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We give a survey of computational class field theory. We first explain how to compute ray class groups and discriminants of the corresponding ray class fields. We then explain the three main methods in use for computing an equation for the class fields themselves: Kummer theory, Stark units and complex multiplication. Using these techniques we can construct many new number fields, including fields of very small root discriminant.

A variety of Euler's sum of powers conjecture

Tianxin Cai, Yong Zhang (2021)

Czechoslovak Mathematical Journal

We consider a variety of Euler’s sum of powers conjecture, i.e., whether the Diophantine system $\{\begin{matrix} n = a_{1} + a_{2} + \dots + a_{s - 1}, \\ a_{1} a_{2} \dots a_{s - 1} (a_{1} + a_{2} + \dots + a_{s - 1}) = b^{s} \end{matrix}$ has positive integer or rational solutions $n$ , $b$ , $a_{i}$ , $i = 1, 2, \dots, s - 1$ , $s \geq 3 .$ Using the theory of elliptic curves, we prove that it has no positive integer solution for $s = 3$ , but there are infinitely many positive integers $n$ such that it has a positive integer solution for $s \geq 4$ . As a corollary, for $s \geq 4$ and any positive integer $n$ , the above Diophantine system has a positive rational solution. Meanwhile, we give conditions such that...

Abelian surfaces of GL₂-type as Jacobians of curves

Josep González, Jordi Guàrdia, Victor Rotger (2005)

Acta Arithmetica

Abelian surfaces over finite fields as Jacobians. With an appendix by Everett W. Howe.

Maisner, Daniel, Nart, Enric (2002)

Experimental Mathematics

Abelian varieties, ...-adic representations, and ...-independence.

M. Larsen, R. Pink (1995)

Mathematische Annalen

Abelian varieties and Galois module structure in global function fields.

A. Agboola (1994)

Mathematische Zeitschrift

Abelian varieties over fields of finite characteristic

Yuri Zarhin (2014)

Open Mathematics

The aim of this paper is to extend our previous results about Galois action on the torsion points of abelian varieties to the case of (finitely generated) fields of characteristic 2.

Abelian varieties over finite fields with a specified characteristic polynomial modulo $ℓ$

Joshua Holden (2004)

Journal de Théorie des Nombres de Bordeaux

We estimate the fraction of isogeny classes of abelian varieties over a finite field which have a given characteristic polynomial $P (T)$ modulo $ℓ$ . As an application we find the proportion of isogeny classes of abelian varieties with a rational point of order $ℓ$ .

Abelian varieties-Galois representation and properties of ordinary reduction

Rutger Noot (1995)

Compositio Mathematica

Abstract class field theory (formatting of modules).

Ershov, Yu.L. (2004)

Sibirskie Ehlektronnye Matematicheskie Izvestiya [electronic only]

Adelic equidistribution of tori and equidistribution of CM points. (Équidistribution adélique des tores et équidistribution des points CM.)

Clozel, L., Ullmo, E. (2007)

Documenta Mathematica

Algebraic covers : field of moduli versus field of definition

Pierre Dèbes, Jean-Claude Douai (1997)

Annales scientifiques de l'École Normale Supérieure

Algebraic leaves of algebraic foliations over number fields

Jean-Benoît Bost (2001)

Publications Mathématiques de l'IHÉS

We prove an algebraicity criterion for leaves of algebraic foliations defined over number fields. Namely, consider a number field $K$ embedded in $C$ , a smooth algebraic variety $X$ over $K$ , equipped with a $K -$ rational point $P$ , and $F$ an algebraic subbundle of the its tangent bundle $T_{X}$ , defined over $K$ . Assume moreover that the vector bundle $F$ is involutive, i.e., closed under Lie bracket. Then it defines an holomorphic foliation of the analytic manifold $X (C)$ , and one may consider its leaf $F$ through $P$ . We prove...

Algebraic points of low degree on the Fermat quintic

Matthew Klassen, Pavlos Tzermias (1997)

Acta Arithmetica

Algebraic S-integers of fixed degree and bounded height

Fabrizio Barroero (2015)

Acta Arithmetica

Let k be a number field and S a finite set of places of k containing the archimedean ones. We count the number of algebraic points of bounded height whose coordinates lie in the ring of S-integers of k. Moreover, we give an asymptotic formula for the number of S̅-integers of bounded height and fixed degree over k, where S̅ is the set of places of k̅ lying above the ones in S.

Algebraic values of G-functions.

F. Beukers (1993)

Journal für die reine und angewandte Mathematik

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