### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 17 (2021), 083, 40 pages      arXiv:2104.00895      https://doi.org/10.3842/SIGMA.2021.083
Contribution to the Special Issue on Mathematics of Integrable Systems: Classical and Quantum in honor of Leon Takhtajan

### Resolvent Trace Formula and Determinants of $n$ Laplacians on Orbifold Riemann Surfaces

Lee-Peng Teo
Department of Mathematics, Xiamen University Malaysia, Jalan Sunsuria, Bandar Sunsuria, 43900, Sepang, Selangor, Malaysia

Received April 07, 2021, in final form September 05, 2021; Published online September 13, 2021

Abstract
For $n$ a nonnegative integer, we consider the $n$-Laplacian $\Delta_n$ acting on the space of $n$-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 $\Delta_n+s(s+2n-1)$, from which we deduce the regularized determinant of $\Delta_n$, denoted by $\det\!'\Delta_n$. Taking into account the contribution from the absolutely continuous spectrum, $\det\!'\Delta_n$ is equal to a constant $\mathcal{C}_n$ times $Z(n)$ when $n\geq 2$. Here $Z(s)$ is the Selberg zeta function of $X$. When $n=0$ or $n=1$, $Z(n)$ is replaced by the leading coefficient of the Taylor expansion of $Z(s)$ around $s=0$ and $s=1$ respectively. The constants $\mathcal{C}_n$ are calculated explicitly. They depend on the genus, the number of cusps, as well as the ramification indices, but is independent of the moduli parameters.

Key words: determinant of Laplacian; $n$-differentials; cocompact Riemann surfaces; Selberg trace formula.

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References

1. Alekseevskii V.P., On functions similar to the gamma function, Comm. Proc. Kharkov Math. Soc. 1 (1889), 169-238.
2. Andrews G.E., Askey R., Roy R., Special functions, Encyclopedia of Mathematics and its Applications, Vol. 71, Cambridge University Press, Cambridge, 1999.
3. Barnes E.W., The theory of the $G$-function, Q. J. Math. 31 (1900), 264-314.
4. D'Hoker E., Phong D.H., On determinants of Laplacians on Riemann surfaces, Comm. Math. Phys. 104 (1986), 537-545.
5. Efrat I., Determinants of Laplacians on surfaces of finite volume, Comm. Math. Phys. 119 (1988), 443-451.
6. Fay J.D., Fourier coefficients of the resolvent for a Fuchsian group, J. Reine Angew. Math. 293-294 (1977), 143-203.
7. Fischer J., An approach to the Selberg trace formula via the Selberg zeta-function, Lecture Notes in Math., Vol. 1253, Springer-Verlag, Berlin, 1987.
8. Freixas i Montplet G., von Pippich A.-M., Riemann-Roch isometries in the non-compact orbifold setting, J. Eur. Math. Soc. (JEMS) 22 (2020), 3491-3564, arXiv:1604.00284.
9. Gong D.G., Zeta-determinant and torsion functions on Riemann surfaces of finite volume, Manuscripta Math. 86 (1995), 435-454.
10. Gradshteyn I.S., Ryzhik I.M., Table of integrals, series, and products, 6th ed., Academic Press Inc., San Diego, CA, 2000.
11. Hejhal D.A., The Selberg trace formula for ${\rm PSL}(2,\mathbb R)$, Vol. 1, Lecture Notes in Math., Vol. 548, Springer-Verlag, Berlin, 1976.
12. Hejhal D.A., The Selberg trace formula for ${\rm PSL}(2, \mathbb R)$, Vol. 2, Lecture Notes in Math., Vol. 1001, Springer-Verlag, Berlin, 1983.
13. Iwaniec H., Spectral methods of automorphic forms, 2nd ed., Graduate Studies in Mathematics, Vol. 53, Amer. Math. Soc., Providence, RI, 2002.
14. Koyama S., Determinant expression of Selberg zeta functions. I, Trans. Amer. Math. Soc. 324 (1991), 149-168.
15. Koyama S., Determinant expression of Selberg zeta functions. III, Proc. Amer. Math. Soc. 113 (1991), 303-311.
16. McIntyre A., Takhtajan L.A., Holomorphic factorization of determinants of Laplacians on Riemann surfaces and a higher genus generalization of Kronecker's first limit formula, Geom. Funct. Anal. 16 (2006), 1291-1323, arXiv:math.CV/0410294.
17. Sarnak P., Determinants of Laplacians, Comm. Math. Phys. 110 (1987), 113-120.
18. Selberg A., Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series, J. Indian Math. Soc. (N.S.) 20 (1956), 47-87.
19. Selberg A., Göttingen lectures on harmonic analysis, in Alte Selberg Collected Papers, Springer Collected Works in Mathematics, Springer-Verlag, Berlin, 1989, 626-675.
20. Takhtajan L.A., Zograf P.G., A local index theorem for families of $\overline\partial$-operators on punctured Riemann surfaces and a new Kähler metric on their moduli spaces, Comm. Math. Phys. 137 (1991), 399-426.
21. Takhtajan L.A., Zograf P.G., Local index theorem for orbifold Riemann surfaces, Lett. Math. Phys. 109 (2019), 1119-1143, arXiv:1701.00771.
22. Teo L.-P., Ruelle zeta function for cofinite hyperbolic Riemann surfaces with ramification points, Lett. Math. Phys. 110 (2020), 61-82, arXiv:1901.07898.
23. Venkov A.B., Spectral theory of automorphic functions, Proc. Steklov Inst. Math. 153 (1982), 1-163.
24. Venkov A.B., Kalinin V.L., Faddeev L.D., A nonarithmetic derivation of the Selberg trace formula, J. Soviet Math. 8 (1977), 177-199.
25. Voros A., Spectral functions, special functions and the Selberg zeta function, Comm. Math. Phys. 110 (1987), 439-465.