In this paper, we prove Hurwitz-Eichler type formulas for Hurwitz class numbers with each level $ M when the modular curve X_0(M) has genus zero. A key idea is to calculate intersection of correspondences in two different ways. generalization Atkin-Lehner involutions \Gamma_0(M) and its subgroup \Gamma_0^{(M')}(M) introduced multiplicities at cusps.

Double Hurwitz numbers count covers of the sphere by genus g curves with assigned ramification profiles over 0 and ∞, and simple ramification over a fixed branch divisor. Goulden, Jackson and Vakil (2005) have shown double Hurwitz numbers are piecewise polynomial in the orders of ramification, and Shadrin, Shapiro and Vainshtein (2008) have determined the chamber structure and wall crossing for...

In this paper we describe explicit generating functions for a large class of Hurwitz-Hodge integrals. These are integrals of tautological classes on moduli spaces of admissible covers, a (stackily) smooth compactification of the Hurwitz schemes. Admissible covers and their tautological classes are interesting mathematical objects on their own, but recently they have proved to be a useful tool f...

In 1986 E. I. Jury conjectured by analogy to the theory of digital filters that a two-dimensional analog filter is BIBO stable if its transfer function is of the form H = 1/P, where P is a very strict Hurwitz polynomial (VSHP). In this article we prove a generalisation of Jury’s conjecture to r -dimensional analog filters (r > 2) with proper transfer function H = Q/P, where the denominator P is...

The set of residue classes modulo an element π in the rings of Gaussian integers, Lipschitz integers and Hurwitz integers, respectively, is used as alphabets to form the words of error correcting codes. An error occurs as the addition of an element in a set E to the letter in one of the positions of a word. If E is a group of units in the original rings, then we obtain the Mannheim, Lipschitz a...

Let Z3 act on C 2 by non-trivial opposite characters. Let X = [C/Z3] be the orbifold quotient, and let Y be the unique crepant resolution. We show the equivariant genus 0 Gromov-Witten potentials FX and F Y are equal after a change of variables — verifying the Crepant Resolution Conjecture for the pair (X , Y ). Our computations involve Hodge integrals on trigonal Hurwitz spaces which are of in...

The Hurwitz space is a compactification of the space of rational functions of a given degree. We study the intersection of various strata of this space with its boundary. A study of the cohomology ring of the Hurwitz space then allows us to obtain recurrence relations for certain numbers of ramified coverings of a sphere by a sphere with prescribed ramifications. Generating functions for these ...

Résumé. Dans ce travail, un codage symbolique des séries génératrices de Dirichlet généralisées est obtenu par les techniques combinatoires des séries formelles en variables non-commutative. Il permet d’expliciter les séries génératrices de Dirichlet généralisées ’périodiques’ – donc notamment les polyzêtas colorés – comme combinaison linéaire de polyzêtas de Hurwitz. De plus, la version non co...

We study “pure-cycle” Hurwitz spaces, parametrizing covers of the projective line having only one ramified point over each branch point. We start with the case of genus-0 covers, using a combination of limit linear series theory and group theory to show that these spaces are always irreducible. In the case of four branch points, we also compute the associated Hurwitz numbers. Finally, we give a...

We present several asymptotic analyses for quantities associated with the Riemann and Hurwitz zeta functions. We first determine the leading asymptotic behavior of the Stieltjes constants γk(a). These constants appear in the regular part of the Laurent expansion of the Hurwitz zeta function. We then use asymptotic results for the Laguerre polynomials Ln to investigate a certain sum Sγ(n) involv...