A resolvable modified group divisible design (RMGD) is a modified group divisible design whose blocks can be partitioned into parallel classes. We show that the necessary conditions for the existence of a 3-RMGDDλ of type g, namely g ≥ 3, u ≥ 3, gu ≡ 0 mod 3 and λ(g−1)(u−1) ≡ 0 mod 2, are sufficient with the two exceptions of (g, u, λ) ∈ {(6, 3, 1), (3, 6, 1)}.

For every prime p we give infinitely many examples of torsors under abelian varieties over Q that are locally trivial but not divisible by p in the Weil-Châtelet group. We also give an example of a locally trivial torsor under an elliptic curve over Q which is not divisible by 4 in the Weil-Châtelet group. This gives a negative answer to a question of Cassels.

Let $F_0$ be an absolutely algebraic field of characteristic $p>0$ and $\kappa$ infinite cardinal. It is shown that there exists a $F$ such $F^*\cong F^*_0\oplus(\oplus_\kappa \mathbb{Q})$ with $Br(F)=\{0\}$. $L$ closure $F$. Then for any finite subextension $K$ $L/F$, we have $K^*\cong T(K^*)\oplus(\oplus_\kappa \mathbb{Q})$, where $T(K^*)$ the group torsion elements $K^*$. In addition,...

A p-divisible group over a field K admits a slope decomposition; associated to each slope λ is an integer m and a representation Gal(K) → GLm(Dλ), where Dλ is the Qp-division algebra with Brauer invariant [λ]. We call m the multiplicity of λ in the p-divisible group. Let G0 be a p-divisible group over a field k. Suppose that λ is not a slope of G0, but that there exists a deformation of G in wh...

In this paper we study abelian varieties and of p-divisible groups in characteristic p. Even though a non-trivial deformation of an abelian variety can produce a non-trivial Galois-representation, say on the Tate-l-group of the generic fiber (in any characteristic), the geometric generic fiber has a “constant” Tate-l-group. The same phenomenon we encounter for the p-structure in positive charac...