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Symmetries in Algebra and Number Theory (SANT) By Ina Kersten, Ralf Meyer (Eds.)

These proceedings contain most of the contributions to the Göttingen-Jerusalem Conference 2008 on „Symmetries in Algebra and Number Theory“ including three addresses given at the conference opening, and two contributions to the Satellite Conference „On the Legacy of Hermann Weyl“. The contributions are survey articles or report on recent work by the authors, for exemple new results on the famous Leopoldt conjecture.

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Contents:

Self-similarity in group theory and algebra

LAURENT BARTHOLDI

The author describes avatars of self-similarity in the worlds of groups, associative and Lie algebras; in each case, he highlights some important examples and the questions or conjectures that they answer.

L-functions, automorphic forms, and arithmetic

VALENTIN BLOMER & GERGELY HARCOS

The authors give a short, informal survey on the role of automorphic L-functions in number theory. They present the strongest currently known subconvexity bounds for twisted L-functions over number fields due to the authors and give various arithmetic applications. This is based on a talk of the first author.

Bounded cohomology of the p-adic upper half plane

EHUD DE SHALIT

The author considers the bounded cohomology of Drinfel’d’s p-adic upper half plane, with values in a local system M. He shows that it has a canonical splitting into the space of bounded one-forms with values in M. The paper is largely expository, and self-contained.

On Arakelov vector bundles over arithmetic curves

NORBERT HOFFMANN

Here ‘arithmetic curve’ means the set of all places X of a number field K. Under the classical analogy between number fields and function fields, Arakelov vector bundles over X correspond to algebraic vector bundles over a smooth projective curve C. The text discusses analogues over X of three results on such bundles over C: Riemann-Roch, the Narasimhan-Seshadri theorem, and Faltings’ criterion for semistability.

A conjectural non-commutative generalization of a volume formula of McMullen-Schneider

TOBIAS FINIS & EREZ LAPID

The Fourier transform of the characteristic function of a convex polytope is given by a localization formula due to Brion. On the other hand, an argument by McMullen-Schneider expresses the mixed volume of d polytopes in a d-dimensional space in terms of volumes of parallelotopes formed by d-tuples of edges, one from each polytope. When these formulas are specialized one gets two seemingly different expressions for the volume of a polytope. The authors propose a non-commutative conjectural generalization of this identity. In the case of a root zonotope, such an identity is known and used in the study of the spectral side of Arthur’s trace formula.

The j-invariant of a plane tropical cubic

HANNAH MARKWIG ( JOINT WITH ERIC KATZ AND THOMAS MARKWIG)

In tropical geometry, algebraic varieties are replaced by certain degenerations called tropical varieties. Tropical varieties are piece-wise linear objects that can be studied using combinatorics and linear algebra methods. One can use tropical geometry to prove theorems about algebraic geometry by means of these new methods. To reach this aim, one has to understand the connection between algebraic geometry and tropical geometry. In this talk, the author wants to understand the tropical analogue of the j -invariant of a smooth elliptic curve.

Leopoldt’s conjecture for some galois extensions

PREDA MIH˘AILESCU

In this paper the author uses Baker theory for proving some special cases of Leopoldt’s conjecture. Thus he shows that if the conjecture is true for some field K, then it is true for solvable extensions thereof. Also, if it holds for arbitrary extensions with simple groups, then it holds for any number field.

The proofs are essentially based on the talk given at the SANT Conference.

On Leopoldt’s conjecture and a special case of Greenberg’s conjecture

PREDA MIH˘AILESCU

The conjecture of Leopoldt states that the p - adic regulator of a number field does not vanish. It was proved for the abelian case in 1967 by Brumer, using Baker theory. The author shows that when K is a totally real extension in which the prime p is totally split, then Leopoldt’s conjecture is equivalent to a special case of Greenberg’s conjecture, which he proves. This case allows thus a proof which does not use transcendence theory. At the end of the paper, he gives a new approach for the general case of Leopoldt’s conjecture, which is currently under peer review. The two papers in these Proceedings gather various approaches for attacking the conjecture of Leopoldt.

Homological algebra for Schwartz algebras

RALF MEYER

Let G be a reductive group over a non-Archimedean local field. For two tempered smooth representations, it makes no difference for the Ext-groups whether one works in the category of tempered smooth representations of G or of all smooth representations of G. Similar results hold for certain discrete groups. The author explains the basic ideas from functional analysis and geometric group theory that are needed to state this result correctly and prove it.

Weights in generalizations of Serre’s conjecture and the mod p local Langlands correspondence.

MICHAEL M. SCHEIN

In this mostly expository article the author gives a survey of some of the generalizations of Serre’s conjecture and results towards them that have been obtained in recent years. He also discuss recent progress towards a mod p local Langlands correspondence for p-adic fields and its connections with Serre’s conjecture. A theorem describing the structure of some mod p Hecke algebras for GLn is proved.

Towards Langlands correspondence over function fields for split reductive groups

YAKOV VARSHAVSKY ( JOINT WITH DAVID KAZHDAN)

In this note the author describes joint work with David Kazhdan on the global Langlands correspondence over function fields for arbitrary split reductive groups. The main result asserts that for every pair (π,ω), where π is a cuspidal representation of G one of whose local components is a cuspidal Deligne-Lusztig representation, and ω is a representation of the dual group, there exists a virtual Galois representation ρπ,ω whose L-function equals the L-function of the pair (π,ω).

Maximal Sobolev regularity at radial points

INGO WITT

The author establishes precise regularity results for solutions to pseudodifferential equations with real principal symbols near radial points, microlocally in the scale of Hs Sobolev spaces. There is the new phenomenon of maximal Sobolev regularity. The author clarifies this point by discussing how the two distinguished parametrices that one has near a radial point, where the pseudodifferential operator under study is of real principal type, extend into that radial point, and he provides an analytic formula for maximal Sobolev regularity in terms of the principal and subprincipal symbols.

Elliptic gamma function provides the ˇCech cocycle of a gerbe

CHENCHANG ZHU

G. Felder and A. Varchenko discovered certain modular formulas for elliptic gamma functions. These identities are generalized to an infinite set of identities for elliptic gamma functions associated to pairs of planes in 3-dimensional space in a previous paper. There the language of stacks and gerbes is used to give a natural framework for a systematic description of these identities and their domain of validity. In this note, the previous work is summarized with an emphasize on the ˇCech open covers.

The affine Macdonald’s formula

DAVID KAZHDAN

In a paper with A. Braverman the author defined the affine spherical Hecke algebra. It is natural then to ask about the spherical eigenfunctionals. In the finite-dimensional case they are described by MacDonald. In this note the author talks about their affine analogue.

Hermann Weyl and the Early History of Gauge Theories

NORBERT STRAUMANN

One of the major developments of twentieth century physics has been the gradual recognition that a common feature of the known fundamental interactions is their gauge structure. In this talk the early history of gauge theory is reviewed, emphasizing especially Weyl’s seminal contributions of 1918 and 1929.

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