An evaluation of the central value of the automorphic scattering determinant

TL;DR

This paper investigates the value of the automorphic scattering determinant at the central point for hyperbolic Riemann surfaces, providing a formula relating it to zeros, poles, and Dirichlet series coefficients.

Contribution

It establishes a precise formula for the central value of the automorphic scattering determinant in terms of its zeros, poles, and Dirichlet series coefficient, addressing a longstanding open problem.

Findings

Derived a formula for (1/2) involving zeros and poles

Connected the sign of (1/2) to the Dirichlet series coefficient

Enhanced understanding of automorphic scattering determinants at the central point

Abstract

Let $M$ be a finite volume, non-compact hyperbolic Riemann surface, possibly with elliptic fixed points, and let $\phi(s)$ denote the automorphic scattering determinant. From the known functional equation $\phi(s)\phi(1-s)=1$ one concludes that $\phi(1/2)^{2} = 1$. However, except for the relatively few instances when $\phi(s)$ is explicitly computable, one does not know $\phi(1/2)$. In this article we address this problem and prove the following result. Let $N$ and $P$ denote the number of zeros and poles, respectively, of $\phi(s)$ in $(1/2,\infty)$, counted with multiplicities. Let $d(1)$ be the coefficient of the leading term from the Dirichlet series component of $\phi(s)$. Then $\phi(1/2)=(-1)^{N+P} \cdot \mathrm{sgn}(d(1))$.