Special Values of Class Group L-Functions for CM Fields

Let H be the Hilbert class field of a CM number field K with maximal totally real subfield F of degree n over ℚ. We evaluate the second term in the Taylor expansion at s = 0 of the Galoisequivariant L-function ΘS∞(s) associated to the unramified abelian characters of Gal(H/K). This is an identity in the group ring C[Gal(H/K)] expressing Θ (n)S∞ (0) as essentially a linear combination of logarithms of special values ﹛Ψ (zσ)﹜, where Ψ: ℍn →ℝ is a Hilbert modular function for a congruence subgroup of SL2 (OF) and ﹛zσ : σ ∈ Gal(H/K)﹜ are CM points on a universal Hilbert modular variety. We apply this result to express the relative class number hH/hK as a rational multiple of the determinant of an (hK − 1) × (hK − 1) matrix of logarithms of ratios of special values Ψ (zσ), thus giving rise to candidates for higher analogs of elliptic units. Finally, we obtain a product formula for Ψ (zσ) in terms of exponentials of special values of L-functions.

On Ihara’s lemma for Hilbert modular varieties

Let ρ be a two-dimensional modulo p representation of the absolute Galois group of a totally real number field. Under the assumptions that ρ has a large image and admits a low-weight crystalline modular deformation we show that any low-weight crystalline deformation of ρ unramified outside a finite set of primes will be modular. We follow the approach of Wiles as generalized by Fujiwara. The main new ingredient is an Ihara-type lemma for the local component at ρ of the middle degree cohomology of a Hilbert modular variety. As an application we relate the algebraic p-part of the value at one of the adjoint L-function associated with a Hilbert modular newform to the cardinality of the corresponding Selmer group.

Cusps of Hilbert modular varieties

Motivated by a question of Hirzebruch on the possible topological types of cusp cross-sections of Hilbert modular varieties, we give a necessary and sufficient condition for a manifoldMto be diffeomorphic to a cusp cross-section of a Hilbert modular variety. Specialized to Hilbert modular surfaces, this proves that every Sol 3–manifold is diffeo morphic to a cusp cross-section of a (generalized) Hilbert modular surface. We also deduce an obstruction to geometric bounding in this setting. Consequently, there exist Sol 3–manifolds that cannot arise as a cusp cross-section of a 1–cusped nonsingular Hilbert modular surface.

Effective resolution of cusps on Hilbert modular varieties

In this paper, we use the Shintani decomposition, known to number theorists, to describe an effective method of finding a resolution of the cusps of a Hilbert modular variety, in any dimension.