Moduli of Q-abelian surfaces with quaternionic multiplication
نویسندگان
چکیده
We prove that the non-CM Q-abelian surfaces whose endomorphism algebra is a quaternion algebra are parametrized, up to isogeny, by the rational points of the quotient of certain Shimura curves by the group of their Atkin-Lehner involutions.
منابع مشابه
Rational 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 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 کاملThe Arithmetic of Qm-abelian Surfaces through Their Galois Representations
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. 1. Abelian surfaces with quaternionic multiplication Fix Q an ...
متن کاملThe field of moduli of quaternionic multiplication on abelian varieties
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.
متن کاملReal Multiplication Abelian Surfaces over Q
Recall: New eigen-cuspforms f of weight 2 on Γ0(N) with Q(f) totally real and [Q(f) : Q] = d ←→ dimension d factors of J0(N) ←→ abelian varieties A/Q of dimension d with real multiplication (=RM) by an order in K = Q(f) up to isogeny. d = 1: elliptic curves, which we understand quite well: moduli (the j-line, ∼= P), explicit formulas (Weierstrass etc.), isogenies, etc. d > 1: we understand thes...
متن کامل