Fields of definition of ℚ -curves

Jordi Quer

Journal de théorie des nombres de Bordeaux (2001)

  • Volume: 13, Issue: 1, page 275-285
  • ISSN: 1246-7405

Abstract

top
Let C be a ℚ -curve with no complex multiplication. In this note we characterize the number fields K such that there is a curve C ' isogenous to C having all the isogenies between its Galois conjugates defined over K , and also the curves C ' isogenous to C defined over a number field K such that the abelian variety Res K / ℚ ( C ' / K ) obtained by restriction of scalars is a product of abelian varieties of GL 2 -type.

How to cite

top

Quer, Jordi. "Fields of definition of $\mathbb {Q}$-curves." Journal de théorie des nombres de Bordeaux 13.1 (2001): 275-285. <http://eudml.org/doc/248694>.

@article{Quer2001,
abstract = {Let $C$ be a $\mathbb \{Q\}$-curve with no complex multiplication. In this note we characterize the number fields $K$ such that there is a curve $C^\{\prime \}$ isogenous to $C$ having all the isogenies between its Galois conjugates defined over $K$, and also the curves $C^\{\prime \}$ isogenous to $C$ defined over a number field $K$ such that the abelian variety Res$_\{K/\mathbb \{Q\}\} (C^\{\prime \}/K)$ obtained by restriction of scalars is a product of abelian varieties of GL$_2$-type.},
author = {Quer, Jordi},
journal = {Journal de théorie des nombres de Bordeaux},
language = {eng},
number = {1},
pages = {275-285},
publisher = {Université Bordeaux I},
title = {Fields of definition of $\mathbb \{Q\}$-curves},
url = {http://eudml.org/doc/248694},
volume = {13},
year = {2001},
}

TY - JOUR
AU - Quer, Jordi
TI - Fields of definition of $\mathbb {Q}$-curves
JO - Journal de théorie des nombres de Bordeaux
PY - 2001
PB - Université Bordeaux I
VL - 13
IS - 1
SP - 275
EP - 285
AB - Let $C$ be a $\mathbb {Q}$-curve with no complex multiplication. In this note we characterize the number fields $K$ such that there is a curve $C^{\prime }$ isogenous to $C$ having all the isogenies between its Galois conjugates defined over $K$, and also the curves $C^{\prime }$ isogenous to $C$ defined over a number field $K$ such that the abelian variety Res$_{K/\mathbb {Q}} (C^{\prime }/K)$ obtained by restriction of scalars is a product of abelian varieties of GL$_2$-type.
LA - eng
UR - http://eudml.org/doc/248694
ER -

References

top
  1. [1] N. Elkies, Remarks on elliptic k-curves. Preprint, 1992. 
  2. [2] E. Pyle, Abelian varieties over Q with large endomorphism algebras and their simple components over Q. Ph.D. Thesis, Univ. of California atBerkeley, 1995. 
  3. [3] J. Quer, Q-curves and Abelian varieties of GL2-type. Proc. London Math. Soc. (3) 81 (2000), 285-317. Zbl1035.11026MR1770611
  4. [4] K. Ribet, Abelian varieties over Q and modular forms. Proceedings of KAIST Mathematics Workshop (1992), 53-79. MR1212980
  5. [5] K. Ribet, Fields of definition of Abelian varieties with real multiplication. Contemp. Math.174 (1994), 107-118. Zbl0877.14030MR1299737

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.