[Paper Review] Determination of $GL(3)$ Hecke-Maass forms from twisted central values

Ritabrata Munshi, Jyoti Sengupta|arXiv (Cornell University)|Jan 30, 2014

Analytic Number Theory Research11 references3 citations

TL;DR

This paper establishes that two $GL(3)$ Hecke-Maass cusp forms for $SL(3,\mathbb{Z})$ are identical if their twisted $L$-functions have equal central values for all primitive Dirichlet characters. Using twisted averages of $L$-functions over factorizable moduli and applying Deligne's bound on hyper-Kloosterman sums, the authors derive an asymptotic formula for the first moment, which implies equality of Fourier coefficients and, via the strong multiplicity one theorem, the identity of the forms.

ABSTRACT

Suppose $π_1$ and $π_2$ are two Hecke-Maass cusp forms for $SL(3,\mathbb{Z})$ such that for all primitive character $χ$ we have $$ L( frac{1}{2},π_1\otimesχ)=L( frac{1}{2},π_2\otimesχ). $$ Then we show that $π_1=π_2$.

Motivation & Objective

• To establish the uniqueness of $GL(3)$ Hecke-Maass cusp forms for $SL(3,\mathbb{Z})$ based on equality of twisted central $L$-values across all primitive characters.
• To extend the classical $GL(2)$ determination result—where central $L$-values of twists determine modular forms—to the higher-rank $GL(3)$ setting.
• To develop a first-moment asymptotic for $GL(3)$ $L$-functions twisted by characters of almost-prime modulus, a key technical challenge in the $GL(3)$ setting.
• To overcome the difficulty of computing twisted moments in $GL(3)$ by using unbalanced approximate functional equations and non-degenerate exponential sums.

Proposed method

• Averaging the twisted central $L$-values over a family of moduli $\mathcal{Q}$ composed of products of two primes in dyadic intervals $Q_1 = Q^{3/4-\delta}$, $Q_2 = Q^{1/4+\delta}$ with $\delta = 1/100$.
• Applying an unbalanced approximate functional equation to express $L(1/2, \pi \otimes \chi)$ as a sum involving $V$-functions and Gauss sums.
• Using Deligne’s bound on hyper-Kloosterman sums and analyzing exponential sums via the non-degeneracy of Laurent polynomials over finite fields.
• Employing the Newton polyhedron method to bound exponential sums $\mathfrak{A}(n; q_1, q_2, q_2')$ and deriving $\ell$-dependent bounds on the twisted average.
• Deriving a main term proportional to $Y = \sum_{q \in \mathcal{Q}} q$ and an error term bounded by $O(\ell^2 Q^{2 - 1/2013 + \varepsilon})$, independent of $\pi$.
• Concluding that equality of twisted central values implies equality of Fourier coefficients $\lambda_\pi(\ell,1)$, and hence $\pi_1 = \pi_2$ via the strong multiplicity one theorem.

Experimental results

Research questions

• RQ1 Can $GL(3)$ Hecke-Maass cusp forms be uniquely determined by the central values of their twists by all primitive Dirichlet characters?
• RQ2 What is the asymptotic behavior of the first moment of $GL(3)$ $L$-functions twisted by characters of factorizable modulus?
• RQ3 How can exponential sums arising in the first moment computation be bounded effectively in the $GL(3)$ setting?
• RQ4 Can the method of twisted averages, adapted from $GL(2)$, be extended to $GL(3)$ despite greater complexity in $L$-function structure and exponential sums?
• RQ5 What conditions on the modulus and character family allow for a non-trivial asymptotic formula in the $GL(3)$ first moment problem?

Key findings

• The central $L$-values of $\pi_1 \otimes \chi$ and $\pi_2 \otimes \chi$ being equal for all primitive characters $\chi$ implies $\pi_1 = \pi_2$, establishing uniqueness via twisted $L$-values.
• An asymptotic formula is proven: $\sum_{q \in \mathcal{Q}} \sum_{\chi \mod q}^\star L(1/2, \pi \otimes \chi)(1 + \chi(-1)) \bar{\chi}(\ell) = \frac{\lambda_\pi(\ell,1)}{\sqrt{\ell}} Y + O(\ell^2 Q^{2 - 1/2013 + \varepsilon})$, with $Y = \sum_{q \in \mathcal{Q}} q$.
• The error term is bounded independently of $\pi$, ensuring uniformity across the family of $GL(3)$ forms.
• The exponential sum $\mathfrak{A}(n; q_1, q_2, q_2')$ is bounded by $O(Q_1^{5/2})$ when $q_1 \nmid n$ and by $O(Q_1^2 (q_1, q_2 - q_2'))$ when $q_1 \mid n$, under non-degeneracy conditions.
• The non-degeneracy of the Laurent polynomial $f(\mathbf{x})$ associated with the exponential sum is verified via Newton polyhedron analysis, enabling the application of the main result from [2].
• The final error bound $\mathfrak{S} \ll Q^{2 - \delta/4 + \varepsilon}$ with $\delta = 1/100$ confirms the asymptotic is strong enough to deduce coefficient equality and hence form identity.