# Report on former Research Area D

## Trace formulas.

In the working group of Rapoport, substantial progress was made on the extension of the Langlands-Kottwitz counting method to determine the local factor of the Hasse-Weil zeta function of a Shimura variety to cases of ramification, at least in the Drinfeld case (Haines-Kottwitz conjecture). As a spectacular application, Rapoport’s PhD student Scholze (BIGS) has given a new proof of the Local Langlands Conjecture for p-adic fields, which is substantially simpler than the previous proofs of Harris/Taylor and of Henniart. Scholze’s threemonth visit in Paris had a strong influence on his thinking about this problem. He has been appointed Clay Research fellow in 2011. The Langlands functoriality was also the topic of the Felix Klein Lectures 2007 by Soudry (Tel Aviv).
In his work on the analytic problems posed by the Arthur trace formula, W. Müller derived in joint work with Finis and Lapid [D:FLM09] a refinement of the spectral expansion of the trace formula. A key feature of this expansion is its absolute convergence with respect to the trace norm. This settles a long standing problem.

## Spectral theory of automorphic forms.

Using the Arthur trace formula, W. Müller established in [D:Mül07] Weyl’s law for the cuspidal spectrum of congruence subgroups of $SL(n,\mathbb{Z})$ for any n≥2. Lapid and W. Müller [D:LM09] determined the distribution of the cuspidal spectrum of $SL(n,\mathbb{Z})$ with an estimation of the remainder term. In June 2010, a joint workshop of Research Area A and Research Area D on ‘Spectral analysis on non-compact manifolds’ took place in Bonn.

## Classical modular forms.

Zagier has studied various problems related to mock modular forms and their applications. The theory of mock modular forms (see [D:Zag09b]) is the key to the  construction of arithmetic theta series associated to indefinite quadratic forms of signature (n;1). Mock modular forms are also closely related to an arithmeticity conjecture concerning perturbative Chern-Simons theory that was formulated and substantiated in a paper by Zagier with Dimofte, Gukov, and Lenells [D:DGLZ09]. Another part of Zagier’s work is concerned with Teichmüller curves and Hilbert modular surfaces. In joint work with Möller, initially inspired by Möller’s joint work with Irene Bouw and Zagier’s earlier work related to Picard-Fuchs differential equations [D:Zag09a], it transpired that (a) there is a new kind of modular form on the Teichmüller curves; (b) the Teichmüller curves are the zero-loci of certain derivatives of theta series; and (c) their study permits one to relate the compactification of Hilbert modular surfaces given by Hirzebruch with a much newer (non-smooth) compactification due to Bainbridge. An international conference ‘Mock theta functions and applications in combinatorics, algebraic geometry, and mathematical physics’ was organized by Katrin Brinkmann, Richter and Zagier in Bonn in May 2009.

## Galois representations.

In work by Rapoport with Pappas [D:PR09], the concept of a coefficient space was developed, which captures the algebro-geometric content of Kisin’s partial resolutions of formal deformation spaces of Galois representations. Forthcoming work by Rapoport’s PhD student Hellmann on the corresponding period domains and period mappings, defined in [D:PR09] represents fascinating progress in this area. His interaction with members of the p-adic trimestre in Paris, financed by HCM, was very beneficial to his research.
There is a fascinating analogy between deformations of Galois representations and affine Deligne-Lusztig varieties, which are prominent subvarieties of affine Grassmannians. Partly in joint work with Hartl, Eva Viehmann, has made a thorough investigation of these varieties [D:Vie08]. In particular, she proved that these varieties are equidimensional of dimension predicted by Rapoport.
Faltings determined in [D:Fal10] the image of the Rapoport-Zink period map in the case where G = GL(n), and when the cocharacter m is minuscule. This confirms conjectures of Hartl and Rapoport/Zink. Hartl has extended this theorem to other minuscule cases of EL-type and PELtype.

# Report on former Research Area E

## Moduli spaces of bundles and Hitchin fibration.

Rapoport, in joint work with Pappas, has considerably enlarged the perspective on the classical theory of G-bundles by going beyond the case of constant group schemes; the conjectures in [E:PR10] have already spurred exciting work of Heinloth on the uniformization of G-bundles on curves, and more is sure to follow.
In [E:Fal09] Faltings uses the Hitchin fibration to compute the dimension of the space of sections of the line bundle of (non-abelian) theta functions on the moduli space of G-bundles on algebraic curves for a linear algebraic group G of ADE type in good characteristics. The central charge of some geometrically given divisors is computed as well.
A summer school on the Hitchin fibration was organized in Bonn by Görtz and Eva Viehmann in 2011 covering various aspects of this fascinating topic, also central for Ngô’s proof of the fundamental lemma.

## Stability conditions and periods.

Period domains of K3 surfaces have been studied via mirror symmetry as moduli spaces of (Bridgeland) stability conditions on certain derived categories. For generic K3 surfaces a complete description of this moduli space has been given in [E:HMS08] of Huybrechts, Macrì (HCM-Postdoc, Bonn Junior Fellow) and Stellari. The result has subsequently been used to complete earlier work by Mukai and Orlov on the group of auto-equivalences of these categories and to prove an analogue of a result of Donaldson on the diffeomorphism group of K3 surfaces [E:HMS09]. The latter was predicted by Kontsevich’s homological mirror symmetry which is also studied in the context of Research Area C. The deformation theory of complexes developed for this purpose was used by Huybrechts to study Chow groups of K3 surfaces from a categorical point of view [E:Huy10]. This new framework allows one to approach conjectures of Bloch and Bloch/Beilinson in the case of K3 surfaces.

## Cluster algebras.

Cluster algebras have, since their invention some ten years ago, quickly developed into an ubiquitous combinatorial framework for very different areas of mathematics (e.g. quantum groups, representation theory of finite dimensional algebras, Calabi-Yau manifolds, Teichmüller theory and many others). Schröer studies the relation between cluster algebras and various aspects of representation theory and in particular uses Euler characteristics of moduli spaces of modules over preprojective algebras to control cluster variables, see [E:GLS06a] and the article ‘Kac-Moody groups and cluster algebras’ both joint with Geiß and Leclerc. This led to a combinatorial description of a part of Lusztig’s geometrically defined dual semicanonical basis.
The categorification of cluster algebras plays a central role in the work of Claire Amiot (HCMPostdoc) who introduced the general framework of ‘Amiot cluster categories’ in her thesis under Keller who himself delivered the Felix Klein Lectures in 2009 on ‘Cluster algebras, cluster categories and periodicity’. In a series of papers, Murfet (HCM-Postdoc) has worked on the foundations of matrix factorizations giving rise to Calabi-Yau categories amenable to computation and yet close to geometric categories by results of Orlov.

## Hyperkähler manifolds.

Higher-dimensional generalization of K3 surfaces are provided by compact hyperkähler manifolds. That their derived categories distinguish them from the two other classes of Ricci-flat manifolds, tori and Calabi-Yau manifolds, has been shown in work by Huybrechts in [E:HNW11]. Maybe the most important conjecture in the area is concerned with the existence of Lagrangian fibrations of hyperkähler manifolds. In forthcoming work of Rollenske (HCM-Postdoc), the existence of such fibrations is deduced from the existence of just one Lagrangian torus. Hyperkähler manifolds also provide interesting examples to test some of the central open conjectures in algebraic geometry. E.g. for deformations of Hilbert schemes of K3 surfaces the Lefschetz standard conjecture was recently proved by Charles and Markman during a visit to Bonn. Huybrechts’ PhD student Schlickewei verified in [E:Sch10] the Hodge conjecture for Hilbert schemes of certain K3 surface beyond the case of complex multiplication.

## Eisenstein series.

Rapoport continued his joint work with Kudla, a frequent visitor to the HCM, on the relation between intersection numbers of arithmetic cycles on Shimura varieties and Fourier coefficients of Eisenstein series. A new perspective was opened up in [E:KR11] with the consideration of Picard moduli schemes of arbitrary dimension. This work makes essential use of certain geometric results due to Inken Vollaard and Wedhorn [E:VW11].
In this area, a very exciting development was the treatment by Rapoport PhD student Terstiege (BIGS) of degenerate intersections (for arithmetic Hirzebruch-Zagier cycles on arithmetic Hilbert modular surfaces in [E:Ter11] and for arithmetic Humbert cycles on arithmetic Picard surfaces).

## Motivic homotopy theory.

In recent work, Hornbostel (Bonn Junior Fellow) uses the language of model categories and symmetric spectra to give a new proof of Lurie’s theorem that topological complex K-theory represents orientations of the derived multiplicative group. This allows him to generalize the result to the motivic situation. In September 2010 he organized with Levine a workshop ‘Geometric Aspects of Motivic Homotopy Theory’ covering various aspects of this central theory of modern mathematics which combines topology, arithmetic and algebraic geometry.