arXiv subjects
Publications and source records attributed to Anton Deitmar.
The analogue of the Bruhat-Tits building of a p-adic group in F1-geometry is a single apartment. In this setting, the trace formula gives rise to a several variable zeta function analogously to the p-adic case. The analogy carries on to the fact that the restriction to certain lines yield zeta functions which are defined in geometrical terms. Also, the classical formula of Ihara has an analogue in this setting.
The higher rank Lefschetz formula for p-adic groups is used to prove rationality of a several-variable zeta function attached to the action of a p-adic group on its Bruhat-Tits building. By specializing to certain lines one gets one-variable zeta functions, which then can be related to geometrically defined zeta functions
A new interpretation of zeta functions is given for F1-schemes which do not satisfy Soulé's condition. Functional equations for reductive groups are computed and a new definition of zeta functions attached to more general counting functions is given which is based on regularization and puts on an equal footing F1-theory on the one hand and spectral theory of Laplace operators on manifolds on the other.
Two applications of nets are given. The first is an extension of the Bochner integral to arbitrary locally convex spaces, leading to an integration theorye of more general vector valued functions then in the classical approach by Gelfand and Pettis. The second application starts with the observation that an operator on a Hilbert space is trace class if and only if the net of "principal trace minors" converges. The notion of a "determinant class operator" then is defined as one for which the net of determinantal principal minors converges. It is shown that for a normal operator A this condition coincides with 1-A being trace class.
The holomorphic torsion of a hermitian locally symmetric space is expressed as a special value of a geometric zeta function.
A representation $π$ of a locally compact group $G$ is called \e{trace class}, if for every test function $f$ the induced operator $π(f)$ is a trace class operator. The group $G$ is called \e{trace class}, if every $π\in G$ is trace class. We show that trace class groups are type I and give a criterion for semi-direct products to be trace class and show that a representation $π$ is trace class if and only if $π\otimesπ'$ can be realized in the space of distributions.
For an eigenfunction of the Laplacian on a hyperbolic Riemann surface, the coefficients of the Fourier expansion are described as intertwining functionals. All intertwiners are classified. A refined growth estimate for the coefficients is given and a summation formula is proved.
In this paper a theory of Hecke operators for higher order modular forms is established. The definition of cusp forms and attached L-functions is extended beyond the realm of parabolic invariants. The role of representation theoretic methods is clarified and, motivated by higher order forms, new convolution products of L-functions are introduced.
A new category of algebro-geometric objects is defined. This construction is a vast generalization of existing F1-theories, as it contains the the theory of monoid schemes on the one hand and classical algebraic theory, e.g. Grothendieck schemes, on the the other. It also gives a handy description of Berkovich subdomains and thus contains Berkovich's approach to abstract skeletons. Further it complements the theory of monoid schemes in view of number theoretic applications as congruence schemes encode number theoretical information as opposed to combinatorial data which are seen by monoid schemes.
A multiple Dirichlet series in two variables is constructed as a Mellin transform of a higher order Eisenstein series. It is shown to extend to a meromorphic function and satisfy two independent functional equations.
We deduce from the work of Chen, that the restriction morphism from closed free iterated integrals to closed iterated integrals on loops is onto. We use this to show that the module of higher order invariants of smooth functions is generated by free closed iterated integrals.
It is shown that the space of invariant trilinear forms on smooth representations of a semisimple Lie group is finite dimensional if the group is a product of Lorentz groups.
We prove a prime geodesic theorem for compact quotients of affine buildings and apply it to get class number asymptotics for global fields of positive characteristic.
A condition is given, under which a general lattice point counting function is asymptotic to the corresponding ball volume growth function. This is then used to give height asymptotics in the style of the Batyrev-Manin Conjecture for certain intrinsically defined heights on semisimple groups.
Strong Bochner type integrals with values in locally convex spaces are introduced. It is shown that the strong integral exists in the same cases as the weak (Gelfand-Pettis) integral is known to exist. The strong integral has better continuity properties that the weak integral.
This paper contains a loose collection of remarks on F1-schemes. Etale morphisms and universal coverings are introduced. The relation to toric varieties, at least for integral schemes, is clarified.In this paper it is shown that integral F1-schemes of finite type are essentially the same as toric varieties. A description of the F1-zeta function in terms of toric geometry is given. Etale morphisms and universal coverings are introduced.
In this work we shall generalize the Selberg trace formula to a non-unitary finite-dimensional complex representation $χ:Γ\rightarrow\operatorname{GL}(V)$ of a uniform lattice $Γ$ of a real Lie group $G$.
Modules for sesquiads and congruence schemes are introduced. It is shown that the corresponding categories are belian and that base change functors establish an ascent datum which allows for a cohomology theory to be established.