Related papers: The $\chi$-part of the analytic class number formu…

Let $F$ be a totally real field and $K$ a finite abelian CM extension of $F$. Using class field theory, we show that our previous result giving a strong form of the Brumer-Stark conjecture implies the minus part of the equivariant Tamagawa…

Among abelian extensions of a congruence function field, an asymptotic relation of class number and genus is established. The proof is classical, employing well-known results from congruence function field theory.

Let F be a global function field and let F^ab be its maximal abelian extension. Following an approach of D.Hayes, we shall construct a continuous homomorphism \rho: Gal(F^ab/F) \to C_F, where C_F is the idele class group of F. Using class…

Let $q$ be a power of a prime number $p$, $k=\mathbb{F}_{q}(t)$ be the rational function field over finite field $\mathbb{F}_{q}$ and $K/k$ be a multi-cyclic extension of prime degree. In this paper we will give an exact formula for the…

Let $k$ be a field containing an algebraically closed field of characteristic zero. If $G$ is a finite group and $D$ is a division algebra over $k$, finite dimensional over its center, we can associate to a faithful $G$-grading on $D$ a…

We prove a formula for the $\infty$-adic special $L$-value of abelian $t$-modules. This gives function field analogues of the class number formula. We also express it in terms of the extension groups of shtukas.

Dirichlet's Lemma states that every primitive quadratic Dirichlet character $\chi$ can be written in the form $\chi(n) = (\frac{\Delta}n)$ for a suitable quadratic discriminant $\Delta$. In this article we define a group, the separant class…

In this article, we prove that a general version of Alladi's formula with Dirichlet convolution holds for arithmetical semigroups satisfying Axiom $A$ or Axiom $A^{\#}$. As applications, we apply our main results to certain semigroups…

Let $K/\Q$ be a cyclic extension of number fields with Galois group $G$. We study the ideal classes of primes $\mathfrak{p}$ of $K$ of residue degree bigger than one in the class group of $K$. In particular, we explore such extensions…

We settle a long-standing problem in the theory of Hecke algebras of complex reflection groups by constructing many (graded) integral cellular bases of these algebras. As applications, we explicitly construct the simple modules of Ariki's…

Let $K$ be a number field. We show that, up to allowing a finite set of denominators in the partial quotients, it is possible to define algorithms for $\mathfrak P$-adic continued fractions satisfying the finiteness property on $K$ for…

We propose a conjecture refining the Stark conjecture St(K/k,S) (Tate's formulation) in the function field case. Of course St(K/k,S) in this case is a theorem due to Deligne and independently to Hayes. The novel feature of our conjecture is…

We characterize the maximum $r$-local index of a Schur algebra over an abelian number field $K$ in terms of global information determined by the field $K$, for $r$ an arbitrary rational prime. This completes and unifies previous results of…

We investigate sections of arithmetic fundamental groups of hyperbolic curves over function fields. As a consequence we prove that the anabelian section conjecture of Grothendieck holds over all finitely generated fields over $\Bbb Q$ if it…

We extend to the context of algebraic groups a classic result on extensions of abstract groups relating the set of isomorphism classes of extensions of $G$ by $H$ with that of extensions of $G$ by the center $Z$ of $H$. The proof should be…

Let $K$ be a totally real number field of degree $r$. Let $K_{\infty}$ denote the cyclotomic $\mathbb{Z}_{2}$-extension of $K$ and let $L_{\infty}$ be a finite extension of $K_{\infty}$, abelian over $K$. The goal of this paper is to…

The well known Andrews-Curtis Conjecture [2] is still open. In this paper, we establish its finite version by describing precisely the connected components of the Andrews-Curtis graphs of finite groups. This finite version has independent…

We associate a rational number $\chi(\mathcal{A})$ to every category $\mathcal{A}$ whose object and morphism sets are finite. We show that the assignment $\chi$ is additive under disjoint union and it preserves products. Hence we consider…

We compute the Picard group of the universal abelian variety over the moduli stack $\mathscr A_{g,n}$ of principally polarized abelian varieties over $\mathbb{C}$ with a symplectic principal level $n$-structure. We then prove that over…

The Steinitz class of a number field extension K/k is an ideal class in the ring of integers O_k of k, which, together with the degree [K:k] of the extension determines the O_k-module structure of O_K. We call R_t(k,G) the classes which are…