Hilbert Modular Group Research Articles

Jacobi-Eisenstein series over number fields

For any given totally real number field K, we compute the Fourier developments of the Jacobi Eisenstein series over K at the cusp at infinity. As main application we prove, for any K with class number 1, that the L-series of the Jacobi Eisenstein series of weight k≥3 for indices with rank and modified level 1 coincide with the L-series of the Eisenstein series of weight 2k−2 on the full Hilbert modular group of K. Moreover, under this correspondence the Fourier coefficients of the Jacobi Eisenstein series are related to the twisted L-series of the Hilbert Eisenstein series at the critical point by a Waldspurger type identity. This is a first step in the proof that Skoruppa's and Zagier's lifting from Jacobi forms over Q to elliptic modular forms holds true over arbitrary totally real number fields too.

Fourier coefficients of Hilbert modular forms at cusps

The aim of this article is to study the fields generated by the Fourier coefficients of Hilbert newforms at arbitrary cusps. Precisely, given a cuspidal Hilbert newform f and a matrix sigma in (a suitable conjugate of) the Hilbert modular group, we give a cyclotomic extension of the field generated by the Fourier coefficients at infinity which contains all the Fourier coefficients of f||_ksigma .

On multiplier systems and theta functions of half-integral weight for the Hilbert modular group [formula omitted

Let F be a totally real number field and o the ring of integers of F. We study theta functions which are Hilbert modular forms of half-integral weight for the Hilbert modular group SL2(o). We obtain an equivalent condition that there exists a multiplier system of half-integral weight for SL2(o). We determine the condition of F that there exists a theta function which is a Hilbert modular form of half-integral weight for SL2(o). The theta function is defined by a sum on a fractional ideal of F.

The Hilbert modular group and orthogonal groups

We derive an explicit isomorphism between the Hilbert modular group and certain congruence subgroups on the one hand and particular subgroups of the special orthogonal group SO(2, 2) on the other hand. The proof is based on an application of linear algebra adapted to number theoretical needs.

A reduction algorithm for Hilbert modular groups

The aim of this paper is to present an explicit reduction algorithm for Hilbert modular groups over arbitrary totally real number fields. An implementation of the algorithm is available to download from [20]. The exposition is self-contained and sufficient details are given for the reader to understand how it works and implement their own version if desired.

Arithmeticity of the monodromy of the Wiman–Edge pencil

The Wiman–Edge pencil is the universal family of projective, genus 6, complex-algebraic curves endowed with a faithful action of the icosahedral group. The goal of this paper is to prove that its monodromy group is commensurable with a Hilbert modular group; in particular is arithmetic. We then give a modular interpretation of this, as well as a uniformization of its base.

Geometry of the Wiman–Edge monodromy

The Wiman–Edge pencil is a pencil of genus 6 curves for which the generic member has automorphism group the alternating group [Formula: see text]. There is a unique smooth member, the Wiman sextic, with automorphism group the symmetric group [Formula: see text]. Farb and Looijenga proved that the monodromy of the Wiman–Edge pencil is commensurable with the Hilbert modular group [Formula: see text]. In this note, we give a complete description of the monodromy by congruence conditions modulo 4 and 5. The congruence condition modulo 4 is new, and this answers a question of Farb–Looijenga. We also show that the smooth resolution of the Baily–Borel compactification of the locally symmetric manifold associated with the monodromy is a projective surface of general type. Lastly, we give new information about the image of the period map for the pencil.

P-adic Measures for Reciprocals of L-functions of Totally Real Number Fields

We generalize the work of Gelbart, Miller, Pantchichkine, and Shahidi on constructing p-adic measures to the case of totally real fields K. This measure is the Mellin transform of the reciprocal of the p-adic L-function which interpolates the special values at negative integers of the Hecke L-function of K. To define this measure as a distribution, we study the non-constant terms in the Fourier expansion of a particular Eisenstein series of the Hilbert modular group of K. Proving the distribution is a measure requires studying the structure of the Iwasawa algebra.

ON THE DIMENSION OF GROUPS THAT SATISFY CERTAIN CONDITIONS ON THEIR FINITE SUBGROUPS

AbstractWe say a group G satisfies properties (M) and (NM) if every nontrivial finite subgroup of G is contained in a unique maximal finite subgroup, and every nontrivial finite maximal subgroup is self-normalizing. We prove that the Bredon cohomological dimension and the virtual cohomological dimension coincide for groups that admit a cocompact model for EG and satisfy properties (M) and (NM). Among the examples of groups satisfying these hypothesis are cocompact and arithmetic Fuchsian groups, one-relator groups, the Hilbert modular group, and 3-manifold groups.

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Erratum

Sums of squares in real quadratic fields and Hilbert modular groups

We use the spectral theory of Hilbert-Maass forms for real quadratic fields to obtain the asymptotics of some sums involving the number of representations as a sum of two squares in the ring of integers.

Cusp shapes of Hilbert–Blumenthal surfaces

We introduce a new fundamental domain for the cusp stabilizer of a Hilbert modular group over a real quadratic field K=Q(sqrt n). This is constructed as the union of Dirichlet domains for the maximal unipotent group, over the leaves in a foliation of the biplane. The region is the Cartesian product of the positive reals with a 3-dimensional tower formed by deformations of lattices in the ring of integers of K, and makes explicit the cusp cross section's Sol 3-manifold structure and Anosov diffeomorphism. We include computer generated images and data illustrating various examples.

Determinants of Laplacians on Hilbert modular surfaces

We study regularized determinants of Laplacians acting on the space of Hilbert–Maass forms for the Hilbert modular group of a real quadratic field. Weshow that these determinants are described by Selberg type zeta functions introduced in [5, 6].

Higher order Dehn functions for horospheres in products of Hadamard spaces

Abstract Let X be a product of r locally compact and geodesically complete Hadamard spaces. We prove that the horospheres in X centered at regular boundary points of X are Lipschitz-(r − 2)-connected. If X has finite Assouad–Nagata dimension, then using the filling construction by R. Young in [10] this gives sharp bounds on higher order Dehn functions for such horospheres. Moreover, if Γ ⊂ Is(X) is a lattice acting cocompactly on X minus a union of disjoint horoballs, then we get a sharp bound on higher order Dehn functions for Γ. We deduce that apart from the Hilbert modular groups already considered by R. Young, every irreducible ℚ-rank one lattice acting on a product of r Riemannian symmetric spaces of the noncompact type is undistorted up to dimension r−1 and has k-th order Dehn function asymptotic to V (k+1)/k for all k ≤ r − 2.

The algebraic and topological $K$-theory of the Hilbert modular group

In this paper we provide descriptions of the Whitehead groups with coefficients in a ring of the Hilbert modular group and its reduced version, as well as for the topological K-theory of $C^*$-algebras, after tensoring with $\mathbb{Q}$, by computing the source of the assembly maps in the Farrell-Jones and the Baum-Connes conjecture respectively. We also construct a model for the classifying space of the Hilbert modular group for the family of virtually cyclic subgroups.

On the algebraic K-theory of the Hilbert modular group

We give formulas for the Whitehead groups and the rational $K$-theory groups of the (integer group ring of the) Hilbert modular group in terms of its maximal finite subgroups.

Presentations of groups acting discontinuously on direct products of hyperbolic spaces

The problem of describing the group of units U(ZG) of the integral group ring ZG of a finite group G has attracted a lot of attention and providing presentations for such groups is a fundamental problem. Within the context of orders, a central problem is to describe a presentation of the unit group of an order O in the simple epimorphic images A of the rational group algebra QG. Making use of the presentation part of Poincare’s Polyhedron Theorem, Pita, del Rio and Ruiz proposed such a method for a large family of finite groups G and consequently Jespers, Pita, del Rio, Ruiz and Zalesskii described the structure of U(ZG) for a large family of finite groups G. In order to handle many more groups, one would like to extend Poincare’s Method to discontinuous subgroups of the group of isometries of a direct product of hyperbolic spaces. If the algebra A has degree 2 then via the Galois embeddings of the centre of the algebra A one considers the group of reduced norm one elements of the order O as such a group and thus one would obtain a solution to the mentioned problem. This would provide presentations of the unit group of orders in the simple components of degree 2 of QG and in particular describe the unit group of ZG for every group G with irreducible character degrees less than or equal to 2. The aim of this paper is to initiate this approach by executing this method on the Hilbert modular group, i.e. the projective linear group of degree two over the ring of integers in a real quadratic extension of the rationals. This group acts discontinuously on a direct product of two hyperbolic spaces of dimension two. The fundamental domain constructed is an analogue of the Ford domain of a Fuchsian or a Kleinian group.

Lipschitz connectivity and filling invariants in solvable groups and buildings

We give some new methods, based on Lipschitz extension theorems, for bounding filling invariants of subsets of nonpositively curved spaces. We apply our methods to find sharp bounds on higher-order Dehn functions of Sol_{2n+1}, horospheres in euclidean buildings, Hilbert modular groups, and certain S-arithmetic groups.

Differences of the Selberg trace formula and Selberg type zeta functions for Hilbert modular surfaces

We present an example of the Selberg type zeta function for non-compact higher rank locally symmetric spaces. This is a generalization of Selberg's unpublished work [26] to non-compact cases. We study certain Selberg type zeta functions and Ruelle type zeta functions attached to the Hilbert modular group of a real quadratic field. We show that they have meromorphic extensions to the whole complex plane and satisfy functional equations. The method is based on considering the differences among several Selberg trace formulas with different weights for the Hilbert modular group. Besides as an application of the differences of the Selberg trace formula, we also obtain an asymptotic average of the class numbers of indefinite binary quadratic forms over the real quadratic integer ring.

The geometric theta correspondence for Hilbert modular surfaces

In a series of papers we have been studying the geometric theta correspondence for non-compact arithmetic quotients of symmetric spaces associated to orthogonal groups. It is our overall goal to develop a general theory of geometric theta liftings in the context of the real differential geometry/topology of non-compact locally symmetric spaces of orthogonal and unitary groups which generalizes the theory of Kudla-Millson in the compact case. In this paper we study in detail the geometric theta lift for Hilbert modular surfaces. In particular, we will give a new proof and an extension (to all finite index subgroups of the Hilbert modular group) of the celebrated theorem of Hirzebruch and Zagier that the generating function for the intersection numbers of the Hirzebruch-Zagier cycles is a classical modular form of weight 2. In our approach we replace Hirzebuch's smooth complex analytic compactification $\tilde{X}$ of the Hilbert modular surface $X$ with the (real) Borel-Serre compactification $\bar{X}$. The various algebro-geometric quantities are then replaced by topological quantities associated to 4-manifolds with boundary. In particular, the "boundary contribution" in Hirzebruch-Zagier is replaced by sums of linking numbers of circles (the boundaries of the cycles) in the 3-manifolds of type Sol (torus bundle over a circle) which comprise the Borel-Serre boundary.