Transfer of Siegel cusp forms of degree 2

Abstract

Let $\pi$ be the automorphic representation of $\GSp_4(\A)$ generated by a full level cuspidal Siegel eigenform that is not a Saito-Kurokawa lift, and $\tau$ be an arbitrary cuspidal, automorphic representation of $\GL_2(\A)$. Using Furusawa's integral representation for $\GSp_4\times\GL_2$ combined with a pullback formula involving the unitary group $\GU(3,3)$, we prove that the $L$-functions $L(s,\pi \times \tau )$ are "nice". The converse theorem of Cogdell and Piatetski-Shapiro then implies that such representations $\pi$ have a functorial lifting to a cuspidal representation of $\GL_4(\A)$. Combined with the exterior-square lifting of Kim, this also leads to a functorial lifting of $\pi$ to a cuspidal representation of $\GL_5(\A)$. As an application, we obtain analytic properties of various $L$-functions related to full level Siegel cusp forms. We also obtain special value results for $\GSp_4\times\GL_1$ and $\GSp_4\times\GL_2$.

99 pages, with a completely re-written introduction; to appear in Memoirs of the AMS

References in corpus (3)

• Bessel models for lowest weight representations of GSp(4,R)
• On the arithmetic of Shalika models and the critical values of $L$-functions for $\mathrm{GL}(2n)$
• A relation between multiplicity one and Bocherer's conjecture

Cited by in corpus (12)

• On the order of vanishing of newforms at cusps
• Doubling constructions: Global functoriality for non-generic cuspidal representations
• Evaluating $L$-functions with few known coefficients
• On the arithmetic of Shalika models and the critical values of $L$-functions for $\mathrm{GL}(2n)$
• Multiplicity one for $L$-functions and applications
• On fundamental Fourier coefficients of Siegel cusp forms of degree 2
• Algebraicity of $L$-values for ${\text{GSp}}_4\times {\text{GL}}_2$ and ${\text{GSp}}_4\times {\text{GL}}_2\times {\text{GL}}_2$
• Spectral summation formula for GSp(4) and moments of spinor L-functions
• Weighted equidistribution theorem for Siegel modular forms of degree 2
• Non-vanishing and sign changes of Hecke eigenvalues for Siegel cusp forms of genus two (with an Appendix by E. Kowalski and A. Saha)
• Large Hecke eigenvalues and an Omega result for non Saito--Kurokawa lifts
• Linear Relations of Siegel Poincare Series and Nonvanshing of the Central Value of Spinor L-functions