Let R be a local Artin ring with maximal ideal $\frak m$ and residue class field of characteristic p > 0. We show that every finite flat group scheme over R is annihilated by its rank, whenever $\frak m$p = p$\frak m$ = 0. This implies that any finite flat group scheme over an Artin ring the square of whose maximal ideal is zero, is annihilated by its rank.

An interesting theory arises when the classical theory of modular forms is expanded to include differential analogs of modular forms. One of the main motivations for expanding the theory of modular forms is the existence of differential modular forms with a remarkable property, called isogeny covariance, that classical modular forms cannot possess. Among isogeny covariant differential modular forms there exists a particular modular form that plays a central role in the theory. The main result presented in the article will be the explicit computation modulo p of this fundamental isogeny covariant differential modular form.

The root discriminant of a number field of degree n is the nth root of the absolute value of its discriminant. Let R2m be the minimal root discriminant for totally complex number fields of degree 2m, and put α0 = lim infmR2m. One knows that α0 [ges ] 4πeγ ≈ 22.3, and, assuming the Generalized Riemann Hypothesis, α0 [ges ] 8πeγ ≈ 44.7. It is of great interest to know if the latter bound is sharp. In 1978, Martinet constructed an infinite unramified tower of totally complex number fields with small constant root discriminant, demonstrating that α0 < 92.4. For over twenty years, this estimate has not been improved. We introduce two new ideas for bounding asymptotically minimal root discriminants, namely, (1) we allow tame ramification in the tower, and (2) we allow the fields at the bottom of the tower to have large Galois closure. These new ideas allow us to obtain the better estimate α0 < 83.9.

We study a dual pair of general linear Lie superalgebras in the sense of R. Howe. We give an explicit multiplicity-free decomposition of a symmetric and skew-symmetric algebra (in the super sense) under the action of the dual pair and present explicit formulas for the highest-weight vectors in each isotypic subspace of the symmetric algebra. We give an explicit multiplicity-free decomposition into irreducible gl(m|n)-modules of the symmetric and skew-symmetric algebras of the symmetric square of the natural representation of gl(m|n). In the former case, we also find explicit formulas for the highest-weight vectors. Our work unifies and generalizes the classical results in symmetric and skew-symmetric models and admits several applications.

Let $\cal X$ be a projective, geometrically irreducible, non-singular, algebraic curve defined over a finite field Fq2 of order q2. If the number of Fq2-rational points of $\cal X$ satisfies the Hasse–Weil upper bound, then $\cal X$ is said to be Fq2-maximal. For a point P0 ∈ $\cal X$(Fq2), let π be the morphism arising from the linear series $\cal D$: = |(q + 1)P0|, and let N: = dim($\cal D$). It is known that N [ges ] 2 and that π is independent of P0 whenever $\cal X$ is Fq2-maximal.