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Modular Shimura varieties and forgetful maps
 Trans. Amer. Math. Soc
"... Abstract. In this note we consider several maps that occur naturally between modular Shimura varieties, HilbertBlumenthal varieties and the moduli spaces of polarized abelian varieties when forgetting certain endomorphism structures. We prove that, up to birational equivalences, these forgetful map ..."
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Cited by 14 (10 self)
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Abstract. In this note we consider several maps that occur naturally between modular Shimura varieties, HilbertBlumenthal varieties and the moduli spaces of polarized abelian varieties when forgetting certain endomorphism structures. We prove that, up to birational equivalences, these forgetful maps coincide with the natural projection by suitable abelian groups of AtkinLehner involutions.
EASY DECISIONDIFFIEHELLMAN GROUPS
 LONDON MATHEMATICAL SOCIETY JOURNAL OF COMPUTATIONAL MATHEMATICS
, 2004
"... The decisionDiffieHellman problem (DDH) is an important computational problem in cryptography. It is known that the Weil and Tate pairings can be used to solve many DDH problems on elliptic curves. Distortion maps are an important tool for solving DDH problems using pairings and it is known that d ..."
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Cited by 12 (0 self)
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The decisionDiffieHellman problem (DDH) is an important computational problem in cryptography. It is known that the Weil and Tate pairings can be used to solve many DDH problems on elliptic curves. Distortion maps are an important tool for solving DDH problems using pairings and it is known that distortion maps exist for all supersingular elliptic curves. We present an algorithm to construct suitable distortion maps. The algorithm is efficient on the curves usable in practice, and hence all DDH problems on these curves are easy. We also discuss the issue of which DDH problems on ordinary curves are easy.
THE ARITHMETIC OF QMABELIAN SURFACES THROUGH THEIR GALOIS REPRESENTATIONS
, 2003
"... 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 ..."
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Cited by 7 (6 self)
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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 several explicit examples.
On finiteness conjectures for endomorphism algebras of abelian surfaces
"... Abstract. 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 GL2type ..."
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Cited by 3 (3 self)
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Abstract. 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 GL2type 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. 1.
On finiteness conjectures for modular quaternion algebras
 Math. Proc. Camb. Philos. Soc
"... Abstract. It is conjectured that there exist only finitely many isomorphism classes of endomorphism algebras of abelian varieties of bounded dimension over a fixed number field. We explore this conjecture when particularized to quaternion endomorphism algebras of abelian surfaces of GL2type over Q ..."
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Cited by 2 (2 self)
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Abstract. It is conjectured that there exist only finitely many isomorphism classes of endomorphism algebras of abelian varieties of bounded dimension over a fixed number field. We explore this conjecture when particularized to quaternion endomorphism algebras of abelian surfaces of GL2type 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. 1.
SHIMURA CURVES EMBEDDED IN IGUSA’S THREEFOLD
, 2003
"... Let O be a maximal order in a totally indefinite quaternion algebra over a totally real number field. In this note we study the locus QO of quaternionic multiplication by O in the moduli space Ag of principally polarized abelian varieties of even dimension g with particular emphasis in the twodimen ..."
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Cited by 1 (0 self)
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Let O be a maximal order in a totally indefinite quaternion algebra over a totally real number field. In this note we study the locus QO of quaternionic multiplication by O in the moduli space Ag of principally polarized abelian varieties of even dimension g with particular emphasis in the twodimensional case. We describe QO as a union of AtkinLehner quotients of Shimura varieties and we compute the number of irreducible components of QO in terms of class numbers of CMfields.