The field of moduli of quaternionic multiplication on abelian varieties
نویسنده
چکیده
We consider principally polarized abelian varieties with quaternionic multiplication over number fields and we study the field of moduli of their endomorphisms in relation to the set of rational points on suitable Shimura varieties. Published in Intern. J. Math. M. Sc. 52 (2004), 2795-2808.
منابع مشابه
Shimura Curves Embedded in Igusa’s Threefold
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 comput...
متن کاملFields of definition of building blocks with quaternionic multiplication
This paper investigates the fields of definition up to isogeny of the abelian varieties called building blocks. In [5] and [3] a characterization of the fields of definition of these varieties together with their endomorphisms is given in terms of a Galois cohomology class canonically attached to them. However, when the building blocks have quaternionic multiplication, then the field of definit...
متن کاملRational Points on Atkin-Lehner Quotients of Shimura Curves
We study three families of Atkin-Lehner quotients of quaternionic Shimura curves: X, X 0 (N), and X D+ 1 (N), which serve as moduli spaces of abelian surfaces with potential quaternionic multiplication (PQM) and level N structure. The arithmetic geometry of these curves is similar to, but even richer than, that of the classical modular curves. Two important differences are the existence of a no...
متن کاملAnti-holomorphic Multiplication and a Real Algebraic Modular Variety
An anti-holomorphic multiplication by the integers Od of a quadratic imaginary number field, on a principally polarized complex abelian variety AC is an action of Od on AC such that the purely imaginary elements act in an anti-holomorphic manner. The coarse moduli space XR of such A (with appropriate level structure) is shown to consist of finitely many isomorphic connected components, each of ...
متن کاملOn Abelian Surfaces with Potential Quaternionic Multiplication
An abelian surface A over a field K has potential quaternionic multiplication if the ring End K̄ (A) of geometric endomorphisms of A is an order in an indefinite rational division quaternion algebra. In this brief note, we study the possible structures of the ring of endomorphisms of these surfaces and we provide explicit examples of Jacobians of curves of genus two which show that our result is...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
- Int. J. Math. Mathematical Sciences
دوره 2004 شماره
صفحات -
تاریخ انتشار 2004