Journal of the Mathematical Society of Japan
- J. Math. Soc. Japan
- Volume 69, Number 2 (2017), 597-671.
Dimension formulas of paramodular forms of squarefree level and comparison with inner twist
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.
J. Math. Soc. Japan Volume 69, Number 2 (2017), 597-671.
First available in Project Euclid: 20 April 2017
Permanent link to this document
Digital Object Identifier
Primary: 11F46: Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms
Secondary: 11F66: Langlands $L$-functions; one variable Dirichlet series and functional equations 11F72: Spectral theory; Selberg trace formula
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