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\(\mathbb Q\)-curves and abelian varieties of \(\text{GL}_ 2\)-type. (English) Zbl 1035.11026

A \(\mathbb{Q}\)-curve is an elliptic curve defined over \(\overline{\mathbb{Q}}\) that is isogenous to all its Galois conjugates. In the paper under review all \(\mathbb{Q}\)-curves will not have complex multiplication. An abelian variety is of \(\mathrm{GL}_2\)-type if it is an abelian variety defined over \(\mathbb{Q}\) and its \(\mathbb{Q}\)-algebra of \(\mathbb{Q}\)-endomorphisms \(\mathbb{Q}\otimes \operatorname{End}_{\mathbb{Q}}(A)\) is a number field of degree equal to the dimension of \(A\). The main examples of \(\mathrm{GL}_2\)-type abelian varieties are the varieties \(A_f\) constructed by G. Shimura [Introduction to the arithmetic theory of automorphic forms. Princeton: Princeton University Press (1971; Zbl 0221.10029)] attached to newforms \(f\) for \(\Gamma_1(N)\).
Recent interest of \(\mathbb{Q}\)-curves with no complex multiplication arouse from results of Elkies and Ribet. N. Elkies [Remarks on elliptic \(k\)-curves. Preprint, Harvard University (1993)] proved that every isogeny class of a \(\mathbb{Q}\)-curve with no CM contains a curve whose \(j\)-invariant corresponds to a rational non-cusp non-CM point of the modular curve \(X^*(N)\) which is the quotient of \(X_0(N)\) by all the Atkin-Lehner involutions, for some square-free integer \(N\). K. Ribet characterized in [Algebra and Topology 1992, Proceedings of the Seventh KAIST Mathematics Workshop, S. G. Hahn (ed.) et al., 53–79 (1992; Zbl 0998.00011)] \(\mathbb{Q}\)-curves as the elliptic curves over \(\overline{\mathbb{Q}}\) that are quotients of some abelian variety of \(\mathrm{GL}_2\)-type. He also gave evidence that the abelian varieties \(A_f\) exhaust (up to isogeny) all abelian varieties of \(\mathrm{GL}_2\)-type. In particular, one had the conjectural characterization of \(\mathbb{Q}\)-curves as those elliptic curves defined over \(\mathbb{Q}\) that are quotients of some \(J_1(N)\).
The fact that an elliptic curve is a \(\mathbb{Q}\)-curve is invariant by isogeny. So it is natural to investigate their arithmetic properties up to isogenies, in particular their fields of definition. K. Ribet [Contemp. Math. 174, 107–118 (1994; Zbl 0877.14030] attached to a \(\mathbb{Q}\)-curve a 2-cocycle class \(\xi(C)\in H^2(G_{\mathbb{Q}},\mathbb{Q}^*)\) which is an invariant of the isogeny class of \(C\). He characterized the fields of definition for the curve up to isogeny in terms of \(\xi(C)\). He used this characterization to show that there is always a smallest such field and this is a field of type \((2,\dots,2)\).
Interesting aspects of the arithmetic of \(\mathbb{Q}\)-curves appear when the curve \(C\) is defined over a field \(k\) over which all its Galois conjugates are defined as well as the isogenies between them. In this case it is said that \(C\) is completely defined over \(k\). The author obtains a criterion characterizing fields over which a \(\mathbb{Q}\)-curve is completely defined up to isogeny in terms of \(\xi(C)\). As a consequence, he shows that every \(\mathbb{Q}\)-curve is isogenous to one completely defined over a field of type \((2,\dots,2)\) strictly containing in general the smallest field of definition up to isogeny. He computes an element \(\xi(C)_{\pm}\in\,\)Br\(_2(\mathbb{Q})\) related to \(\xi(C)\) and reformulates the criterion in terms of this element which is easier to apply in practice.
An abelian variety is said to be attached to a \(\mathbb{Q}\)-curve if it is an abelian variety of \(\mathrm{GL}_2\)-type of which the \(\mathbb{Q}\)-curve is a quotient. The \(\mathbb{Q}\)-endomorphism algebras [K. Ribet (1992; loc. cit.) and E. E. Pyle, Abelian varieties over \(\mathbb{Q}\) with large endomorphism algebras and their simple components over \(\overline{\mathbb{Q}}\), PhD thesis, University of California, Berkeley (1995)] of abelian varieties attached to a \(\mathbb{Q}\)-curve are shown to be of the type \(E=\mathbb{Q}(\{\beta(\sigma)\})\), where \(\beta:G_{\mathbb{Q}}\to\overline{\mathbb{Q}}^*\) is a splitting map. The author shows that the fields \(E\) depend only on certain Galois characters \(\varepsilon:G_{\mathbb{Q}}\to \overline{\mathbb{Q}}^*\) attached to \(\xi(C)\) which can be computed in terms of \(\xi(C)_{\pm}\). This result yields information on the abelian varieties attached to \(\mathbb{Q}\)-curves, in particular their dimensions are computed.
The abelian varieties attached to a \(\mathbb{Q}\)-curve \(C\) constructed by Ribet in [K. Ribet (1992; loc. cit.) were obtained as factors of \(B=\text{Res}_{K/\mathbb{Q}}(C/K)\) which are restriction of scalars to a field \(K\) where \(C\) is completely defined. The author shows how to control the factors in the \(\mathbb{Q}\)-decomposition of \(B\) as a product of simple abelian varieties in the case where \(B\) factors up to isogeny as a product of non \(\mathbb{Q}\)-isogenous abelian varieties of \(\mathrm{GL}_2\)-type.

MSC:

11G10 Abelian varieties of dimension \(> 1\)
11G16 Elliptic and modular units
14K15 Arithmetic ground fields for abelian varieties
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