The Hurwitz space approach to the regular Inverse Galois Problem was the only successful approach to Galois group realizations beyond nilpotent groups. It gave regular realizations of many series of groups. More significantly, the M(odular) T(ower) program identified obstructions to systematically finding regular realizations. Finding a way around those obstructions generalize renown results on...

In this chapter, we will see a proof of the analytic continuation of the Riemann zeta function ζ(s) and the Dirichlet L function L(s, χ) via the Hurwitz zeta function. This then gives rise to a functional equation for ζ(s) and a direct computation for the value of this function at negative integer points. 1. The Hurwitz zeta function We have already seen the definition of the Riemann zeta funct...

In this paper, we define a (p, v)-extension of Hurwitz-Lerch Zeta function by considering an extension of beta function defined by Parmar et al. [J. Classical Anal. 11 (2017) 81106]. We obtain its basic properties which include integral representations, Mellin transformation, derivative formulas and certain generating relations. Also, we establish the special cases of the main results. keywords...

We investigate the Hurwitz action of the braid group Br n on the n-fold Cartesian product Br n 3 and determine some stabilisers of its Artin systems. Our algebraic result is complemented by a geometric study of families of plane polynomial coverings of degree 3. Together they lead to characterisations of the set of paths realised by degenerations of the polynomials as defined by Donaldson [Do].

This survey grew out of notes accompanying a cycle of lectures at the workshop Modern Trends in Gromov-Witten Theory, in Hannover. The lectures are devoted to interactions between Hurwitz theory and Gromov-Witten theory, with a particular eye to the contributions made to the understanding of the Double Ramification Cycle, a cycle in the moduli space of curves that compactifies the double Hurwit...

100 years ago exactly, in 1906, Hartogs published a celebrated extension phenomenon (birth of Several Complex Variables), whose global counterpart was understood later: holomorphic functions in a connected neighborhood V(∂Ω) of a connected boundary ∂Ω b C (n > 2) do extend holomorphically and uniquely to the domain Ω. Martinelli in the early 1940’s and Ehrenpreis in 1961 obtained a rigorous pro...

1.1 The classical Hurwitz space and the moduli of curves The classical Hurwitz space first appeared in the work of Clebsch [5] and Hurwitz [17] as an auxiliary object to study the moduli space of curves. Let X be a smooth projective curve of genus g over C. A rational function f : X → P of degree n is called simple if there are at least n − 1 points on X over every point of P. Such a cover has ...

We prove the second order vanishing of a new family of character combinations of Hurwitz zeta functions, thus answering a query of Anderson in [And02]. Along the way, we prove a family of new trigonometric identities.

We study properties of the tropical double Hurwitz loci defined by Bertram, Cavalieri and Markwig. We show that all such loci are connected in codimension one. If we mark preimages of simple ramification points, then for a generic choice of such points the resulting cycles are weakly irreducible, i.e. an integer multiple of an irreducible cycle. We study how Hurwitz cycles can be written as div...

We consider the analogue of Hurwitz curves, smooth projective curves C of genus g ≥ 2 that realize equality in the Hurwitz bound |Aut(C)| ≤ 84(g − 1), to smooth compact quotients S of the unit ball in C. When S is arithmetic, we show that |Aut(S)| ≤ 288e(S), where e(S) is the (topological) Euler characteristic, and in the case of equality show that S is a regular cover of a particular Deligne–M...