Selberg Trace Formula Research Articles

On Selberg's trace formula

On Selberg's trace formula

The Selberg trace formula for congruence subgroups

The Selberg trace formula for congruence subgroups

The Selberg Trace Formula for Groups of F-rank One

The Selberg Trace Formula for Groups of F-rank One

L-series, modular imbeddings, and signatures

For each cusp singularity of the Hilbert modular variety of a totally real algebraic number field there is associated an L-series which determines the corresponding to the version of Selberg's trace formula worked out by Shimizu. For a real quadratic field Meyer [9, 10] has given an elementary algorithm by Dedekind sums for the computation of the parabolic contribution (see also Siegel [14]) from which a simple formula for the parabolic contribution in terms of the continued fraction associated with the cyclic resolution of the corresponding cusp singularity can be derived ([5, 7, 8]). In fact, the parabolic contribution equals ~2(3r-bo b l . . . . . br-1) where bo .. . . . br-1 are the characteristic numbers of the dual normal bundles of the rational curves which form the exceptional fibre of the cyclic resolution. The first purpose of this note is to give a simple formula for the total parabolic contribution, i.e., the sum of the parabolic contributions for the different cusps. This formula is deduced from a general identity satisfied by certain L-series in any real algebraic number field. In the quadratic case one obtains an expression for the total parabolic contribution in terms of the class numbers of imaginary quadratic fields. Further, in the quadratic case the total parabolic contribution vanishes if and only if the discriminant of the field is the sum of two squares and is negative otherwise. The total parabolic contribution is of additional interest in the quadratic case for two reasons: first, it is one-fourth the signature (adjusted for defects coming from quotient singularities in certain special cases) of the open rational homology 4-manifold obtained as the quotient of the product of two upper-half planes by the non-symmetric Hilbert modular group. This result, announced to some extent in [5], will be proved here. Thus, from [2] the vanishing of this (adjusted) signature is a necessary and sufficient condition for the existence of modular imbeddings. Second, it will be shown that this signature is twice the difference of the arithmetic genera for the ordinary and mixed Hilbert modular groups.

The Parabolic Contribution to the Number of Linearly Independent Automorphic Forms on a Certain Bounded Domain

then r\D is a compact complex manifold. The dimension of the vector space of r-automorphic forms of weight m has been determined in [9] by means of the Riemann-Roch theorem. Hirzebruch [10] has used the AtiyahBott fixed point theorem to eliminate condition ii). Not much is known about the dimension of the space of automorphic forms when r\D is not compact. Leslie Cohn [5] has applied the Selberg trace formula to a certain domain D = G/K and a discrete group r =-Gz that does not satisfy either ii) or iii). In this paper, the index theorem is used to determine the dimension of the space of r-automorphic forms of sufficiently large weight m for certain groups r that satisfy i) and ii) but not iii). The domain D in this paper is the same as the one considered by Cohn [5]. Here, now, is an outline of the procedure to be followed. Let D===SU(2,1)/S(U(2) X U(1)) and let r C S(2,1) be a discrete subgroup which acts freely on D. Suppose also that r\D has finite invariant volume. By the assumptions on r, r\D is a complex manifold with a cusps. If r is sufficiently nice, then it is possible to compactify rFD as a complex manifold by adding a torus to each cusp. One can then define a complex analytic line bundle Em-r\D such fhat dim HO(r\D, Q(Em)) is the dimension of the space of r-automorphic forms of weight m. Here Q(Em) denotes the

Selberg's trace formula as applied to a compact riemann surface

Selberg's trace formula as applied to a compact riemann surface

On some applications of Selberg's trace formula

On some applications of Selberg's trace formula

On Discontinuous Groups Operating on the Product of the Upper Half Planes

Let H be the direct product of the n upper half planes, and let be the connected component of the identity of the group of all analytic automorphisms of Hn. is the direct product of n subgroups G1, G2, . . .* Gq each of which is isomorphic to the group of all analytic automorphisms of the upper half plane H. Consider a discrete subgroup of such that the factor space 17\G is of finite measure. (The notations Hn, or will keep, unless otherwise stated, these meanings throughout this paper.) For the study of the groups of this type, it is important to investigate the case where is an irreducible subgroup of in the following sense. A discrete subgroup of is said to be irreducible if is not commensurable' with any direct product F' x F, where F' and F are respectively discrete subgroups of the partial products G' and G of =G, x G2 x ... x Ga with = G' x G, G' I {1}, G # {1}. The main purpose of this paper was originally to calculate the dimension of the space of cusp foruis for an irreducible group by means of Selberg's trace formula; however, for the sake of completeness, and in view of the fact that no proof has been published as yet for the results stated by Pyatetzki-Shapiro in [6], it has been found desirable to prove these results here, following the ideas indicated by Pyatetzki-Shapiro himself. Therefore, no new results will be found in our ?? 1-3 except for some supplementary results such as Theorem 1, and for Theorems 6 and 7, which are proved under an additional condition on the fundamental domain of (Assumption (F) in No. 11, ?3). Theorems 2, 3, 4, 5 are the restatements of the results of Pyatetzki-Shapiro; but, for the sake of simplicity, we state here the latter three theorems only for irreducible case. For these results, Pyatetzki-Shapiro has indicated in [6] only a sketch of proof for Theorem 3 as well as the implications between them. However, our proofs will be probably more or less the same as the ones he has. In ??4-5, we shall calculate the dimension of the space of cusp forms for an irreducible group under the assumption (F). The main result is

On the behaviour of eigenvalues of Hecke operators

In this paper we use the Selberg trace formula for Hecke operators in order to obtain information on the distribution of the eigenvalues of Hecke operators in the situation of the hyperbolic 3-space. We prove that the eigenvalues are equidistributed with respect to a measure that tends to the Sato-Tate measure.