, 1995.

10.
V. Gritsenko and K. Hulek, Minimal Siegel modular threefolds, Math. Proc. Cambridge Philos. Soc. 123 (1998), no. 3, 461-485. crossref(new window)

11.
V. Gritsenko and K. Hulek, The modular form of the Barth-Nieto quintic, Internat. Math. Res. Notices 17 (1999), no. 17, 913-937.

12.
V. Gritsenko and V. Nikulin, Automorphic Forms and Lorentzian Kac-Moody Algebras. II, Internat. J. Math. 9 (1998), no. 2, 201-275. crossref(new window)

13.
V. Gritsenko, N. Skoruppa, and D. Zagier, Theta blocks, preprint.

14.
M. Gross and S. Popescu, Calabi-Yau three-folds and moduli of abelian surfaces II, Trans. Amer. Math. Soc. 363 (2011), no. 7, 3573-3599. crossref(new window)

15.
E. Freitag, Siegelsche Modulfunktionen, Grundlehren der mathematischen Wissenschaften, Band 254, Berlin-Heidelberg-New York: Springer-Verlag, 1983.

16.
K. Hashimoto and T. Ibukiyama, On Relations of Dimensions of Automorphic Forms of Sp(2,${\mathbb{R}}$) and Its Compact Twist Sp(2) (II), Automorphic forms and number theory (Sendai 1983), 31-102, Adv. Stud. Pure Math., 7, North-Holland, Amsterdam, 1985.

17.
T. Ibukiyama, On symplectic Euler factors of genus two, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 30 (1984), no. 3, 587-614.

18.
T. Ibukiyama, On relations of dimensions of automorphic forms of Sp(2,${\mathbb{R}}$) and its compact twist Sp(2). I., Automorphic forms and number theory (Sendai 1983), 7-30, Adv. Stud. Pure Math., 7, North-Holland, Amsterdam, 1985.

19.
T. Ibukiyama, Dimension formulas of Siegel modular forms of weight 3 and supersingular abelian surfaces, Siegel Modular Forms and Abelian Varieties, 39-60, Proceedings of the 4-th Spring Conference on Modular Forms and Related Topics, 2007.

20.
T. Ibukiyama and H. Kitayama, Dimensions of paramodular forms of squarefree level and comparison with inner twist, in preparation.

21.
T. Ibukiyama, C. Poor, and D. Yuen, Jacobi forms that characterize paramodular forms, Abh. Math. Semin. Univ. Hambg. 83 (2013), no. 1, 111-128. crossref(new window)

22.
H. Iwaniec, On the problem of Jacobsthal, Demonstratio Math. 11 (1978), no. 1, 225-231.

23.
D. Mumford, Tata Lectures on Theta I, Progress in Mathematics 28, Birkhauser, Boston 1983.

24.
C. Poor and D. Yuen, Estimates for dimensions of spaces of Siegel modular cusp forms, Abh. Math. Sem. Univ. Hamburg 66 (1996), 337-354. crossref(new window)

25.
C. Poor and D. Yuen, Linear dependence among Siegel modular forms, Math. Ann. 318 (2000), no. 2, 205-234. crossref(new window)

26.
C. Poor and D. Yuen, The Extreme Core, Abh. Math. Sem. Univ. Hamburg 75 (2005), 51-75. crossref(new window)

27.
C. Poor and D. Yuen, The Berge-Martinet constant and slopes of Siegel cusp forms, Bull. London Math. Soc. 38 (2006), no. 6, 913-924. crossref(new window)

28.
C. Poor and D. Yuen, Computations of spaces of Siegel modular cusp forms, J. Math. Soc. Japan 59 (2007), no. 1, 185-222. crossref(new window)

29.
C. Poor and D. Yuen, Dimensions of Cusp Forms for ${\Gamma}_0$(p) in Degree Two and Small Weights, Abh. Math. Sem. Univ. Hamburg 77 (2007), 59-80. crossref(new window)

30.
C. Poor and D. Yuen, The cusp structure of the paramodular groups for degree two, J. Korean Math. Soc. 50 (2013), no. 2, 445-464.

31.
C. Poor and D. Yuen, Paramodular cusp forms, Math. Comp. 84 (2015), no. 293, 1401-1438.

32.
C. Poor and D. Yuen, Paramodular newforms via Borcherds products, in preparation.

33.
M. Raum, Formal Fourier-Jacobi expansions and special cycles of codimension 2, arXiv:1302.0880v2 [math.NT].

34.
H. Reefschlager, Berechnung der Anzahl der 1-Spitzen der Paramodularen Gruppen 2-ten Grades, Dissertation, Georg-August-Universitat zu Gottingen, 1973.

35.
I. Satake, Compactification de espaces quotients de Siegel II, Seminaire Cartan 10 (1957/58), no. 2, 1-10.

36.
N.-P. Skoruppa and D. Zagier, A trace formula for Jacobi forms, J. Reine Angew. Math. 393 (1989), 168-198.