W. Casselman

Bio: W. Casselman is an academic researcher. The author has contributed to research in topics: Automorphic form & Finite volume method. The author has an hindex of 2, co-authored 2 publications receiving 605 citations.

Automorphic Forms, Representations, and L-Functions

Abstract: Contains sections on Automorphic representations and L-functions as well as Arithmetical algebraic geometry and L-functions.

Higgs bundles and local systems

Abstract: © Publications mathématiques de l’I.H.É.S., 1992, tous droits réservés. L’accès aux archives de la revue « Publications mathématiques de l’I.H.É.S. » ( http://www. ihes.fr/IHES/Publications/Publications.html), implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

Automorphic forms with singularities on Grassmannians

Abstract: We construct some families of automorphic forms on Grassmannians which have singularities along smaller sub Grassmannians, using Harvey and Moore's extension of the Howe (or theta) correspondence to modular forms with poles at cusps. Some of the applications are as follows. We construct families of holomorphic automorphic forms which can be written as infinite products, which give many new examples of generalized Kac-Moody superalgebras. We extend the Shimura and Maass-Gritsenko correspondences to modular forms with singularities. We prove some congruences satisfied by the theta functions of positive definite lattices, and find a sufficient condition for a Lorentzian lattice to have a reflection group with a finite volume fundamental domain. We give some examples suggesting that these automorphic forms with singularities are related to Donaldson polynomials and to mirror symmetry for K3 surfaces.

Classifying Space for Proper Actions and K-Theory of Group C*-algebras

Abstract: We announce a reformulation of the conjecture in [8,9,10]. The advantage of the new version is that it is simpler and applies more generally than the earlier statement. A key point is to use the universal example for proper actions introduced in [10]. There, the universal example seemed somewhat peripheral to the main issue. Here, however, it will play a central role.

Noncommutative Geometry, Quantum Fields and Motives

Abstract: Quantum fields, noncommutative spaces, and motives The Riemann zeta function and noncommutative geometry Quantum statistical mechanics and Galois symmetries Endomotives, thermodynamics, and the Weil explicit formula Appendix Bibliography Index.

Points on some Shimura varieties over finite fields

Abstract: The Hasse-Weil zeta functions of varieties over number fields are conjecturally products (and quotients) of automorphic L-functions. For a Shimura variety S associated to a connected reductive group Gover Q one can hope to be more specific about which automorphic L-functions appear in the zeta function. In fact Eichler, Shimura, Kuga, Sato, and Ihara, who studied GL2 and its inner forms, found in those cases that it was enough to use automorphic L-functions for the group G itself. In the general case Langlands [L2-L4] has given a conjectural description of the zeta function in terms of automorphic L-functions for G and its endoscopic groups [LS] (see also [KS] for the contribution of non-tempered representations), and a description of this type has been verified in certain cases, beginning with [L3]. For GL2 it was possible to use the Eichler-Shimura congruence relation in order to make the connection between the zeta function and automorphic Lfunctions. In general one needs more information than the Eichler-Shimura congruence relation gives, and it seems to be necessary to describe the points on S over finite fields in terms of group-theoretical data (for the group G), in a way that is adapted to an eventual comparison of the number of points modulo p with the Selberg trace formula for G (actually with the stable trace formulas for G and its endoscopic groups), as is explained in [L2, L3, KS]. Ihara [11, 12] gave such a group-theoretical description of points modulo p in the case of GL2(Q) and its inner forms, and Deligne [D!] gave a related description of the category of ordinary abelian varieties over a finite field. Langlands [Ll] conjectured a group-theoretical description in the general case, based on a detailed though incomplete study of Shimura varieties of PEL type [S]; Milne [Mil] gave a simplified exposition of this work of Langlands in a special case, using the description of the category of abelian varieties up to isogeny over a finite field due to Honda [H] and Tate [T2, T3]. Zink [Z2] gave complete proofs for part of Langlands's conjectures for Shimura varieties of PEL type, but by that time it was clear that a new idea was needed to give a complete proof for the full conjecture, even for the case of the group of symplectic similitudes. In fact the conjecture itself needed some refinement; this was one of the objects of some work by Langlands-Rapoport [LR], whose main goal, however, was to put the conjecture into a Tannakian framework, in which it became conceptually clearer. The final step was taken independently by