# Achievements of 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.