arXiv subjects
Publications and source records attributed to Anton Deitmar.
We give a definition of higher dimensional iterated integrals based on integration over membranes. We prove basic properties of this definition and formulate a conjecture which extends Chen's de Rham Theorem for iterated integrals to the membrane case.
We show that every admissible irreducible representation of a product of two locally compact groups is a tensor product of admissible irreducible representations of the factors.
Higher order cohomology of arithmetic groups is expressed in terms of (g,K)-cohomology. Generalizing results of Borel, it is shown that the latter can be computed using functions of (uniform) moderate growth. A higher order versions of Borel's conjecture is stated, asserting that the cohomology can be computed using automorphic forms.
Higher order group cohomology is defined and first properties are given. Using modular symbols, an Eichler-Shimura homomorphism is constructed mapping spaces of higher order cusp forms to higher order cohomology groups.
Extending ideas of A.N. Parshin and D.V. Osipov, Harmonic Analysis is developed for filtered infinite dimensional modules over a ring. We establish Pontryagin duality, the Fourier inversion formula, Plancherel formula and Poisson summation formula for all dimensions.
The connection between Lefschetz formulae and zeta function is explained. As a particular example the theory of the generalized Selberg zeta function is presented. Applications are given to the theory of Anosov flows and prime geodesic theorems.
It is shown that the modular symbol of a cusp form of weight two has logarithmic growth.
Using the approach of Kurokawa, Ochiai, and Wakayama to 'absolute mathematics' we define a corresponding notion of schemes.
We show that the notion of zeta functions over F1, as given in special cases by Soule', extends naturally to all F1-schemes as defined earlier by the author. We further give two constructions of K-theory for affine schemes or F1-rings, we show that these coincide in the group case, but not in general.
We prove an asymptotic formula for class numbers of totlally imaginary quartic number fields, ie for number fields of degree 4 over Q with only complex embeddings. After previous work for real quadratic fields (Sarnak) and complex cubic fields (Deitmar), this is the last remaining case of number fields with unit rank one.
A prime geodesic theorem is proven for singular geodesics in quotients of SL(4). This is a case where regularity assumptions of previous papers fail. As a consequence, the analysis becomes much more involved. For applications in number theory (class number asymptotics) it is, however, necessary to consider this case, too.
A general Lefschetz formula for the geodesic action on locally symmetric spaces is proven.
The prime geodesic theorem for regular geodesics in a higher rank locally symmetric space is proved. An application to class numbers is given. The proof relies on a Lefschetz formula that is based on work of Andreas Juhl.
In this essay I will give a strictly subjective selection of different types of zeta functions. Instead of providing a complete list, I will rather try to give the central concepts and ideas underlying the theory. This article is going to appear in the collected works of Erich Kähler.
The cuspidal cohomology groups of arithmetic groups in certain infinite dimensional Modules are computed. As a result we get a simultaneous generalization of the Patterson-Conjecture and the Lewis-Correspondence.
A "simple trace formula" is used to derive an asymptotic result for class numbers of complex cubic orders.
Lefschetz formulae for torus actions on p-adic groups are proven.
We prove an approximation to an Ihara-formula for the zeta function of an arithmetic quotient of the Bruhat-Tits building of a p-adic PGL(3).