Hyperbolic polynomials and rigid orders of moduli

Arithmetic of singular moduli and class polynomials

We investigate divisibility properties of the traces and Hecke traces of singular moduli. In particular we prove that, if p is prime, these traces satisfy many congruences modulo powers of p which are described in terms of the factorization of p in imaginary quadratic fields. We also study generalizations of Lehner’s classical congruences j(z)|Up ≡ 744 (mod p) (where p 11 and j(z) is the usual ...

Hyperbolic Polynomials and Spectral Order

Hyperbolic Polynomials and Convex Analysis

A homogeneous real polynomial p is hyperbolic with respect to a given vector d if the univariate polynomial t → p(x − td) has all real roots for all vectors x. Motivated by partial differential equations, Gårding proved in 1951 that the largest such root is a convex function of x, and showed various ways of constructing new hyperbolic polynomials. We present a powerful new such construction, an...

Peripheral Polynomials of Hyperbolic Knots

If K is a hyperbolic knot in the oriented S3, an algebraic component of its character variety containing the holonomy of the complete hyperbolic structure of finite volume of S3 \ K is an algebraic curve (excellent component K). The traces of the peripheral elements of K define polynomial functions in K. These functions are related in pairs by canonical polynomials. These peripheral polynomials...

The Moduli Space of Hyperbolic Cone Structures

Introduction. Roughly speaking, a cone structure is a manifold together with a link each of whose component has a cone angle attached. It is a kind of singular manifold structure. If each cone angle is of the form 2π/n, for some integer n, the cone structure becomes an orbifold structure. Unlike an orbifold structure, the cone structure is not a natural concept, but it turns out to be very impo...