Resolvent Trace Formula and Determinants of Laplacians on Orbifold Riemann Surfaces

• Veröffentlicht in: Symmetry, Integrability and Geometry: Methods and Applications
• Datum: 2021
• ISSN: 1815-0659
• 1. Verfasser: Teo, Lee-Peng
• Format: Artikel
• Sprache: Englisch
• Veröffentlicht: Інститут математики НАН України 2021
• Online Zugang: https://nasplib.isofts.kiev.ua/handle/123456789/211444
• Назва журналу: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Zitieren: Resolvent Trace Formula and Determinants of Laplacians on Orbifold Riemann Surfaces. Lee-Peng Teo. SIGMA 17 (2021), 083, 40 pages

For a nonnegative integer, we consider the -Laplacian Δₙ acting on the space of -differentials on a confinite Riemann surface X which has ramification points. The trace formula for the resolvent kernel is developed along the line à la Selberg. Using the trace formula, we compute the regularized determinant of Δₙ + ( + 2 − 1), from which we deduce the regularized determinant of Δₙ, denoted by det′Δₙ. Taking into account the contribution from the absolutely continuous spectrum, det′Δₙ is equal to a constant Cₙ times () when ≥ 2. Here () is the Selberg zeta function of . When = 0 or = 1, () is replaced by the leading coefficient of the Taylor expansion of () around = 0 and = 1, respectively. The constants Cn are calculated explicitly. They depend on the genus, the number of cusps, as well as the ramification indices, but are independent of the moduli parameters.