A Subconvex Metaplectic Prime Geodesic Theorem and the Shimura Correspondence

TL;DR

This paper advances the prime geodesic theorem by achieving subconvex error bounds in the context of metaplectic coverings and the Shimura correspondence, breaking previous barriers and providing new theoretical insights.

Contribution

It introduces subconvex error bounds for the prime geodesic theorem in the setting of higher metaplectic coverings, linking the main term to residual Laplace eigenvalues.

Findings

Breaks the 3/4 barrier for nontrivial multiplier systems.

Provides the first theoretical evidence for the optimal exponent 1+ε with Kubota characters.

Achieves polynomial power-saving in error terms relative to Shimura correspondents.

Abstract

We investigate the prime geodesic theorem with an error term dependent on the varying weight and its higher metaplectic coverings in the arithmetic setting, each admitting subconvex refinements despite the softness of our input. The former breaks the $\frac{3}{4}$-barrier due to Hejhal (1983) when the multiplier system is nontrivial, while the latter represents the first theoretical evidence supporting the prevailing consensus on the optimal exponent $1+\varepsilon$ when the multiplier system specialises to the Kubota character. Our argument relies on the elegant phenomenon that the main term in the prime geodesic theorem is governed by the size of the largest residual Laplace eigenvalue, thereby yielding a simultaneous polynomial power-saving in the error term relative to its Shimura correspondent where the multiplier system is trivial.