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. Sept. 2006.

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.

by Thomas Ueckerdt, Holomorphic torsion for fibre bundles. Jan. 2014.

Abstract: This thesis is dedicated to develop a generalised product formula for the equivariant holomorphic torsion of a holomorphic, Hermitian line bundle over a certain kind of fibre bundle. Furthermore, we study an example which is given on the one hand, by a holomorphic fibre bundle, consisting of a compact, connected, even-dimensional Lie group modded out by a maximal torus and on the other hand, by flat complex line bundles over this Lie group. In both parts of this thesis, we generalise a non-equivariant result from Stanton.
Take a holomorphic line bundle L over a holomorphic fibre bundle E -> M. There are certain conditions that guaranty a splitting of the Dolbeault-Laplacian on L into a horizontal part and a vertical part. In the first part of this thesis, we show that this splitting sometimes extends to a splitting of the spectral equivariant zeta-function into a part that depends only on the kernel of the horizontal Laplacian, consisting of a sum over various indexes of certain holomorphic vector bundles over M, and a part the depends only on the kernel of the vertical Laplacian. The latter part is given by a sum over equivariant holomorphic torsions of holomorphic vector bundles over M.
For the special case of an admissible action that induces an action on M which has only non-degenerated fixed points, we obtain an even simpler result. This is due to the fact that we can apply the Atiyah-Bott fixed point formula to the sum over the indexes occurring in the first part of the expression for the equivariant holomorphic torsion of L.
In the second part of this thesis, we study the example of the holomorphic fibre bundle, induced by a compact, even-dimensional Lie group G and a maximal torus T therein. We show that for certain flat line bundles over G the theory of the first part is applicable. Let g0 be an element of the universal covering group G0 that covers an element g in G which generates a maximal torus. For the special case of an equivariant action that is essentially given by left multiplication with g0, we obtain an expression for the equivariant holomorphic torsion for the flat line bundle over G that depends only on the roots of G and on the equivariant holomorphic torsions of the line bundle restricted to the maximal torus with g0 induced action.

by Pascal Teßmer, On certain aspects of the asymptotics of the holomorphic torsion forms. Juli 2022.

Abstract: The purpose of this thesis is to investigate the asymptotic behavior of the holomorphic analytic torsion forms and its equivariant version associated with increasing powers p of a given fibrewise positive line bundle. We prove that the asymptotic expansion of the holomorphic analytic torsion forms of degree 2k consists of terms of the form p^{k+n-i}log p, p^{k+n-i}, and local coefficients where n is the complex dimension of the fibres. For the case that when the familiy of vector bundles arise from a principle bundle we give concrete formulas for the first coefficients in the asymptotic of the heat kernel of the curvature of the Bismut superconnection. These results are family versions of the results of Finski. We also study the asymptotic behavior of the equivariant holomorphic analytic torsion forms and generalize a result of Puchol for the equivariant case.

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