Shimura Curves Embedded in Igusa’s Threefold
نویسنده
چکیده
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 byO in the moduli spaceAg of principally polarized abelian varieties of even dimension g with particular emphasis in the two-dimensional case. We describe QO as a union of Atkin-Lehner quotients of Shimura varieties and we compute the number of irreducible components of QO in terms of class numbers of CM-fields.
منابع مشابه
Equations of Shimura Curves of Genus Two
LetBD be the indefinite quaternion algebra overQ of reduced discriminantD=p1· · · · ·p2r for pairwise different prime numbers pi and let XD/Q be the Shimura curve attached to BD. As it was shown by Shimura [23], XD is the coarse moduli space of abelian surfaces with quaternionic multiplication by BD. Let W = {ωm : m | D} ⊆ Aut Q(XD) be the group of Atkin-Lehner involutions. For any m | D, we wi...
متن کاملDetermination of the real poles of the Igusa zeta function for curves
The numerical data of an embedded resolution determine the candidate poles of Igusa’s p-adic zeta function. We determine in complete generality which real candidate poles are actual poles in the curve case.
متن کاملOn Finiteness Conjectures for Modular Quaternion Algebras
It is conjectured that there exist finitely many isomorphism classes of simple endomorphism algebras of abelian varieties of GL2-type over Q of bounded dimension. We explore this conjecture when particularized to quaternion endomorphism algebras of abelian surfaces by giving a moduli interpretation which translates the question into the diophantine arithmetic of Shimura curves embedded in Hilbe...
متن کاملOn Finiteness Conjectures for Endomorphism Algebras of Abelian Surfaces
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 S...
متن کاملec 2 00 4 Rational curves of degree 10 on a general quintic threefold ∗
We prove the “strong form” of the Clemens conjecture in degree 10. Namely, on a general quintic threefold F in P, there are only finitely many smooth rational curves of degree 10, and each curve C is embedded in F with normal bundle O(−1) ⊕ O(−1). Moreover, in degree 10, there are no singular, reduced, and irreducible rational curves, nor any reduced, reducible, and connected curves with ration...
متن کامل