The $n^{th}$ centered moments of a large orthogonal family of automorphic $L$-functions

TL;DR

We study $n$-th centered moments of the one-level density for the orthogonal family of $L$-functions attached to holomorphic Hecke newforms of level $q$, averaged over $q \sim Q$. Under GRH and with test functions whose Fourier transforms have total support $\sum_i \widehat{\Phi}_i$ in $(-4,4)$, we prove that the limit of the $n$-th moment matches the random matrix theory prediction $C(n)$ for the orthogonal group. A key novelty is the appearance of off-diagonal main terms and a phantom continuous-spectrum contribution that must be carefully extracted and canceled to achieve agreement with $C(n)$; this necessitates a precise combinatorial sieve and Kuznetsov–Petersson analysis. The paper provides an explicit integral representation for $C(n)$ in terms of the $\Phi_i$ and establishes a detailed correspondence with the corresponding random matrix calculation, advancing understanding of symmetry types and high-order non-vanishing phenomena in families of $L$-functions.

Abstract

We obtain the $n$th centered moments of one level densities of a large orthogonal family of $L$-functions associated with holomorphic Hecke newforms of level $q$, averaged over $q\sim Q$. We verify the Katz-Sarnak conjecture for these statistics, in the range where the sum of the supports of the Fourier transforms of test functions lies in $(-4, 4)$. In so doing, we need to understand certain phantom oversized terms, which allow us to extract the right off-diagonal contributions. We further need to resolve the combinatorial problem that arises when matching our main terms with random matrix predictions.

Theorems & Definitions (92)

• Theorem 1.1
• Definition 1
• Theorem 1.2
• Corollary 1.3
• Remark 1
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5