A Fourier-Jacobi Dirichlet series for cusp forms on orthogonal groups

Area of Science:

• Number Theory
• Representation Theory
• Automorphic Forms

Background:

• Cusp forms are fundamental objects in number theory with deep connections to modular forms.
• Dirichlet series and L-functions encode arithmetic information and are crucial in analytic number theory.
• Orthogonal groups and their representations play a significant role in various areas of mathematics.

Purpose of the Study:

• To investigate a Dirichlet series associated with Fourier-Jacobi coefficients of cusp forms for orthogonal groups.
• To establish connections between this Dirichlet series and standard L-functions.
• To derive Euler product expressions for the Dirichlet series under specific conditions.

Main Methods:

• Utilizing Fourier-Jacobi coefficients of cusp forms F and G.
• Analyzing Dirichlet series involving these coefficients for orthogonal groups of signature (2, n+2).
• Applying techniques from the theory of automorphic forms and L-functions.

Main Results:

• Established a connection between the Dirichlet series and the standard L-function for a Hecke eigenform F and a Maass lift G.
• Derived explicit Euler product expressions for the Dirichlet series for certain orthogonal groups.
• Recovered classical results for Siegel modular forms and introduced new examples.

Conclusions:

• The study provides new insights into the structure of Dirichlet series related to cusp forms and orthogonal groups.
• The findings extend existing knowledge on L-functions and their properties.
• The research opens avenues for exploring connections with other types of modular forms, including paramodular, Hermitian, and quaternionic forms.