ESI Programme on
Automorphic Forms: Arithmetic and Geometry

January - February 2012

Organising Committee

James W. Cogdell (Ohio State U, Columbus)
Colette Moeglin (U Paris VII)
Goran Muić (U Zagreb)
Joachim Schwermer (U Vienna)

Overview Programme List of Participants


The theory of automorphic forms has its roots in the early nineteenth century in the works of Gauss, Jacobi, Eisenstein and others. The subject experienced a vast expansion and reformulation following the work of Selberg, Harish-Chandra, and Langlands, in the 1970's, and remains a major focal point of development in number theory and algebraic geometry, with applications in many diverse areas, including combinatorics and mathematical physics. In the last decade there have been a number of advances in the theory of automorphic forms that have resolved long outstanding problems, for example, Ngo Bao Chau's proof of the fundamental lemma, Arthur's work on endoscopy and the classification of automorphic representations for classical groups, and the work of Taylor, Harris, Clozel and others that has provided a proof of the Sato-Tate conjecture. At the same time these advances - through both the results obtained and the innovative methods introduced - have opened up very important new directions for research. It is the main goal of this program to survey work on some of these new directions, their crossroads and their possible applications to problems in number theory and geometry.


The focal points of the programme will be:

1. Langlands functoriality beyond endoscopy

Among the many fundamental insights of Langlands are the following:

- Automorphic representations of a given reductive group G defined over an algebraic number field k occur in packets (L-packets or Arthur packets), parametrized by representations of the Weil-Deligne group in the Langlands dual group LG. A local version of this should describe the (irreducible, admissible) representations of the group G(F) for any local field F.

- It is necessary to consider the automorphic representations of all reductive groups together, and, in particular, their relations, the most important of which are predicted by the principle of functoriality.

These insights are fundamental though their complete realization is still a very distant dream. Nonetheless, they have provided a guide for much of the subsequent research in this area and a number of the most important techniques that have been brought to bear will become major points for the discussion during the program. These will include the Arthur-Selberg trace formula, various variants of the fundamental lemma (weighted, twisted), local and global descent, converse theorems, and theta correspondence. As predicted by Langlands, the general principle of functoriality would imply relations between automorphic representations that lie much deeper than the relations of endoscopy whose understanding has been the main focus of research for the last decades. Serious, but very preliminary, work, has now begun on these relations.

2. The classification of automorphic representations for classical groups - A-packets

The work of Arthur on endoscopy and the classification of global automorphic representations for classical groups has natural local implications. More precisely, in order to establish a classification for the discrete spectrum of split classical groups defined over a number field one divides the ''potential local candidates'' for the global automorphic representations in question into A-packets. The local multiplicity formula in A-packets is fundamental to compute the multiplicity formula in the discrete spectrum. If the case of the finite local places is now understood for classical p-adic groups, the case of the archimedean places is still under consideration. In particular, the link between the recent works of Adams, Barbash, and Vogan and Arthur's construction is not fully understood. Moreover the same circle of questions has to be investigated in the case of exceptional groups. Up to now one only finds the results of Wee Teck Gan and Gurevich for groups of type G2. One observes that only unitary representations of local groups G(kv) contribute to the automorphic spectra, therefore A-packets should contain only unitary representations. The question of unitarity for a given local representation is a difficult problem. One way to prove unitarity of a local representation is finding a global representation which occurs in the discrete spectrum of L2(G(k)\G(A)) and contains as a local component. This approach requires a good deal of understanding of global methods. One would like to work on the questions of unitarity of representations in the local A-packets. We remark that a great deal of progress on the computational side of unitarity has been established for real groups by Vogan and his collaborators. But the questions of unitarity of representations in A-packets is open either for real or p-adic groups. We remark that the unitarity may give bounds of Hecke eigenvalues and give some information on the decomposition of L2(G(k)\G(A)) (Burger-Sarnak method, and extension due to Clozel and Ulmo).

3. Applications of endoscopy to the geometry and arithmetic of Shimura varieties

The recent work of Arthur on the structure of automorphic representations of classical groups should have important applications to the understanding of the cohomology of Shimura varieties, both deRham and čtale. In addition, new results can be expected on base change for classical groups and for the transfer of automorphic representations between inner forms. Finally, these results will have applications in the cohomology theory of arithmetic groups, using its interpretation in terms of the automorphic spectrum. Indeed, automorphic forms have a deep connection with the geometry of the underlying locally symmetric spaces and vice versa, where, for example, the boundary behavior of cohomology classes represented by derivatives of Eisenstein series or residues of such can be applied to the study of special values of automorphic L-functions. Due to recent work of Harder and one of his former students this research area continues to be a vibrant subject in which many exciting developments can be expected in the future.

4. The Sato-Tate conjecture

Consider an elliptic curve, E, with rational coefficients and without complex multiplication, and for any prime number p denote by Np(E) the number of points of E modulo p. Thanks to Hasse, one knows that (p + 1-Np(E))/2p1/2 is a real number in [-1,1] and the Sato-Tate conjecture predicts the distribution of theses numbers in [-1,1]. First cases of this conjecture were proved in 2006 by Harris, Taylor and their collaborators Clozel and Shepherd-Barron. The reciprocity which links Galois l-adic families of representations arising from elliptic curves to the same kind of objects arising from some automorphic representations of GL(2), known as the Shimura-Taniyama-Weil conjecture, is not sufficient to obtain a proof of the Sato-Tate conjecture. One needs some information which is encoded in Langlands functoriality, in fact, a reciprocity law for all the self products of E. More precisely, by a deep result of Taylor, only reciprocity after base change, so-called potential automorphy, is enough. In the last few years, this scheme has been designed, extending the initial ideas of Wiles in his proof of the Shimura-Taniyama-Weil conjecture. In the most recent paper by Barnet-Lamb, Gee and Geraghty one finds a proof of the Sato-Tate conjecture for Hilbert modular forms. Thus Langlands functoriality is now known, after suitable base change, for regular algebraic cuspidal automorphic representations of GL(2). The proof of Langlands functoriality in this case is subject to some restrictions, the most serious of which is the assumption of regular algebraicity. However, it appears tractable, using the full stabilization of the twisted trace formula for all unitary groups and certain methods of p-adic automorphic forms, to discard the base change restriction. Another very promising perspective is to pass from GL(2) to more general reductive groups.

5. The Gross-Prasad conjecture

In 1986, Gross and Zagier described the height of Heegner points in terms of some special values of derivatives of L-functions. Since that time, in joint works, D. Prasad and B. Gross have formulated some conjectures to generalize that picture; this amounts to interpret the value of the L-function or derivative of it, at s = 1/2, for the Rankin-Selberg product of two automorphic cuspidal representations, one of a classical group acting on a space of dimension n and the other of the connected stabilizer of a non-isotropic vector in that space. Assuming that the L-function does not vanish at s = 1/2, Ichino and Ikeda have refined this conjecture, at least in the case of orthogonal groups and with certain technical hypotheses on the cuspidal representations involved. Some time ago, Jacquet and Rallis have also formulated a refined version of the Gross-Prasad conjecture in the case of unitary groups which links it to the functoriality between unitary groups and their base changes to general linear groups. This approach uses the relative trace formula of Jacquet and makes clear that the trace formula machinery is a powerful tool to prove the conjecture. Very recently, using in particular methods and results as announced by Arthur regarding the trace formula and twisted endoscopy lifting, Waldspurger proved the local version of the Gross-Prasad conjecture for orthogonal groups and tempered representations. The perspectives in this domain, are, of course, an eventual proof of the global conjecture but also the extension from the orthogonal groups case to a much more general setting as formulated by Gan, Gross and Prasad. This gives a new push on the link between automorphic forms and arithmetic geometry as understood by Kudla, Rappoport and others and we want to focus on it.

List of participants

James Arthur (Toronto) 08.01.-14.01
Thomas Barnet-Lamb (Waltham)
Tobias Berger (Sheffield)
Przemyslaw Chojecki (Paris)
Laurent Clozel (Paris)
James W. Cogdell (Columbus) 03.01.-28.02
Laurent Fargues (Strasbourg)
Daniel File (SFS) 10.1.-21.02
Wee Teck Gan (Songapoure) 2nd workshop conference
Toby Gee (London)
Ulrich Goertz (Essen)
Neven Grbac (Rijeka)
Marcela Hanzer (Zagreb) 13.02.-28.02
Guenter Harder (Bonn) 30.01.-28.02
Guy Henniart (Paris) 12.02.-27.02
Haruzo Hida (Los Angeles) 10.02.-20.02
Fritz Hoermann (Freiburg)
Atsushi Ichino (Kyoto) 11.02.-23.02
Dihua Jiang (Minneapolis) 08.01.-28.01
Vitzeslav Kala (West Lafayette)
Henry Kim (Toronto) 13.02.-23.02
Krystof Klosin (Flushing)
Stephen S. Kudla (Toronto)
Jean-Pierre Labesse (Marseille) 08.01.-22.01
Erez Lapid (Jerusalem) 05.02.-26.02
Jing Feng Lau (West Lafayette)
Gerard Laumon (Paris) 13.02.-19.02
Jaime Lust (Iowa City)
Ivan Matic (Osijek) 11.02.-25.02
Colette Moeglin (Paris) 28.01.-28.02
Sophie Morel (Boston) 12.01.-21.01
Goran Muic (Zagreb) 08.01.-25.02
Chufeng Nien (SFS)
Michael Rapoport(Bonn) 07.02-18.02
Juergen Rohlfs (Eichstatt)
Gordan Savin (Salt Lake City) 12.02.-21.02
Joachim Schwermer (Vienna)
Anthony Scholl (Cambridge) 04.01.-21.01
Freydoon Shahidi (West Lafayette) 03.01.-27.01
Sug Woo Shin (Boston)
David Soudry (Tel Aviv) 05.02-26.02
Marko Tadic (Zagreb) 11.02-26.02
Ulrich Terstiege (Bonn) 07.02.-18.02
Tung-Lin Tsai (West Lafayette)
Jean-Loup Waldpurger (Paris) 12.02.-26.02
Tong-Hai Yang (Madison) 07.01.-21.02