Functional equations in number theory are relations between an L-function and some sort of dual L-function, and in general, the L-function and its dual need not coincide. For example, if χ is a primitive Dirichlet character then the functional equation relates L(s, χ) to L(1− s, χ), and L(s, χ) = L(s, χ) if and only if χ = 1. Or if f is a primitive cusp form of weight two for Γ1(N) and f∨ is th...

The algebraic closure of R is C, which is a finite extension. Are there other fields which are not algebraically closed but have an algebraic closure which is a finite extension? Yes. An example is the field of real algebraic numbers. Since complex conjugation is a field automorphism fixing Q, and the real and imaginary parts of a complex number can be computed using field operations and comple...

Let k be a complete discrete valuation field of equal characteristic p > 0 with possibly imperfect residue field. Let Gk be its Galois group. We will prove that the conductors computed by the arithmetic ramification filtrations on Gk defined in [2] coincide with the differential Artin conductors and Swan conductors of Galois representations of Gk defined in [15]. As a consequence, we give a Has...

The solvability of monomial groups is a well-known result in character theory. Certain properties of Artin L-series suggest a generalization of these groups, namely to such groups where every irreducible character has some multiple which is induced from a character φ of U with solvable factor group U/ker(φ). Using the classification of finite simple groups, we prove that these groups are also s...

Let A be an Artin group with standard generating set Σ. Tits conjectured that the only relations in A amongst the squares of the generators are the obvious ones, namely that σ2 and τ2 commute whenever σ and τ commute, for σ, τ ∈ Σ. In this paper we prove Tits’ conjecture for all Artin groups. In fact, given a number m(σ) ≥ 2 for each σ ∈ Σ, we show that the elements {T (σ) = σm(σ) : σ ∈ Σ} gene...

Let k be a complete discretely valued field of equal characteristic p > 0 with possibly imperfect residue field and let Gk be its Galois group. We prove that the conductors computed by the arithmetic ramification filtrations on Gk defined in [3] coincide with the differential Artin conductors and Swan conductors of Galois representations of Gk defined in [17]. As a consequence, we give a Hasse-...

It is well-known that the Artin-Mazur dynamical zeta function of a hyperbolic or quasi-hyperbolic toral automorphism is a rational function, which can be calculated in terms of the eigenvalues of the corresponding integer matrix. We give an elementary proof of this fact that extends to the case of general toral endomorphisms without change. The result is a closed formula that can be calculated ...

We implement new techniques involving Artin fans to study the realizability of tropical stable maps in superabundant combinatorial types. Our approach is to understand the skeleton of a fundamental object in logarithmic Gromov–Witten theory—the stack of prestable maps to the Artin fan. This is used to examine the structure of the locus of realizable tropical curves and derive three principal co...

Let G be a group endowed with a solution to the conjugacy problem and with an algorithm which computes the centralizer in G of any element of G. Let H be a subgroup of G. We give some conditions on H , under which we provide a solution to the conjugacy problem in H . We apply our results to some Artin groups and braid type groups. In particular, we give explicit solutions to the conjugacy probl...

We first introduce the notion of an Auslander-Reiten sequence in a Krull-Schmidt category. This unifies the notion of an almost split sequence in an abelian category and that of an Auslander-Reiten triangle in a triangulated category. We then define the Auslander-Reiten quiver of a Krull-Schmidt category and describe the shapes of its semi-stable components. The main result generalizes those fo...