A Subconvex Metaplectic Prime Geodesic Theorem and the Shimura Correspondence
Ikuya Kaneko
TL;DR
This work advances the prime geodesic theorem by treating weight-varying and metaplectic settings with a vector-valued Selberg trace framework. It upper-bounds error terms through subconvex refinements, achieving $oldsymbol{oldsymbol{ u}}_{1}^{(2)}\le \tfrac34-\tfrac{|k|}{4}$ in the $2$-dimensional case and, in the cubic metaplectic setting, obtaining unconditional $oldsymbol{oldsymbol{ u}}_{1}^{(3)}\le \tfrac{25}{21}$, while revealing a Shimura-type correspondence bridging trivial and nontrivial multipliers. The results hinge on a spectral explicit formula driven by the largest residual Laplace eigenvalue and are extended to vector-valued multiplier systems, enabling a direct metaplectic analogue of the Shimura correspondence without Kohnen’s plus-space restrictions. The paper thus connects half-integral weight phenomena (via the 3-fold theta multiplier) and cubic Kubota character theory to sharpen error terms in the prime geodesic counting, with implications for optimal exponents and arithmetic explicit formulas in higher metaplectic covers.
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.