A quantitative sharpening of Moriwaki's arithmetic Bogomolov inequality

A. Moriwaki proved the following arithmetic analogue of the Bogomolov unstability theorem. If a torsion-free hermitian coherent sheaf on an arithmetic surface has negative discriminant then it admits an arithmetically destabilising subsheaf. In the geometric situation it is known that such a subsheaf can be found subject to an additional numerical constraint and here we prove the arithmetic analogue. We then apply this result to slightly simplify a part of C. Soul\'e's proof of a vanishing theorem on arithmetic surfaces.Comment: final version, to appear in Math. Res. Let

Arithmetically defined dense subgroups of Morava stabilizer groups

For every prime $p$ and integer $n\ge 3$ we explicitly construct an abelian variety A/\F_{p^n} of dimension $n$ such that for a suitable prime $l$ the group of quasi-isogenies of A/\F_{p^n} of $l$-power degree is canonically a dense subgroup of the $n$-th Morava stabilizer group at $p$. We also give a variant of this result taking into account a polarization. This is motivated by a perceivable generalization of topological modular forms to more general topological automorphic forms. For this, we prove some results about approximation of local units in maximal orders which is of independent interest. For example, it gives a precise solution to the problem of extending automorphisms of the $p$-divisible group of a simple abelian variety over a finite field to quasi-isogenies of the abelian variety of degree divisible by as few primes as possible.Comment: major revision, main results slightly changed; final version, to appear in Compositio Mat

Beta-elements and divided congruences

The f-invariant is an injective homomorphism from the 2-line of the Adams-Novikov spectral sequence to a group which is closely related to divided congruences of elliptic modular forms. We compute the f-invariant for two infinite families of beta-elements and explain the relation of the arithmetic of divided congruences with the Kervaire invariant one problem.Comment: minor changes; final version, to appear in Amer. J. Mat