Parametrization of Abelian K-surfaces with quaternionic multiplication

Guitart, Xavier; Molina, Santiago

doi:10.1016/j.crma.2009.09.025

Cited by 4 publications

(4 citation statements)

References 6 publications

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“…Let us consider ω m (P ) = [A ′ , i ′ ] ∈ CM(R). By [10,Corollary 2], the abelian surfaces with QM (A, i) and (A ′ , i ′ ) are isogenous. By this we mean that there exists an isogeny λ : A→A ′ making, for all α ∈ O, the following diagram commutative:…”

Section: Specialization Of Heegner Pointsmentioning

confidence: 99%

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Equations of hyperelliptic Shimura curves

Molina

2012

Proceedings of the London Mathematical Society

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We describe a method for computing explicit models of hyperelliptic Shimura curves attached to an indefinite quaternion algebra over Q and Atkin-Lehner quotients of them. It exploits Cerednik-Drinfeld's non-archimedean uniformisation of Shimura curves, a formula of Gross and Zagier for the endomorphism ring of Heegner points over Artinian rings and the connection between Ribet's bimodules and the specialization of Heegner points, as introduced in [22]. As an application, we provide a list of equations of Shimura curves and quotients of them obtained by our method that had been conjectured by Kurihara.

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Section: Specialization Of Heegner Pointsmentioning

confidence: 99%

“…Let us consider ω m (P ) = [A , i ] ∈ CM(R). By[13, Corollary 2], the abelian surfaces with QM (A, i) and (A , i ) are isogenous. By this, we mean that there exists an isogeny λ : A→A making, for all α ∈ O, the following diagram commutative:Write I n m = Hom Wn ((A, i), (A , i ))for the set of isogenies between (A, i)/W n and (A , i )/W n .…”

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confidence: 99%

Equations of hyperelliptic Shimura curves

Molina

2012

Proceedings of the London Mathematical Society

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“…there is a natural map δ : X 0 (D, N ) → X 0 (D, M ); composing with the Atkin-Lehner involution w N/M , we obtain a second map δ • w N/M : X 0 (D, N ) → X 0 (D, M ) and the product of both yields an embedding  : X 0 (D, N ) ֒→ X 0 (D, M ) × X 0 (D, M ) (cf. [9] for more details).…”

Section: Appendix A: Moduli Interpretations Of Shimura Curvesmentioning

confidence: 99%

Ribet bimodules and the specialization of Heegner points

Molina

2011

Isr. J. Math.

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Abstract. For a given order R in an imaginary quadratic field K, we study the specialization of the set CM(R) of Heegner points on the Shimura curve X = X 0 (D, N ) at primes p | DN . As we show, if p does not divide the conductor of R, a point P ∈ CM(R) specializes to a singular point (resp. a connected component) of the special fiberX of X at p if p ramifies (resp. does not ramify) in K. Exploiting the moduli interpretation of X 0 (D, N ) and K. Ribet's theory of bimodules, we give a construction of a correspondence Φ between CM(R) and a set of conjugacy classes of optimal embeddings of R into a suitable order in a definite quaternion algebras that allows the explicit computation of these specialization maps. This correspondence intertwines the natural actions of Pic(R) and of an Atkin-Lehner group on both sides. As a consequence of this and the work of P. Michel, we derive a result of equidistribution of Heegner points inX. We also illustrate our results with an explicit example.

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“…In the number field setting, as a higher-dimensional generalization of Q-curves, the notion of abelian k-varieties are studied by many people. For example, using Galois cohomological method, Ribet [Rib94] and Pyle [Pyl04] show that any abelian k-variety with some conditions (socalled building block) can be defined up to isogeny over a polyquadratic extension of k. In [GM09], Guitart and Molina parametrize abelian k-surfaces with quaternionic multiplication by k-rational points of the quotient of a Shimura curve by all Atkin-Lehner involutions. In the function field setting, as higher-dimensional generalizations of Drinfeld A-modules and analogues of abelian varieties, Anderson [And86] defined abelian t-modules and the dual notion of t-motives.…”

Section: Introductionmentioning

confidence: 99%

Parametrization of virtually $K$-rational Drinfeld modules of rank two

Okumura¹

2019

Preprint

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For an algebraic extension K of the rational function field Fq(T ) over a finite field, we introduce the notion of K-virtual Drinfeld modules as a function field analogue of Q-curves, which are elliptic curves over Q isogenous to all its Galois conjugates. Our goal in this article is to prove that all K-virtual Drinfeld modules of rank two with no complex multiplication are parametrized up to isogeny by K-rational points of a quotient curve of the Drinfeld modular curve Y0(n) with some square-free level n, which is an analogue of Elkies' well-known result on Q-curves.

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Parametrization of Abelian K-surfaces with quaternionic multiplication

Cited by 4 publications

References 6 publications

Equations of hyperelliptic Shimura curves

Equations of hyperelliptic Shimura curves

Ribet bimodules and the specialization of Heegner points

Parametrization of virtually $K$-rational Drinfeld modules of rank two

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