The arithmetic of QM-abelian surfaces through their Galois representations (original) (raw)
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Acta Arithmetica, 2013
An abelian variety over a number field is called L-abelian variety if, for any element of the absolute Galois group of a number field L, the conjugated abelian variety is isogenous to the given one by means of an isogeny that preserves the Galois action on the endomorphism rings. We can think of them as generalizations of abelian varieties defined over L with endomorphisms also defined over L. In the one dimensional case, an elliptic curve defined over L gives rise to a Galois representation provided by the Galois action on its Tate module. This classical Galois representation has been a central object of study in Number Theory over the last decades. Besides, given an elliptic L-curve one can construct a projective analogue of the previous Galois representation. In this work we construct similar projective representations in the two-dimensional case, namely, attached to abelian L-surfaces with quaternionic multiplication or fake elliptic curves. Moreover, we prove that such projecti...
Modularity of Abelian Surfaces with Quaternionic Multiplication
Mathematical Research Letters, 2003
We prove that any abelian surface defined over Q of GL 2-type having quaternionic multiplication and good reduction at 3 is modular. We generalize the result to higher dimensional abelian varieties with "sufficiently many endomorphisms"
On finiteness conjectures for endomorphism algebras of abelian surfaces
Mathematical Proceedings of the Cambridge Philosophical Society, 2006
It is conjectured that there exist only finitely many isomorphism classes of endomorphism algebras of abelian varieties of bounded dimension over a number field of bounded degree. We explore this conjecture when restricted to quaternion endomorphism algebras of abelian surfaces of GL 2type over Q by giving a moduli interpretation which translates the question into the diophantine arithmetic of Shimura curves embedded in Hilbert surfaces. We address the resulting problems on these curves by local and global methods, including Chabauty techniques on explicit equations of Shimura curves.
Abelian Varieties with Quaternion Multiplication
2005
In this article we use a Prym construction to study low dimensional abelian varieties with an action of the quaternion group. In special cases we describe the Shimura variety parameterizing such abelian varieties, as well as a map to this Shimura variety from a natural parameter space of quaternionic abelian varieties. Our description is based on the moduli of cubic
Parametrization of Abelian K-surfaces with quaternionic multiplication
Comptes Rendus Mathematique, 2009
We prove that the abelian K-surfaces whose endomorphism algebra is an indefinite rational quaternion algebra are parametrized, up to isogeny, by the K-rational points of the quotient of certain Shimura curves by the group of their Atkin-Lehner involutions.
On the image of l-adic Galois representations for abelian varieties of type I and II
2004
In this paper we investigate the image of the lll-adic representation attached to the Tate module of an abelian variety over a number field with endomorphism algebra of type I or II in the Albert classification. We compute the image explicitly and verify the classical conjectures of Mumford-Tate, Hodge, Lang and Tate, for a large family of abelian varieties of type I and II. In addition, for this family, we prove an analogue of the open image theorem of Serre.
Acta Arithmetica, 2018
Motivated by our arithmetic applications, we required some tools that might be of independent interest. Let E be an absolutely irreducible group scheme of rank p 4 over Zp. We provide a complete description of the Honda systems of p-divisible groups G such that G[p n+1 ]/G[p n ] ≃ E for all n. Then we find a bound for the abelian conductor of the second layer Qp(G[p 2 ])/Qp(G[p]), stronger in our case than can be deduced from Fontaine's bound. Let π : Sp 2g (Zp) → Sp 2g (Fp) be the reduction map and let G be a closed subgroup of Sp 2g (Zp) with G = π(G) irreducible and generated by transvections. We fill a gap in the literature by showing that if p = 2 and G contains a transvection, then G is as large as possible in Sp 2g (Zp) with given reduction G, i.e. G = π −1 (G). One simple application arises when A = J(C) is the Jacobian of a hyperelliptic curve C : y 2 + Q(x)y = P (x), where Q(x) 2 + 4P (x) is irreducible in Z[x] of degree m = 2g + 1 or 2g + 2, with Galois group Sm ⊂ Sp 2g (F 2). If the Igusa discriminant I 10 of C is odd and some prime q exactly divides I 10 , then G = Gal(Q(A[2 ∞ ])/Q) isπ −1 (Sm), whereπ : GSp 2g (Zp) → Sp 2g (Fp). When m = 5, Q(x) = 1 and I 10 = N is a prime, A = J(C) is an example of a favorable abelian surface. We use the machinery above to obtain nonexistence results for certain favorable abelian surfaces, even for large N .
Documenta Mathematica Extra Volume: : John H. Coates’ Sixtieth Birthday (2006), pp.35-75., 2006
In this paper we investigate the image of the l-adic representation attached to the Tate module of an abelian variety over a number field with endomorphism algebra of type I or II in the Albert classification. We compute the image explicitly and verify the classical conjectures of Mumford-Tate, Hodge, Lang and Tate for a large family of abelian varieties of type I and II. In addition, for this family, we prove an analogue of the open image theorem of Serre. 2000 Mathematics Subject Classification: 11F80, 11G10