Jürg Kramer

COHOMOLOGICAL ARITHMETIC CHOW RINGS

We develop a theory of abstract arithmetic Chow rings, where the role of the fibres at infinity is played by a complex of abelian groups that computes a suitable cohomology theory. As particular cases of this formalism we recover the original arithmetic intersection theory of Gillet and Soule for projective varieties. We introduce a theory of arithmetic Chow groups, which are covariant with respect to arbitrary proper morphisms, and we develop a theory of arithmetic Chow rings using a complex of differential forms with log-log singularities along a fixed normal crossing divisor. This last theory is suitable for the study of automorphic line bundles. In particular, we generalize the classical Faltings height with respect to logarithmically singular hermitian line bundles to higher dimensional cycles. As an application we compute the Faltings height of Hecke correspondences on a product of modular curves.

Cerebral correlates of analogical processing and their modulation by training

There is increasing interest in understanding the neural systems that mediate analogical thinking, which is essential for learning and fluid intelligence. The aim of the present study was to shed light on the cerebral correlates of geometric analogical processing and on training-induced changes at the behavioral and brain level. In healthy participants a bilateral fronto-parietal network was engaged in processing geometric analogies and showed greater blood oxygenation dependent (BOLD) signals as resource demands increased. This network, as well as fusiform and subcortical brain regions, additionally showed training-induced decreases in the BOLD signal over time. The general finding that brain regions were modulated by the amount of resources demanded by the task, and/or by the reduction of allocated resources due to short term training, reflects increased efficiency--in terms of more focal and more specialized brain activation--to more economically process the geometric analogies. Our data indicate a rapid adaptation of the cognitive system which is efficiently modulated by short term training based on a positive correlation of resource demands and brain activation.

Jacobiformen und Thetareihen

We give a characterisation of Jacobi forms by classical modular forms from which we obtain dimension formulas for the spaces of Jacobi forms in certain cases. Then we consider the ordinary theta series to the quaternary quadratic forms of discriminant q2 (q an odd prime) representing 2; these possess a ‘natural’ continuation to Jacobi forms for which we give a sufficient condition of linear independence. If this condition is fulfilled and if there is no cusp form of weight 4 with respect to Γo(q) which vanishes at the cusp 0 with a certain order then the classical theta series are also linear independent.

Bounds on canonical Green's functions

A fundamental object in the theory of arithmetic surfaces is the Green’s function associated to the canonical metric. Previous expressions for the canonical Green’s function have relied on general functional analysis or, when using specific properties of the canonical metric, the classical Riemann theta function. In this article, we derive a new identity for the canonical Green’s function involving the hyperbolic heat kernel. As an application of our results, we obtain bounds for the canonical Green’s function through covers and for families of modular curves.

Uniform Sup-Norm Bounds on Average for Cusp Forms of Higher Weights

Let \(\Gamma \subset \mathrm{ PSL}_{2}(\mathbb{R})\) be a Fuchsian subgroup of the first kind acting by fractional linear transformations on the upper half-plane \(\mathbb{H}\). Consider the d-dimensional space of cusp forms \(\mathcal{S}_{2k}^{\Gamma }\) of weight 2k for \(\Gamma\), and let {f1, …, f d } be an orthonormal basis of \(\mathcal{S}_{2k}^{\Gamma }\) with respect to the Petersson inner product. In this paper we show that the sup-norm of the quantity \(S_{2k}^{\Gamma }(z):=\sum _{ j=1}^{d}\vert f_{j}(z)\vert ^{2}\,\mathrm{Im}(z)^{2k}\) is bounded as \(O_{\Gamma }(k)\) in the cocompact setting, and as \(O_{\Gamma }(k^{3/2})\) in the cofinite case, where the implied constants depend solely on \(\Gamma\). We also show that the implied constants are uniform if \(\Gamma\) is replaced by a subgroup of finite index.

Das Deutsche Zentrum für Lehrerbildung Mathematik – DZLM

Im vorangegangenen Kapitel wurde die Vorgeschichte beschrieben, wie es durch Initiative der Deutsche Telekom Stiftung zur Grundung des DZLM gekommen ist. In diesem zweiten Kapitel werden die Tatigkeitsfelder und Konzepte des DZLM in seiner ersten Forderphase vom 01.10.2011 bis 30.09.2016 beschrieben.

Star products of Green's currents and automorphic forms

In previous work, the authors computed archimedian heights of hermitian line bundles on families of polarized, n-dimensional abelian varieties. In this paper, a detailed analysis of the results obtained in the setting of abelian fibrations is given, and it is shown that the proofs can be modified in such a way that they no longer depend on the specific setting of abelian fibrations and hence extend to a quite general situation. Specifically, we let f : X → Y be any family of smooth, projective, ndimensional complex varieties over some base, and consider a line bundle on X equipped with a smooth, hermitian metric. To this data is associated a hermitian line bundle M on Y characterized by conditions on the first Chern class. Under mild additional hypotheses, it is shown that, for generically chosen sections of L, the integral of the (n + 1)-fold star product of Green’s currents associated to the sections, integrated along the fibers of f , is the log-norm of a global section of M . Furthermore, it is proven that in certain general settings the global section of M can be explicitly expressed in terms of point evaluations of the original sections. A particularly interesting example of this general result appears in the setting of polarized Enriques surfaces when M is a moduli space of degree-2 polarizations. In this setting, the global section constructed via Green’s currents is equal to a power of the -function first studied by R. Borcherds. Additional examples and problems are presented.

The singularities of the invariant metric on the Jacobi line bundle

A theorem by Mumford implies that every automorphic line bundle on a pure open Shimura variety, equipped with an invariant smooth metric, can be uniquely extended as a line bundle on a toroidal compactification of the variety, in such a way that the metric acquires only logarithmic singularities. This result is the key of being able to compute arithmetic intersection numbers from these line bundles. Hence, it is natural to ask whether Mumford’s result remains valid for line bundles on mixed Shimura varieties. In this paper we examine the simplest case, namely the Jacobi line bundle on the universal elliptic curve, whose sections are the Jacobi forms. We will show that Mumford’s result cannot be extended directly to this case and that a new type of singularity appears. By using the theory of b-divisors, we show that an analogue of Mumford’s extension theorem can be obtained. We also show that this extension is meaningful because it satisfies Chern-Weil theory and a Hilbert-Samuel type of formula.

An effective bound for the Huber constant for cofinite Fuchsian groups

Let Γ be a cofinite Fuchsian group acting on hyperbolic two-space ℍ. Let M = Γ\ℍ be the corresponding quotient space. For γ, a closed geodesic of M, let l(γ) denote its length. The prime geodesic counting function π M (u) is defined as the number of Γ-inconjugate, primitive, closed geodesics γ such that e l(γ) 1. π M (u) - 0≤λ M,j ≤1/4 We call the (absolute) constant C M the Huber constant. The objective of this paper is to give an effectively computable upper bound of C M for an arbitrary cofinite Fuchsian group. As a corollary we bound the Huber constant for PSL(2,Z), showing that C M < 16,607,349,020,658 ≈ exp(30.44086643).

Die Riemannsche Vermutung

In dem hier vorzustellenden Millenniumsproblem handelt es sich um eine zahlentheoretische Fragestellung aus dem 19. Jahrhundert, die seit ein paar Jahren auch uberraschende Zusammenhange zu anderen Gebieten der Mathematik und der theoretischen Physik erkennen lasst. Die Problemstellung hat ihren Ursprung bei der Frage nach der Dichte der Primzahlen im Bereich der naturlichen Zahlen. Um den Leser in den Problemkreis einzufuhren, wollen wir einfach beginnen. Wir bezeichnen mit ℕ die Menge der naturlichen Zahlen, d.h.