p –adische Zahlen

Im Abschnitt 8 haben wir quadratische Gleichungen im Korper 𝔽p=𝔽/p𝔽gelost.Wie kann man Gleichungen in den Ringen 𝔽/p k𝔽,die ja noch nicht einmal Integritatsringe sind, losen?

A note on the unirationality of a moduli space of double covers

In this note we look at the moduli space $\cR_{3,2}$ of double covers of genus three curves, branched along 4 distinct points. This space was studied by Bardelli, Ciliberto and Verra. It admits a dominating morphism $\cR_{3,2} \to {\mathcal A}_4$ to Siegel space. We show that there is a birational model of $\cR_{3,2}$ as a group quotient of a product of two Grassmannian varieties. This gives a proof of the unirationality of $\cR_{3,2}$ and hence a new proof for the unirationality of ${\mathcal A}_4$.

From motives to differential equations for loop integrals

In this talk we discuss how ideas from the theory of mixed Hodge structures can be used to find differential equations for Feynman integrals. In particular we discuss the two-loop sunrise graph in two dimensions and show that these methods lead to a differential equation which is simpler than the ones obtained from integration-by-parts.

General Set-Up

Die Ringeℤ/nℤ

In diesem Abschnitt wollen wir die Ergebnisse des letzten abstrahieren und vertiefen. Wir starten mit der folgenden offensichtlichen Bemerkung.

The unequal mass sunrise integral expressed through iterated integrals on

We solve the two-loop sunrise integral with unequal masses systematically to all orders in the dimensional regularisation parameter ε. In order to do so, we transform the system of differential equations for the master integrals to an ε-form. The sunrise integral with unequal masses depends on three kinematical variables. We perform a change of variables to standard coordinates on the moduli space M1,3 of a genus one Riemann surface with three marked points. This gives us the solution as iterated integrals on M‾1,3. On the hypersurface τ=const our result reduces to elliptic polylogarithms. In the equal mass case our result reduces to iterated integrals of modular forms.

Applications to Algebraic Cycles: Nori's Theorem

Deligne cohomology is a tool that makes it possible to unify the study of cycles through an object that classifies extensions of ( p , p )-cycles by points in the p -th intermediate Jacobian (which is the target of the Abel–Jacobi map on cycles of codimension p ). This is treated in Section 10.1 with applications to normal functions. Before giving the proof of Nori's theorem in Section 10.6, we need some results from mixed Hodge theory. These are proven in Section 10.2 where we also state different variants of the theorem. Sections 10.3 and 10.4 treat a localto- global principle and an extension of the method of Jacobian representations of cohomology which are both essential for the proof. …

Hodge Theory and Algebraic Cycles

Algebraic cycles and Hodge theory, in particular Chow groups, Deligne cohomology and the study of cycle class maps were some of the themes of the Schwerpunkt ”Globale Methoden in der Komplexen Geometrie”. In this survey we report about several projects around the structure of (higher) Chow groups CH(X,n) [3] which the author has studied with his coauthors during this time by using different methods. In my opinion there are two interesting view points: first the internal structure of higher Chow groups, i.e., the existence of interesting elements and nontriviality of parts of their Bloch-Beilinson filtrations. This case has arithmetic and geometric features, and the groups in question show d…

The Period Isomorphism

The aim of this section is to define well-behaved isomorphisms between singular and de Rham cohomology of algebraic varieties.

Algebraic de Rham Cohomology

Let k be a field of characteristic zero. We are going to define relative algebraic de Rham cohomology for general varieties over k, not necessarily smooth.

Holomorphic de Rham Cohomology

We are going to define a natural comparison isomorphism between algebraic de Rham cohomology and singular cohomology of varieties over the complex numbers with coefficients in \(\mathbb {C}\). The link is provided by holomorphic de Rham cohomology, which we study in this chapter.

Der Satz von Hasse–Minkowski

Dedekinds Untersuchungen zum Zahlbegriff

Der Zahlbegriff steht im Mittelpunkt von Dedekinds Werk. Auch sein Habilitationsvortrag von 1854 und der Briefwechsel mit Cantor, die wir beide nur gestreift haben, zeigen dies recht deutlich. Wir wollen in diesem Kapitel auf seine restlichen Untersuchungen zu den Grundlagen des Zahlbegriffs intensiver eingehen.

Die Struktur der Einheitengruppen Un

Nachdem wir im letzten Abschnitt samtliche endlichen abelschen Gruppen kennengelernt haben, stellt sich naturlich die Frage, welche Struktur die Gruppe Un hat. Wegen des chinesischen Restsatzes in der Form von Lemma 5.13 konnen wir uns auf den Fall n =pr beschranken. Wir werden zeigen, dass alle diese Gruppen Upr zyklisch sind — mit Ausnahme der U 2r fur r≥3.

Miscellaneous Periods: An Outlook

In this chapter, we collect several other important examples of periods in the literature for the convenience of the reader.

Der SFB/Transregio 45 „Perioden, Modulräume und Arithmetik algebraischer Varietäten” der Deutschen Forschungsgemeinschaft

Im Sommer 2007 wurde von der Deutschen Forschungsgemeinschaft der SFB/Transregio 45 „Perioden, Modulraume und Arithmetik algebraischer Varietaten” an den Standorten Bonn, Duisburg-Esen und Mainz (Sprecherhochschule) eingerichtet. Thematisch ist der Transregio im Gebiet der Algebraischen und Arithmetischen Geometrie verankert. In diesem Artikel werden die mathematischen Forschungsthemen und einige strukturelle Aspekte beschrieben.

The Abel–Jacobi map for higher Chow groups

We construct a map between Bloch's higher Chow groups and Deligne homology for smooth, complex quasiprojective varieties on the level of complexes. For complex projective varieties this results in a formula which generalizes at the same time the classical Griffiths Abel–Jacobi map and the Borel/Beilinson/Goncharov regulator type maps.

Wirkungsgeschichte und Positionen der Forschung

In diesem Kapitel wollen wir die Wirkungsgeschichte von Dedekinds Werk beschreiben und Themen und Fragestellungen aus der heutigen Forschung erlautern, die eine Beziehung zu Dedekinds Arbeiten haben.

Erklärung der Texte in heutiger Sprache

Die beiden abgedruckten Texte wollen wir in diesem Kapitel ausfuhrlich kommentieren. Da der kurzere Text Stetigkeit und Irrationale Zahlen vergleichsweise wenige mathematische Details enthalt, fallt seine Inhaltsangabe in Abschnitt 4.1 kurz aus und wir geben zum Ausgleich eine ausfuhrliche Beschreibung des Inhaltes aus heutiger Sicht in Abschnitt 4.2. Im Falle des umfangreicheren Buches Was sind und was sollen die Zahlen? geben wir jedoch nur eine kurze Zusammenfassung aus moderner Sicht in Abschnitt 4.4, da die Inhaltsangabe in Abschnitt 4.3 naturgemas langer ausfallt und wir darin bereits aus inhaltlichen Grunden auf modernere Sichtweisen eingehen mussen.

Periods of Varieties

A period, or more precisely, a period number may be thought of as the value of an integral that occurs in a geometric context. In their papers [Kon99, KZ01], Kontsevich and Zagier list various ways of defining a period. We show that all these variants give the same notion.

Weights and Pure Nori Motives

In this chapter, we explain how Nori motives relate to other categories of motives. By the work of Harrer, the realisation functor from geometric motives to absolute Hodge motives factors via Nori motives. We then use this in order to establish the existence of a weight filtration on Nori motives with rational coefficients. The category of pure Nori motives turns out to be equivalent to Andre’s category of motives via motivated cycles.

More on Diagrams

The aim of this chapter is to introduce and study additional structures on a diagram such that its diagram category becomes a rigid tensor category. The assumptions are tailored to the application to Nori motives.

Kontsevich–Zagier Periods

We compare the set of Kontsevich–Zagier periods defined by integrals over semi-algebraic subsets of \(\mathbb {R}^n\) with cohomological periods.

Nori’s Diagram Category

We explain Nori’s construction of an abelian category attached to the representation of a diagram and establish some properties for it. The construction is completely formal. It mimics the standard construction of the Tannakian dual of a rigid tensor category with a fibre functor . Only, we do not have a tensor product or even a category but only what we should think of as the fibre functor.

The unequal mass sunrise integral expressed through iterated integrals on M‾1,3

Abstract We solve the two-loop sunrise integral with unequal masses systematically to all orders in the dimensional regularisation parameter e. In order to do so, we transform the system of differential equations for the master integrals to an e-form. The sunrise integral with unequal masses depends on three kinematical variables. We perform a change of variables to standard coordinates on the moduli space M 1 , 3 of a genus one Riemann surface with three marked points. This gives us the solution as iterated integrals on M ‾ 1 , 3 . On the hypersurface τ = const our result reduces to elliptic polylogarithms. In the equal mass case our result reduces to iterated integrals of modular forms.

Abdruck der beiden Texte

Auf den folgenden Seiten sind die beiden Bucher Stetigkeit und Irrationale Zahlen und Was sind und was sollen die Zahlen? von Dedekind in den letzten Au agen von 1965 abgedruckt.

Die Klassenzahl quadratischer Zahlkörper

Ziel des Abschnittes ist es, einen Algorithmus zu entwickeln, mit dem wir ein Reprasentantensystem der Idealklassengruppe eines quadratischen Zahlkorpers bestimmen konnen. Dazu bringen wir zunachst die ganzen Ideale von ℴ K in eine Normalform.

Normal Functions and Their Applications

Specialization of cycles and the K-theory elevator

A general specialization map is constructed for higher Chow groups and used to prove a "going-up" theorem for algebraic cycles and their regulators. The results are applied to study the degeneration of the modified diagonal cycle of Gross and Schoen, and of the coordinate symbol on a genus-2 curve.

Der ggT und der euklidische Algorithmus

Formal Periods and the Period Conjecture

Following Kontsevich (see Kontsevich in Operads and motives in deformation quantization. Lett. Math. Phys. 48(1):35–72, 1999), we now introduce another algebra \(\tilde{\mathbb {P}}(k)\) of formal periods from the same data we have used in order to define the actual period algebra of a field in Chap. 11. The main aim of this chapter is to give conceptual interpretation of this algebra of formal periods. We then use it to formulate and discuss the period conjecture.

Teilertheorie im Ring ganzer Zahlen

In diesem Abschnitt wollen wir die Einheiten des Ringes ganzer Zahlen eines Zahlkorpers bestimmen — oder zumindest Aussagen uber die Struktur dieser Gruppe machen. Wir werden das fur quadratische Zahlkorper genau durchfuhren und die Ergebnisse uber beliebige Zahlkorper zitieren.

Abelian varieties and theta functions associated to compact Riemannian manifolds; constructions inspired by superstring theory

We look into a construction of principal abelian varieties attached to certain spin manifolds, due to Witten and Moore-Witten around 2000 and try to place it in a broader framework. This is related to Weil intermediate Jacobians but it also suggests to associate abelian varieties to polarized even weight Hodge structures. The latter construction can also be explained in terms of algebraic groups which might be useful from the point of view of Tannakian categories. The constructions depend on moduli much as in Teichm\"uller theory although the period maps in general are only real analytic. One of the nice features is how the index for certain differential operators canonically associated to …

Endlich erzeugte abelsche Gruppen

In den letzten Abschnitten haben wir die Ringe ℤ und ℤn ℤ kennengelernt, in diesem Abschnitt betrachten wir nur noch ihre additive Gruppenstruktur. Ziel des Abschnittes ist es zu zeigen, dass jede endlich erzeugte abelsche Gruppe ein direktes Produkt aus diesen Gruppen ist. Starten wir mit einigen Definitionen.

Multiple Zeta Values

We study in some detail the very important class of periods called multiple zeta values (MZV). These are periods of mixed Tate motives, which we discussed in Sect. 6.4. Multiple zeta values are in fact periods of unramified mixed Tate motives, a full subcategory of all mixed Tate motives.

Categories of (Mixed) Motives

Quadratrestklassen und Hilbert–Symbole

A second-order differential equation for the two-loop sunrise graph with arbitrary masses

We derive a second-order differential equation for the two-loop sunrise graph in two dimensions with arbitrary masses. The differential equation is obtained by viewing the Feynman integral as a period of a variation of a mixed Hodge structure, where the variation is with respect to the external momentum squared. The fibre is the complement of an elliptic curve. From the fact that the first cohomology group of this elliptic curve is two-dimensional we obtain a second-order differential equation. This is an improvement compared to the usual way of deriving differential equations: Integration-by-parts identities lead only to a coupled system of four first-order differential equations.