Picard groups of higher real -theory spectra at height

Using the descent spectral sequence for a Galois extension of ring spectra, we compute the Picard group of the higher real $K$-theory spectra of Hopkins and Miller at height $n=p-1$, for $p$ an odd prime. More generally, we determine the Picard groups of the homotopy fixed points spectra $E_{n}^{hG}$, where $E_{n}$ is Lubin–Tate $E$-theory at the prime $p$ and height $n=p-1$, and $G$ is any finite subgroup of the extended Morava stabilizer group. We find that these Picard groups are always cyclic, generated by the suspension.

A DESCENT SPECTRAL SEQUENCE FOR ARBITRARY K(n)-LOCAL SPECTRA WITH EXPLICIT E2-TERM

Let n be any positive integer and p be any prime. Also, let X be any spectrum and let K(n) denote the nth Morava K-theory spectrum. Then we construct a descent spectral sequence with abutment π∗(LK(n)(X)) and E2-term equal to the continuous cohomology of Gn, the extended Morava stabilizer group, with coefficients in a certain discrete Gn-module that is built from various homotopy fixed point spectra of the Morava module of X. This spectral sequence can be contrasted with the K(n)-local En-Adams spectral sequence for π∗(LK(n)(X)), whose E2-term is not known to always be equal to a continuous cohomology group.

Arithmetically defined dense subgroups of Morava stabilizer groups

For every prime p and integer n≥3 we explicitly construct an abelian variety $A/\mathbb {F}_{p^n}$ of dimension n such that for a suitable prime l the group of quasi-isogenies of $A/\mathbb {F}_{p^n}$ of l-power degree is canonically a dense subgroup of the nth Morava stabilizer group at p. We also give a variant of this result taking into account a polarization. This is motivated by the recent construction by Behrens and Lawson of topological automorphic forms which generalizes topological modular forms. For this, we prove some arithmetic results of independent interest: a result about approximation of local units in maximal orders of global skew fields which also 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.