# Fields of definition of $\mathbb{Q}$-curves

Journal de théorie des nombres de Bordeaux (2001)

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

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topQuer, 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] N. Elkies, Remarks on elliptic k-curves. Preprint, 1992.
- [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] J. Quer, Q-curves and Abelian varieties of GL2-type. Proc. London Math. Soc. (3) 81 (2000), 285-317. Zbl1035.11026MR1770611
- [4] K. Ribet, Abelian varieties over Q and modular forms. Proceedings of KAIST Mathematics Workshop (1992), 53-79. MR1212980
- [5] K. Ribet, Fields of definition of Abelian varieties with real multiplication. Contemp. Math.174 (1994), 107-118. Zbl0877.14030MR1299737

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