Geometric methods in monogenic extensions

Authors

  • Francesc Pedret

Keywords:

monogenity, diophantine equations, elliptic curves, Galois cohomology.

Abstract

Un cos de nombres K és monogen si el seu anell d’enters està generat per un sol element com a Z-àlgebra. En el cas cúbic, determinar si K és monogen o no és equivalent a resoldre l’equació diofàntica |IK (X,Y )| = 1 sobre Z, on IK és la forma índex del cos. Una solució entera determina un punt racional a la corba de gènere 1 IK (X,Y ) = Z3. Mitjançant aquesta construcció, es pot demostrar que K determina una F3-òrbita en H1(Q, E[3]), on E és la corba el·líptica definida per Y 2 = 4X3 + Disc(K). Donem la construcció explícita d’aquesta òrbita pel cas de cossos cúbics purs i caracteritzem la suma de cocicles associats a cossos no isomorfs.

Downloads

Download data is not yet available.

Downloads

How to Cite

Pedret, F. (2024). Geometric methods in monogenic extensions. Reports@SCM, 9(1), 101–102. Retrieved from https://revistes.iec.cat/index.php/reports/article/view/154350

Issue

Vol. 9 No. 1 (2024)

Section

Extended abstracts

License

The intellectual property of articles belongs to the respective authors.
On submitting articles for publication to the journal Reports@SCM, authors accept the following terms:
  1. Authors assign to Societat Catalana de Matemàtiques (a subsidiary of Institut d’Estudis Catalans) the rights of reproduction, communication to the public and distribution of the articles submitted for publication to Reports@SCM.
  2. Authors answer to Societat Catalana de Matemàtiques for the authorship and originality of submitted articles.
  3. Authors are responsible for obtaining permission for the reproduction of all graphic material included in articles.
  4. Societat Catalana de Matemàtiques declines all liability for the possible infringement of intellectual property rights by authors.
  5. The contents published in the journal, unless otherwise stated in the text or in the graphic material, are subject to a Creative Commons Attribution-NonCommercial-NoDerivs (by-nc-nd) 3.0 Spain licence, the complete text of which may be found at https://creativecommons.org/licenses/by-nc-nd/3.0/es/deed.en. Consequently, the general public is authorised to reproduce, distribute and communicate the work, provided that its authorship and the body publishing it are acknowledged, and that no commercial use and no derivative works are made of it.
  6. The journal Reports@SCM is not responsible for the ideas and opinions expressed by the authors of the published articles.