Low-lying zeros of Hilbert modular $L$-functions weighted by powers of central $L$-values

TL;DR

This paper investigates the distribution of low-lying zeros of Hilbert modular L-functions weighted by powers of their central values, confirming predictions from Random Matrix Theory for small powers and proposing a general conjecture.

Contribution

It provides the first rigorous analysis of weighted low-lying zeros for Hilbert modular forms and introduces a conjectural formula for arbitrary weights based on the recipe method.

Findings

For r=1,2,3, the distributions match Random Matrix Theory predictions.

Formulated a conjectural general formula for all r ≥ 1.

Established connections between low-lying zeros and central L-value weights.

Abstract

Let $\mathcal{F}(\textbf{k},\mathfrak{q})$ be the set of primitive Hilbert modular forms of weight $\textbf{k}$ and prime level $\mathfrak{q}$, with trivial central character. We study the one-level density of low-lying zeros of $L(s,\pi)$ weighted by powers of central $L$-values $L(1/2,\pi)^r$, where $\pi$ runs through $\mathcal{F}(\textbf{k},\mathfrak{q})$. For $r=1,2,3$, we show that the resulting distributions $W_r$ match with predictions from Random Matrix Theory. For general $r \geq 1$, we also formulate a conjectural formula for $W_r$ based on the ``recipe'' method.