Modularity of Abelian Surfaces with Quaternionic Multiplication (original) (raw)
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The arithmetic of QM-abelian surfaces through their Galois representations
Journal of Algebra, 2004
This note provides an insight to the diophantine properties of abelian surfaces with quaternionic multiplication over number fields. We study the fields of definition of the endomorphisms on these abelian varieties and the images of the Galois representations on their Tate modules. We illustrate our results with an explicit example.
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...
Which quaternion algebras act on a modular abelian variety?
Mathematical Research Letters, 2008
Let A/Q be a modular abelian variety. We establish criteria to prevent a given quaternion algebra over a totally real number field to be the endomorphism algebra of A overQ. We accomplish this by analyzing the representation of Gal (Q/Q) on the points of N -torsion of A at primes N which ramify in B and by applying descent techniques to certain covers of Shimura varieties. Our result also applies to show that many Atkin-Lehner quotients of Shimura curves fail to have rational points over Q.
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
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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.
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.
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Algebra & Number Theory
Generalizing the method of Faltings-Serre, we rigorously verify that certain abelian surfaces without extra endomorphisms are paramodular. To compute the required Hecke eigenvalues, we develop a method of specialization of Siegel paramodular forms to modular curves. Contents 1. Introduction 1 2. A general Faltings-Serre method 5 3. Core-free subextensions 12 4. Abelian surfaces, paramodular forms, and Galois representations 17 5. Group theory and Galois theory for GSp 4 (F 2) 25 6. Computing Hecke eigenvalues by specialization 29 7. Verifying paramodularity 39 References 44
Examples of abelian surfaces with everywhere good reduction
Mathematische Annalen, 2015
We describe several explicit examples of simple abelian surfaces over real quadratic fields with real multiplication and everywhere good reduction. These examples provide evidence for the Eichler-Shimura conjecture for Hilbert modular forms over a real quadratic field. Several of the examples also support a conjecture of Brumer and Kramer on abelian varieties associated to Siegel modular forms with paramodular level structures.