Related papers: Genus fields of finite abelian extensions

In this paper we obtain the extended genus field of a finite abelian extension of a global rational function field. We first study the case of a cyclic extension of prime power degree. Next, we use that the extended genus fields of a…

In this paper we first obtain the genus field of a finite abelian non-Kummer $l$--extension of a global rational function field. Then, using that the genus field of a composite of two abelian extensions of a global rational function field…

Let $G$ be a finite abelian $p$-group. We count \'etale $G$-extensions of global rational function fields $\mathbb F_q(T)$ of characteristic $p$ by the degree of what we call their Artin-Schreier conductor. The corresponding (ordinary)…

In this paper we obtain the genus field of a general Kummer extension of a global rational function field. We study first the case of a general Kummer extension of degree a power of a prime. Then we prove that the genus field of a composite…

We obtain the extended genus field of an abelian extension of a rational function field. We follow the definition of Angl\`es and Jaulent, which uses class field theory. First we show that the natural definition of extended genus field of a…

In this paper we obtain the extended genus field of a global field. First we define the extended genus field of a global function field and we obtain, via class field theory, the description of the extended genus field of an arbitrary…

We study the arithmetic aspects of the finite group of extensions of abelian varieties defined over a number field. In particular, we establish relations with special values of L-functions and congruences between modular forms.

In this work we present some arithmetic properties of families of abelian $p$--extensions of global function fields, among which are their generators and their type of ramification and decomposition.

We give an exact formula for the number of $G$-extensions of local function fields $\mathbb{F}_q((t))$ for finite abelian groups $G$ up to a conductor bound. As an application we give a lower bound for the corresponding counting problem by…

In the present work we give the construction of the genus field and the extended genus field of an elementary abelian $l$-extension of a field of rational functions, where $l$ is a prime number. In the Kummer case, if $K$ is contained in a…

In the published version of this paper [Finite Fields and Their Applications {\bf 20} (2013) 40--54], there is an error in the proof of Theorem 4.2 of the paper. Here we correct the error and give the right statments for Theorems 4.2, 4.5…

Let $U/L$ be a finite abelian extension of number fields. We first construct a universal primitive generator of $U$ over $L$ whose relative trace to any intermediate field $F$ becomes a generator of $F$ over $L$, too. We also develop a…

Degree bounds for algebra generators of invariant rings are a topic of longstanding interest in invariant theory. We study the analogous question for field generators for the field of rational invariants of a representation of a finite…

In the present work, we determine explicitly the genus of any separable cubic extension of any global function field given the minimal polynomial of the extension. We give algorithms computing the ramification data and the genus of any…

The article at hand contains exact asymptotic formulas for the distribution of conductors of abelian p-extensions of global function fields of characteristic p. These yield a new conjecture for the distribution of discriminants fueled by an…

We determine the distribution of discriminants of wildly ramified elementary-abelian extensions of local and global function fields in characteristic $p$. For local and rational function fields, we also give precise formulae for the number…

We study degree bounds on rational but not necessarily polynomial generators for the field $\mathbf{k}(V)^G$ of rational invariants of a linear action of a finite abelian group. We show that lattice-theoretic methods used recently by the…

We extend the well-known Cassels-Tate dual exact sequence for abelian varieties A over global fields K in two directions: we treat the p-primary component in the function field case, where p is the characteristic of K, and we dispense with…

The article at hand contains exact asymptotic formulas for the distribution of conductors of elementary abelian p-extensions of global function fields of characteristic p. As a consequence for the distribution of discriminants, this leads…

We study on finite unramified extensions of global function fields (function fields of one valuable over a finite field). We show two results. One is an extension of Perret's result about the ideal class group problem. Another is a…