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

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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

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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

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  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

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