Robert P. Langlands

On the functional equations satisfied by Eisenstein series

The assumptions.- Cusp forms.- Eisenstein series.- Miscellaneous lemmas.- Some functional equations.- The main theorem.

Conformal invariance in two-dimensional percolation

The word percolation, borrowed from the Latin, refers to the seeping or oozing of a liquid through a porous medium, usually to be strained. In this and related senses it has been in use since the seventeenth century. It was introduced more recently into mathematics by S. R. Broadbent and J. M. Hammersley ([BH]) and is a branch of probability theory that is especially close to statistical mechanics. Broadbent and Hammersley distinguish between two types of spreading of a fluid through a medium, or between two aspects of the probabilistic models of such processes: diffusion processes, in which the random mechanism is ascribed to the fluid; and percolation processes, in which it is ascribed to the medium. A percolation process typically depends on one or more probabilistic parameters. For example, if molecules of a gas are absorbed at the surface of a porous solid (as in a gas mask) then their ability to penetrate the solid depends on the sizes of the pores in it and their positions, both conceived to be distributed in some random manner. A simple mathematical model of such a process is often defined by taking the pores to be distributed in some regular manner (that could be determined by a periodic graph), and to be open (thus very large) or closed (thus smaller than the molecules) with probabilities p and 1 − p. As p increases the probability of deeper penetration of the gas into the interior of the solid grows.

THE DIMENSION OF SPACES OF AUTOMORPHIC FORMS.

1. The trace formula of Selberg reduces the problem of calculating the dimension of a space of automorphic forms, at least when there is a compact fundamental domain, to the evaluation of certain integrals. Some of these integrals have been evaluated by Selberg. An apparently different class of definite integrals has occurred in iarish-Chandras investigations of the representations of semi-simple groups. These integrals have been evaluated. In this paper, after clarifying the relation between the two types of integrals, we go on to complete the evaluation of the integrals appearing in the trace formula. Before the formula for the dimension that results is described let us review iarish-Chandras construction of bounded symmetric domains and introduce the automorphic forms to be considered. If (7 is the connected component of the identity in the group of pseudoconformal mappings of a bounded symmetric domain then (7 has a trivial centre and a maximal compact subgroup of any simple component has nondiscrete centre. Conversely if (7 is a connected semi-simple group with these two properties then (7 is the connected component of the identity in the group of pseudo-conformal mappings of a bounded symmetric domain [2(d)]. Let g be the Lie algebra of (7 and gc its complexification. Let GC be the simplyconnected complex Lie group with Lie algebra gc; replace G by the connected subgroup G of GC with Lie algebra g. Let K be a maximal compact sub

On the universality of crossing probabilities in two-dimensional percolation

Six percolation models in two dimensions are studied: percolation by sites and by bonds on square, hexagonal, and triangular lattices. Rectangles of widtha and heightb are superimposed on the lattices and four functions, representing the probabilities of certain crossings from one interval to another on the sides, are measured numerically as functions of the ratioa/b. In the limits set by the sample size and by the conventions and on the range of the ratioa/b measured, the four functions coincide for the six models. We conclude that the values of the four functions can be used as coordinates of the renormalization-group fixed point.

A Combinatorial Laplacian with Vertex Weights

One of the classical results in graph theory is the matrix-tree theorem which asserts that the determinant of a cofactor of the combinatorial Laplacian is equal to the number of spanning trees in a graph (see 1, 17, 11, 15]). The usual notion of the combinatorial Laplacian for a graph involves edge weights. Namely, a Laplacian L forGis a matrix with rows and columns indexed by the vertex setVofG, and the (u,v)-entry of L, foru,vinG,u?v, is associated with the edge-weight of the edge (u,v). It is not so obvious to consider Laplacians with vertex weights (except for using some symmetric combinations of the vertex weights to define edge-weights). In this note, we consider a vertex weighted Laplacian which is motivated by a problem arising in the study of algebro-geometric aspects of the Bethe Ansatz 18]. The usual Laplacian can be regarded as a special case with all vertex-weights equal. We will generalize the matrix-tree theorem to matrix-tree theorems of counting “rooted” directed spanning trees. In addition, the characteristic polynomial of the vertex-weighted Laplacian has coefficients with similar interpretations. We also consider subgraphs with non-trial boundary. We will shown that the Laplacian with Dirichlet boundary condition has its determinant equal to the number of rooted spanning forests. The usual matrix-tree theorem is a special case with the boundary consisting of one single vertex.

Modular Forms and ℓ-Adic Representations

This report is another attempt on the part of its author to come to terms with the circumstance that L-functions can be introduced not only in the context of automorphic forms, with which he has had some experience, but also in the context of diophantine geometry. That this circumstance can be the source of deep problems was, I believe, first perceived by E. Artin. He was, to be sure, concerned with forms on GL(1) and with varieties of dimension 0. This remains the only case in which results of any profundity have been obtained. These have been hard won. Their mathematical germ is the theory of cyclotomic fields; itself easy-only in comparison to the general theory.

Stable conjugacy: definitions and lemmas

The purpose of the present note is to introduce some notions useful for applications of the trace formula to the study of the principle of functoriality, including base change, and to the study of zeta-functions of Shimura varieties. In order to avoid disconcerting technical digressions I shall work with reductive groups over fields of characteristic zero, but the second assumption is only a matter of convenience, for the problems caused by inseparability are not serious.

On the Zeta-Functions of Some Simple Shimura Varieties

In an earlier paper [14] I have adumbrated a method for establishing that the zero-function of a Shimura variety associated to a quaternion algebra over a totally real field can be expressed as a product of L-functions associated to automorphic forms. Now I want to add some body to that sketch. The representation-theoretic and combinatorial aspects of the proof will be given in detail, but it will simply be assumed that the set of geometric points has the structure suggested in [13]. This is so at least when the algebra is totally indefinite, but it is proved by algebraic-geometric methods that are somewhat provisional in the context of Shimura varieties. However, contrary to the suggestion in [13] the general moduli problem has yet to be treated fully. There are unresolved difficulties, but they do not arise for the problem attached to a totally indefinite quaternion algebra, which is discussed in detail in [17]. It does not add to the essential difficulties if we enlarge our perspective a little and consider not only the zeta-function defined by the constant sheaf but also that defined by the sheaves associated to finite dimensional representatives of the group defining the variety, and we might even dissipate some of the current misconceptions about the nature of these sheaves. Their existence is a formal consequence of Shimura’s conjecture. We should moreover not confine ourselves to the multiplicative group of the quaternion algebra, but should in addition consider subgroups lying between the full multiplicative group and the kernel of the norm, for then we can see the effect of L-indistinguishability [7] in the place where it was first noticed. In this introduction the results of [7], to which [22] is meant to serve as a kind of exegesis, are used in conjunction with facts about continuous cohomology to arrive at an assertion about the zeta-function which the remainder of the paper is devoted to proving. Some readers will find that I have given too free rein to a lamentable tendency to argue from the general to the particular, and have obfuscated them by interjecting unfamiliar concepts of representation theory into what could be a purely geometric discussion. My intention is not that, but rather to equip myself, and perhaps them as well, for a serious study of the Shimura varieties in higher dimensions. We are in a forest whose trees will not fall with a few timid hatchet blows. We have to take up the double-bitted axe and the cross-cut saw, and hope that our muscles are equal to them. The method of proof has already been described in [14]. It is ultimately combinatoric. The Bruhat-Tits buildings, which arise naturally in the study of orbital integrals and Shimura varieties, are used systematically. However the automorphic L-functions used to express the zeta-functions of the varieties are unusual and most of §2 is taken up with the attempt to understand them and express their coefficients in manageable, elementary terms. The appearance of L-indistinguishability complicates the task considerably. The meaning of the conjectures of [13] is also obscure, even to their author, and considerable effort is necessary before it is revealed sufficiently that a concrete expression for the coefficients of the zeta-functions is obtained. Once this is done, in §3 and the appendix, the equality to be proved is reduced to elementary assertions which are proved by combinatorial arguments in §4. A connected reductive group G over Q and a weight μ of the associate group G are the principal data specifying a Shimura variety. The conditions they must satisfy are described in [4]. If Af is the group of finite adeles one needs an open compact subgroup K of G(Af ) as well. μ is the weight of G defined by the co-weight h0 of [13]. The primary datum is h0, rather than μ. To completely define S(K) one needs h0. The variety will be denoted by S(K) and only K will appear explicitly, for G and μ are usually fixed. The group G comes provided with a Borel subgroup B and a Cartan subgroup T 0 in B. We may suppose that μ is a positive

Descent for Transfer Factors

In [I] we introduced the notion of transfer from a group over a local field to an associated endoscopic group, but did not prove its existence, nor do we do so in the present paper. Nonetheless we carry out what is probably an unavoidable step in any proof of existence: reduction to a local statement at the identity in the centralizer of a semisimple element, a favorite procedure of Harish Chandra that he referred to as descent.

Universality and conformal invariance for the Ising model in domains with boundary

The partition function with boundary conditions for various two-dimensional Ising models is examined and previously unobserved properties of nonformal invariance and universality are established numerically.