Doktorarbeit

by Tobias Ebel, Equivariant analytic torsion on hyperbolic Riemann surfaces and the arithmetic Lefschetz trace of an Atkin-Lehner involution on a compact Shimura curve.

Abstract: In this thesis, we compute the equivariant analytic torsion of a Hermitian vector bundle over a hyperbolic Riemann surface given by a factor of automorphy of arbitrary weight and rank in terms of an equivariant Selberg zeta function and derivatives of Lerch's Phi function (Theorem 1.3). We also specialise this result to the case of powers of the canonical bundle (Corollary 1.4). We accomplish this by comparing the functional determinant of the auto morphic Laplacian for a cocompact Fuchsian group with elliptic elements with the completed Selberg zeta function (Theorem 1.1) and employing a Fourier transform argument. As a byproduct, we also compute the ordinary analytic torsion of very ample powers of the canonical bundle (Corollary 1.12). Using Eichler's theory of indefinite rational quaternion algebras, we succeed in computing the equivariant Selberg zeta function (Proposition 2.10) with re spect to an Atkin-Lehner involution acting on a compact Shimura curve. With the help of the moduli interpretation and the generalised Chowla-Selberg for mula (Theorem 2.14), we also manage to compute the height of the fixed point scheme of an Atkin-Lehner involution (Proposition 2.13). Combined with these two results, the arithmetic Lefschetz fixed point for mula of Köhler and Roessler then yields an explicit formula for the arithmetic Lefschetz trace of an Atkin-Lehner involution (Theorem 0.1). Finally we point out a curious identity on arithmetic surfaces of genus two (Proposition 2.18) that can be obtained from a simultaneous application of the arithmetic Lefschetz fixed point theorem and the arithmetic Riemann-Roch theorem of Gillet and Soulé. All results about Shimura curves are illustrated by means of the example of discriminant 26.