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Anton Deitmar

Publications and source records attributed to Anton Deitmar.

At least 73 records · Page 4Linked to original sources
arXiv2004

The Lewis Correspondence for submodular groups

The Lewis Correspondence attaching a unique "period function" to each Maass wave form for SL(2,Z) is extended to subgroups of finite index. This is achieved by using vector-valued forms.

math.NT↗
arXiv2003

A discrete Lefschetz formula

A Lefschetz formula is given that relates loops in a regular finite graph to traces of a certain representation. As an application the poles of the Ihara/Bass zeta function are expressed as dimensions of global section spaces of locally constant sheaves.

math.NT↗
arXiv2002

A dynamical Lefschetz trace formula for algebraic Anosov diffeomorphisms

For algebraic Anosov diffeomorphisms we first express the reduced leafwise cohomology with respect to the unstable foliation in terms of finite dimensional Lie algebra cohomology. We then prove a dynamical Lefschetz trace formula for the induced action on this cohomology using a result of Nomizu. We also consider generalized Anosov maps for which the cohomology in question is no longer finite dimensional in general. Using the representation theory of nilpotent Lie groups we still arrive at a meaningful definition of traces on cohomology. Again we establish a dynamical Lefschetz trace formula.

math.DS↗
arXiv2002

Euler-Poincar'e functions

A new construction of Euler-Poincaré functions for real reductive groups is given. This construction also works for non-connected groups and representations that do not lift.

math.RT↗
arXiv2001

Mellin transforms of p-adic Whittaker functions

It is shown that Mellin transforms of p-adic Whittaker functions exist for generic characters. For good choices of vectors they are rational functions. For class one vectors they can be calculated explicitly. It turns out that they are automorphic L-factors times normalization factors.

math.NT↗
arXiv2000

Mellin transform of Whittaker functions

In this note we show that for an arbitrary semisimple Lie group and any admissible irreducible Banach representation the Mellin transforms of Whittaker functions extend to meromorphic functions. We locate the possible poles.

math.NT↗
arXiv2000

Differential operators on equivariant vector bundles over symmetric spaces

Generalizing the algebra of motion-invariant differential operators on a symmetric space we study invariant operators on equivariant vector bundles. We show that the eigenequation is equivalent to the corresponding eigenequation with respect to the larger algebra of all invariant operators. We compute the possible eigencharacters and show that for invariant integral operators the eigencharacter is given by the Abel transform. We show that sufficiently regular operators are surjective, i.e. that equations of the form $Df=u$ are solvable for all $u$.

math.AP↗
arXiv1999

Combinatorial $L^2$-determinants

We show that the zeta function of a regular graph admits a representation as a quotient of a determinant over a $L^2$-determinant of the combinatorial Laplacian.

math.NT↗
arXiv1999

Harmonic Analysis on the quotient $\Q^\times\ltimes\Q\bs\A^1\ltimes \A$

Let $G$ be the semidirect product $\A^1\ltimes \A$ of the adeles and the norm 1 ideles. Let $\Ga$ be the discrete subgroup $\Q^\times\ltimes\Q$. In this paper the trace formula for this setting is established and used to give the complete decomposition of the $G$-representation on $L^2(\Ga\bs G)$. It turns out that every character of the norm-1 idele class group gives a one dimensional isotype and the complement of those consists of one irreducible representation.

math.NT↗
arXiv1999

A Polya-Hilbert operator for automorphic L-functions

We generalize the first part of A. Connes paper (math/9811068) on the zeroes of the Riemann zeta function from a number field $k$ to any simple algebra $M$ over $k$. To a given automorphic representation $π$ of the reductive group $M^\times$ of invertible elements of $M$ we find a Hilbert space $H_π$ and an operator $D_π$ (Polya-Hilbert operator), which is the infinitesimal generator of a canonical flow such that the spectrum of $D_π$ coincides with the purely imaginary zeroes of the function $L(π,\rez{2} +z)$. As a byproduct we get holomorphicity of all automorphic $L$-functions, not only the cuspidal ones.

math.NT↗
arXiv1997

Geometric zeta-functions of locally symmetric spaces

The theory of geometric zeta functions for locally symmetric spaces as initialized by Selberg and continued by numerous mathematicians is generalized to the case of higher rank spaces. We show analytic continuation, describe the divisor in terms of tangential cohomology and in terms of group cohomology which generalizes the Patterson conjecture. We also extend the range of zeta functions in considering higher dimensional flats.

dg-ga↗