a Néron model over , which is a smooth group scheme over , but the Néron model is not proper and hence is not an abelian scheme over . Non-existence V. A. Abrashkin and Jean-Marc Fontaine independently proved that there are no nonzero abelian varieties over Q with good reduction at all primes. Equivalently, there are no nonzero abelian schemes over Spec Z. The proof involves showing that the coordinates of pn-torsion points generate number fields with very little ramification and hence of small discriminant, while, on the other hand, there are lower bounds on discriminants of number fields. Semiabelian variety A semiabelian variety is a commutative group variety which is an extension of an abelian variety by a torus. See also Complex torus Motives Timeline of abelian varieties Moduli of abelian varieties Equations defining abelian varieties Horrocks–Mumford bundle References Sources . A comprehensive treatment of the complex theory, with an overview of the history of the subject. . Online course notes. . Description of the Jacobian of the Covering Curves Algebraic curves Geometry of divisors Algebraic surfaces Niels Henrik Abel