## Journal of the Mathematical Society of Japan

### Dimension formulas of paramodular forms of squarefree level and comparison with inner twist

#### Abstract

In this paper, we give an explicit dimension formula for the spaces of Siegel paramodular cusp forms of degree two of squarefree level. As an application, we propose a conjecture on symplectic group version of Eichler–Jacquet–Langlands type correspondence. It is a generalization of the previous conjecture of the first named author for prime levels published in 1985, where inner twists corresponding to binary quaternion hermitian forms over definite quaternion algebras were treated. Our present study contains also the case of indefinite quaternion algebras. Additionally, we give numerical examples of $L$ functions which support the conjecture. These comparisons of dimensions and examples give also evidence for conjecture on a certain precise lifting theory. This is related to the lifting theory from pairs of elliptic cusp forms initiated by Y. Ihara in 1964 in the case of compact twist, but no such construction is known in the case of non-split symplectic groups corresponding to quaternion hermitian groups over indefinite quaternion algebras and this is new in that sense.

#### Article information

Source
J. Math. Soc. Japan Volume 69, Number 2 (2017), 597-671.

Dates
First available in Project Euclid: 20 April 2017

https://projecteuclid.org/euclid.jmsj/1492653641

Digital Object Identifier
doi:10.2969/jmsj/06920597

#### Citation

IBUKIYAMA, Tomoyoshi; KITAYAMA, Hidetaka. Dimension formulas of paramodular forms of squarefree level and comparison with inner twist. J. Math. Soc. Japan 69 (2017), no. 2, 597--671. doi:10.2969/jmsj/06920597. https://projecteuclid.org/euclid.jmsj/1492653641

#### References

• T. Asai, The conjugacy classes in the unitary, symplectic, and orthogonal groups over an algebraic number field, J. Math. Kyoto Univ., 16 (1976), 325–350.
• M. Eichler, Über die darstellbarkeit von Modulformen durch Theta Reihen, J. Reine Angew. Math., 195 (1956), 159–171.
• M. Eichler, Quadratische Formen und Modulformen, Acta arith., 4 (1958), 217–239.
• V. Gritsenko, Arithmetical lifting and its applications, (ed. David, Sinnou), Number Theory, Séminaire de théorie des nombres de Paris, (1992–93), Cambridge University Press. Lond. Math. Soc. Lect. Note Ser., 215 (1995), 103–126.
• K. Hashimoto, On Brandt matrices associated with the positive definite quaternion Hermitian forms, J. Fac. Sci. Univ. Tokyo sect. IA Math., 27 (1980), 227–245.
• K. Hashimoto, The dimension of the spaces of cusp forms on Siegel upper half plane of degree two I, J. Fac. Sci. Univ. Tokyo sect. IA Math., 30 (1983), 403–488.
• K. Hashimoto, The dimension of the spaces of cusp forms on Siegel upper half plane of degree two II, The $\mathbb Q$-rank one case, Math. Ann., 266 (1984), 539–559.
• K. Hashimoto and T. Ibukiyama, On class numbers of positive definite binary quaternion hermitian forms, J. Fac. Sci. Univ. Tokyo sect. IA Math., 27 (1980), 549–601.
• K. Hashimoto and T. Ibukiyama, On class numbers of positive definite binary quaternion hermitian forms (II), J. Fac. Sci. Univ. Tokyo sect. IA Math., 28 (1982), 695–699.
• K. Hashimoto and T. Ibukiyama, On class numbers of positive definite binary quaternion hermitian forms (III), J. Fac. Sci. Univ. Tokyo sect. IA Math., 30 (1983), 393–401.
• 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, Adv. Stud. Pure Math., 7 (1985), 31–102.
• Y. Hirai, On Eisenstein series on quaternion unitary groups of degree 2, J. Math. Soc. Japan, 51 (1999), 93–128.
• T. Ibukiyama, On symplectic Euler factors of genus two, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 30 (1984), 587–614.
• 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, Adv. Stud. Pure Math., 7 (1985), 7–30.
• T. Ibukiyama, Paramodular forms and compact twist, in Automorphic Forms on GSp(4), Proceedings of the 9th Autumn Workshop on Number Theory, (ed. M. Furusawa), 2007, 37–48.
• T. Ibukiyama, Dimension formulas of Siegel modular forms of weight 3 and supersingular abelian surfaces, in Proceedings of the 4-th Spring Conference, Abelian Varieties and Siegel Modular Forms, 2007, 39–60.
• T. Ibukiyama and Y. Ihara, On automorphic forms on the unitary symplectic group $Sp(n)$ and $SL_{2}(R)$, Math. Ann., 278 (1987), 307–327.
• T. Ibukiyama, On some alternating sums of dimensions of Siegel modular forms of general degree and cusp configurations. J. Fac. Sci. Univ. Tokyo Sect. IA Math., 40 (1993), 245–283
• Y. Ihara, On certain arithmetical Dirichlet series, J. Math. Soc. Japan, 16 (1964), 214–225.
• H. Jacquet and R. P. Langlands, Automorphic forms on GL(2), Lecture Notes in Math., 260, Springer, 1972.
• H. Kitayama, An explicit dimension formula for Siegel cusp forms with respect to the non-split symplectic groups, J. Math. Soc. Japan, 63 (2011), 1263–1310.
• H. Kitayama, On the graded ring of Siegel modular forms of degree two with respect to a non-split symplectic group, Internat. J. Math., 23 (2012), 15pp.
• H. Kitayama, On explicit dimension formulas for spaces of Siegel cusp forms of degree two and their applications, Automorphic Forms, (ed. B. Heim, M. Al-Baali, T. Ibukiyama and F. Rupp), Springer Proceedings in Mathematics & Statistics, 115, Springer, New York, 2014, 137–150.
• R. P. Langlands, Problems in the theory of automorphic forms, Lecture Notes in Math., 170, Springer, 1970, 18–61.
• Y. Morita, An explicit formula for the dimension of spaces of Siegel modular forms of degree two, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 21 (1974), 167–248.
• C. Poor, J. Shurman and D. Yuen, Computations with Siegel modular forms and an introduction to the geometry of numbers, in preparation.
• C. Poor and D. Yuen, The cusp structure of the paramodular groups for degree two, J. Korean Math. Soc., 50 (2013), 445–464.
• B. Roberts and R. Schmidt, On modular forms for the paramodular group, Automorphic forms and zeta functions, Proceedings of the Conference in Memory of Tsuneo Arakawa, World Sci. Publ., Hackensack, NJ, 2006, 334–364.
• B. Roberts and R. Schmidt, Local newforms for GSp(4), Lecture Notes in Mathematics, 1918, Springer, Berlin, 2007.
• R. Schmidt, On classical Saito–Kurokawa liftings. J. Reine Angew. Math., 604 (2007), 211–236.
• N.-P. Skoruppa and D. Zagier, Jacobi forms and a certain space of modular forms, Invent. Math., 94 (1988), 113–146.
• T. Sugano, On holomorphic cusp forms on quaternion unitary groups of degree 2, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 31 (1984), 521–568.
• S. Wakatsuki, Dimension formulas for spaces of vector-valued Siegel cusp forms of degree two, J. Number Theory, 132 (2012), 200–253.
• M. Yamauchi, On the traces of Hecke operators for a normalizer of $\Gamma_0(N)$, J. Math. Kyoto Univ., 13 (1973), 403–411.