A spectral theory of zeta functions of function fields
Abstract
We study Hilbert spaces constructed from the Weil quadratic form associated with the zeta function of a curve over a finite field, together with self-adjoint extensions of the differentiation operator on these spaces. For a distinguished family of these extensions, we prove that the eigenvalues converge to the ordinates of the zeros of the zeta function and that the corresponding entire spectral functions, after normalization, converge locally uniformly to an infinite product determined by these zeros. These limits occur as the interval parameter approaches a finite endpoint, rather than infinity as in the setting of the Riemann zeta function. We also realize the global Hilbert space through the Rosati trace pairing and show that translation by log q is unitarily equivalent to multiplication by normalized Frobenius. This provides a geometric model for the analogous analytic constructions associated with the Riemann zeta function.
Related
- Representation theory and massive spectral zeta functions on spherical space forms
- Zeta functions of quasi-split reductive algebraic groups over local fields
- Cohen-Lenstra heuristics and vanishing of zeta functions for superelliptic curves over finite fields
- p-adic spectral zeta functions via the inverse Stieltjes transform
- Examples of Bessel Functions of Order Zero Related to the Functional Equation of the Riemann Zeta Function
BibTeX
@misc{suzuki2026spectral,
title = {{A spectral theory of zeta functions of function fields}},
author = {Masatoshi Suzuki},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.41115},
eprint = {2610.41115},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.41115},
}
RIS
TY - UNPB
TI - A spectral theory of zeta functions of function fields
AU - Masatoshi Suzuki
PY - 2026
DA - 2026/10/30
PB - arXiv
DO - 10.48550/arxiv.2610.41115
UR - https://arxiv.org/abs/2610.41115
AB - We study Hilbert spaces constructed from the Weil quadratic form associated with the zeta function of a curve over a finite field, together with self-adjoint extensions of the differentiation operator on these spaces. For a distinguished family of these extensions, we prove that the eigenvalues converge to the ordinates of the zeros of the zeta function and that the corresponding entire spectral functions, after normalization, converge locally uniformly to an infinite product determined by these zeros. These limits occur as the interval parameter approaches a finite endpoint, rather than infinity as in the setting of the Riemann zeta function. We also realize the global Hilbert space through the Rosati trace pairing and show that translation by log q is unitarily equivalent to multiplication by normalized Frobenius. This provides a geometric model for the analogous analytic constructions associated with the Riemann zeta function.
ER -