Kristian Strommen

Abstract:

We develop a notion of a `canonical $\mathcal{C}$-henselian valuation' for a class $\mathcal{C}$ of field extensions, generalizing the construction of the canonical henselian valuation of a field. We use this to show that the $p$-adic valuation on a finite extension $F$ of $\mathbb{Q}_p$ can be recovered entirely (or up to some indeterminacy of the residue field) from various small quotients of $G_F$, the absolute Galois group of $F$. In particular, it can be recovered fully from the maximal solvable quotient. We use this to prove several versions of the birational section conjecture for varieties over $p$-adic fields.

Abstract:

In recent years, numerical weather prediction models have begun to show notable levels of skill at predicting the average winter North Atlantic Oscillation (NAO) when initialised one month ahead. At the same time, these model predictions exhibit unusually low signal-to-noise ratios, in what has been dubbed a `signal-to-noise paradox'. We analyse both the skill and signal-to-noise ratio of the Integrated Forecast System (IFS), the European Center for Medium-range Weather Forecasts (ECMWF) model, in an ensemble hindcast experiment. Specifically, we examine the contribution to both from the regime dynamics of the North Atlantic eddy-driven jet. This is done by constructing a statistical model which captures the predictability inherent to to the trimodal jet latitude system, and fitting its parameters to reanalysis and IFS data. Predictability in this regime system is driven by interannual variations in the persistence of the jet latitude regimes, which determine the preferred state of the jet. We show that the IFS has skill at predicting such variations in persistence: because the position of the jet strongly influences the NAO, this automatically generates skill at predicting the NAO. We show that all of the skill the IFS has at predicting the winter NAO over the period 1980-2010 can be attributed to its skill at predicting regime persistence in this way. Similarly, the tendency of the IFS to underestimate regime persistence can account for the low signal-to-noise ratio, giving a possible explanation for the signal-to-noise paradox. Finally, we examine how external forcing drives variability in jet persistence, as well as highlight the role played by transient baroclinic eddy feedbacks to modulate regime persistence.

Abstract:

Let $K$ be a field with $G_K(2) \simeq G_{\mathbb{Q}_2}(2)$, where $G_F(2)$ denotes the maximal pro-2 quotient of the absolute Galois group of a field $F$. We prove that then $K$ admits a (non-trivial) valuation $v$ which is 2-henselian and has residue field $\mathbb{F}_2$. Furthermore, $v(2)$ is a minimal positive element in the value group $\Gamma_v$ and $[\Gamma_v:2\Gamma_v]=2$. This forms the first positive result on a more general conjecture about the structure of pro-$p$ Galois groups. As an application, we prove a strong version of the birational section conjecture for smooth, complete curves $X$ over $\mathbb{Q}_2$, as well as an analogue for varieties.