[Paper Review] Voronoi summation for ${ m GL}_n$: collusion between level and modulus

Andrew Corbett|arXiv (Cornell University)|Jul 2, 2018

Advanced Algebra and Geometry1 references3 citations

TL;DR

This paper establishes the most general Vorono"{i} summation formula for $Γν$-automorphic forms in the level aspect for $n \geq 2$, resolving long-standing challenges in the joint ramification of level and modulus at primes. It extends prior work by Ichino–Templier through a novel $p$-adic Bessel transform derived from local functional equations, enabling precise control over Fourier coefficients in arithmetic progressions and dual sums with cancellation via explicit Bessel transforms.

ABSTRACT

We investigate the Voronoi summation problem for ${ m GL}_n$ in the level aspect for $n\geq 2$. Of particular interest are those primes at which the level and modulus are jointly ramified - a common occurrence in analytic number theory when using techniques such as the Petersson trace formula. Building on previous legacies, our formula stands as the most general of its kind; in particular we extend the results of Ichino-Templier. We also give (classical) refinements of our formula and study the $p$-adic generalisations of the Hankel transform.

Motivation & Objective

• To resolve the lack of a general Vorono"{i} summation formula for $Γν$ in the level aspect when level and modulus are jointly ramified at primes.
• To extend the results of Ichino–Templier by constructing a fully general formula valid for all $n \geq 2$ and arbitrary level $N$.
• To provide classical refinements of the summation formula, including a choice of test vector that simplifies the Bessel transform and a formulation for sums over arithmetic progressions.
• To develop $p$-adic generalizations of the Hankel transform via local functional equations for $Γν \times \u0393Δ$ $L$-functions.
• To overcome the obstruction from the lack of Atkin–Lehner theory for $n > 2$ by analyzing dual summands locally through $p$-adic transforms.

Proposed method

• Derives a general Vorono"{i} summation formula for $\u0393ν$-automorphic forms using the Whittaker model and local functional equations at all places, including $p$-adic ones.
• Constructs a $p$-adic Bessel transform from the local functional equation of $\u0393ν \times \u0393Δ$ $L$-functions at primes dividing the level $N$, enabling control over joint ramification.
• Applies the Chinese remainder theorem and inductive change of variables to unwrap hyper-Kloosterman sums into classical Kloosterman sums with explicit exponential terms.
• Introduces a refined test vector choice that simplifies the Bessel transform and allows for a clean formulation of the summation in arithmetic progressions.
• Uses the $p$-adic transform to analyze the dual sum via local components $\mathcal{B}_{\pi_p, \Phi_p^{a/\ell q}}(\gamma)$, ensuring non-vanishing only when $\gamma_0$ lies in a specific rational lattice.
• Establishes a precise duality between the original sum over $m$ and a dual sum over $d_{n-1}, \dots, d_2, m$ with coefficients involving $\operatorname{KL}_{n-1}$, $\chi$, and the Bessel transforms.

Experimental results

Research questions

• RQ1 How can the Vorono"{i} summation formula for $\u0393ν$ be extended to the full level aspect when level and modulus are jointly ramified at primes?
• RQ2 What is the structure of the Bessel transform in the $p$-adic setting when both level and modulus are ramified?
• RQ3 Can the general formula be refined to yield a classical formulation with explicit test vectors and arithmetic progression sums?
• RQ4 How do the local components $\mathcal{B}_{\pi_p, \Phi_p^{a/\ell q}}(\gamma)$ behave under joint ramification, and what constraints do they impose on the dual sum?
• RQ5 What is the precise relationship between the original sum over $m$ and the dual sum over $d_{n-1}, \dots, d_2, m$ in terms of Kloosterman sums and Bessel transforms?

Key findings

• The paper establishes Theorem 1.1, a fully general Vorono"{i} summation formula for $\u0393ν$-automorphic forms in the level aspect for $n \geq 2$, valid for arbitrary level $N$ and modulus $M$, with joint ramification at primes.
• The dual sum is expressed as a sum over $m$ with $\gcd(m, MN) = 1$, involving a classical Kloosterman sum $\operatorname{KL}_{n-1}(\bar{a}\ell\gamma_0, m; q, c, d)$, where $a\bar{a} \equiv 1 \pmod{q}$.
• The Bessel transform is explicitly computed via a $p$-adic generalization derived from the local functional equation, yielding a precise expression involving $\mathcal{B}_{\pi_p, \Phi_p^{a/\ell q}}(\gamma)$ for $p \mid MN$.
• A classical refinement (Corollary 1.4) provides a simplified Bessel transform for a specific test vector choice, enabling cleaner analysis of Fourier coefficients.
• A second refinement (Corollary 1.6) formulates the summation in arithmetic progressions, with the dual sum indexed by $m$ satisfying $m \equiv 1 \pmod{N}$ and $\gcd(m, MN) = 1$, and involving a character twist $\chi(\bar{m} \cdot \text{denom})^{-1}$.
• The final formula (equation 41) expresses the original sum as a dual sum over $m$, $r$, $d_{n-1}, \dots, d_2$, with the dual summands involving $\operatorname{KL}_{n-1}$, $\chi$, and the Bessel transforms, with $\lambda_\ell = [\ell, N]\ell^{n-1}L^n$ and $L$ the largest square-free divisor of $MN$.