[Paper Review] On Petersson norms of generic cusp forms and special values of adjoint $L$-functions for ${ m GSp}_4$

Shih-Yu Chen, Atsushi Ichino|arXiv (Cornell University)|Feb 18, 2019

Advanced Algebra and Geometry41 references4 citations

TL;DR

This paper establishes an explicit formula for the Petersson norm of normalized generic cusp forms on $\mathrm{GSp}_4(\mathbb{A}_\mathbb{Q})$ in terms of special values of adjoint $L$-functions and local constants, under the assumption that the archimedean component is either a discrete series or spherical principal series representation. The authors circumvent the difficulty of computing archimedean Whittaker integrals by using endoscopic reduction and the Rallis inner product formula, yielding a fully explicit arithmetic formula for the norm.

ABSTRACT

We prove an explicit formula for the Petersson norms of some normalized generic cuspidal newforms on ${ m GSp}_4$ whose archimedean components belong to either discrete series representations or spherical principal series representations. Our formula expresses the Petersson norms in terms of special values of adjoint $L$-functions and some elementary constants depending only on local representations.

Motivation & Objective

• To derive an explicit formula for the Petersson norm of generic cusp forms on $\mathrm{GSp}_4$ over $\mathbb{Q}$, extending known results for $\mathrm{GL}_n$ and $\mathrm{Mp}_{2n}$.
• To overcome the challenge of computing archimedean Whittaker integrals $\alpha_\infty(f_\infty)$, which are intractable via standard Rankin-Selberg methods.
• To establish a precise link between the Petersson norm and special values of adjoint $L$-functions, valid for both discrete series and spherical principal series archimedean components.
• To provide a concrete arithmetic formula for the norm that is suitable for applications in arithmetic geometry and number theory.

Proposed method

• Use of the Lapid–Mao conjecture as a framework, assuming its validity for $\mathrm{GSp}_4$, and focus on computing the local factor $\alpha_v(f_v)$ at archimedean places.
• Adoption of a reduction to the endoscopic case via the stable base change lift $\pi \to \Pi$ to $\mathrm{GL}_4(\mathbb{A}_\mathbb{Q})$, enabling the use of known inner product formulas.
• Application of the Rallis inner product formula to relate the Petersson norm to special values of $L$-functions via automorphic representations.
• Computation of Whittaker functions at the archimedean place using explicit integral representations and convergence analysis in the $p$-adic and archimedean settings.
• Use of the Siegel-Weil formula and explicit integral evaluations to control convergence and derive uniform bounds in the parameter space.
• Verification of absolute convergence of key integrals in the archimedean case via majorization by exponential and power functions, ensuring the formula's validity.

Experimental results

Research questions

• RQ1 Can the Petersson norm of a generic cusp form on $\mathrm{GSp}_4$ be expressed explicitly in terms of special values of adjoint $L$-functions and local constants?
• RQ2 How can the archimedean Whittaker integral $\alpha_\infty(f_\infty)$, which is notoriously difficult to compute, be bypassed to obtain an explicit norm formula?
• RQ3 What is the precise relationship between the Petersson norm and the adjoint $L$-function $L(s, \pi, \mathrm{Ad})$ for $\mathrm{GSp}_4$ cusp forms with discrete series or spherical principal series archimedean components?
• RQ4 To what extent can the Lapid–Mao conjecture be made explicit for $\mathrm{GSp}_4$ by reducing to endoscopic cases and using the Rallis inner product formula?
• RQ5 What are the convergence properties of the key integrals arising in the inner product formula when the archimedean component is of discrete series or principal series type?

Key findings

• The Petersson norm $\langle f, f \rangle$ of a normalized generic cusp form $f$ on $\mathrm{GSp}_4(\mathbb{A}_\mathbb{Q})$ is expressed as a product of a global $L$-value $L^S(1, \pi, \mathrm{Ad})$, a constant $\Delta_G^S$, and local factors $\alpha_v(f_v)^{-1}$, with $\mathcal{S}_\pi = 1$ for the cases considered.
• For archimedean components of type (DS), the norm is given by $\langle f, f \rangle = C \cdot L(1, \pi, \mathrm{Ad})$, where $C$ is a product of elementary constants depending only on the Blattner parameters $\lambda_1, \lambda_2$, and the normalization of the minimal $\mathrm{U}(2)$-type.
• For archimedean components of type (PS), the norm formula is similarly explicit, with $C$ depending on the complex parameters $\lambda_1, \lambda_2$ and the spherical vector normalization.
• The convergence of all relevant integrals, including those in the archimedean setting, is established uniformly for parameters in compact sets, ensuring the formula's validity across families of representations.
• The authors verify that the key integrals in the Rallis inner product formula converge absolutely for $\mathrm{Re}(s) > 4|\mathrm{Re}(\lambda)| + 4\epsilon - 1$, with uniform bounds in $\lambda$, confirming the analytic continuation of the $L$-function factor.
• The final formula is fully explicit and arithmetic in nature, providing a computable expression for the Petersson norm that can be used in number-theoretic and geometric applications.