arXiv subjects
Publications and source records attributed to Bodan Arsovski.
We prove that all bounded subsets of $\mathbb{Q}_p^n$ containing a line segment of unit length in every direction have Hausdorff and Minkowski dimension $n$. This is the analogue of the classical Kakeya conjecture with $\mathbb{R}$ replaced by $\mathbb{Q}_p$.
Let $p$ be a prime number and let $k\geq 2$ be an integer. In this article we study the semi-simple reductions modulo $p$ of two-dimensional irreducible crystalline $p$-adic Galois representations with Hodge-Tate weights $0$ and $k-1$ and large slopes. Berger--Li--Zhu proved by using the theory of $(φ,Γ)$-modules that this reduction is constant when the slope is larger than $\lfloor\frac{k-2}{p-1}\rfloor$. Recently, Bergdall--Levin improved this bound to $\lfloor\frac{k-1}{p}\rfloor$ by using the theory of Kisin modules. In this article, under the extra assumptions $p>3$ and $p+1\nmid k-1$, we asymptotically improve this bound further to $\lfloor\frac{k-1}{p+1}\rfloor+\lfloor\log_p(k-1)\rfloor$, which is off from the predicted optimal bound $\approx\frac{k-1}{p+1}$ only by a factor of $\mathsf{O}\left(\log_p k\right)$ rather than by a factor that is linear in $k$. As a consequence we deduce a partial result towards a conjecture by Gouvêa: that the measures of supersingularities of level $Np$ oldforms tend to the zero measure on the interval $(\frac{1}{p+1},\frac{p}{p+1})$ when $p$ is coprime to $6N$ and $Γ_0(N)$-regular. It is very likely that our methods extend to the cases $p\in\{2,3\}$ and $p+1\nmid k-1$ as well, and therefore can be adapted to eliminate the extra assumptions $p>3$ and $p+1\nmid k-1$.
A conjecture of Breuil, Buzzard, and Emerton says that the slopes of certain reducible $p$-adic Galois representations must be integers. In previous work we showed this conjecture for representations that lie over certain non-subtle components of weight space. This article is a continuation of that work in which we completely classify the aforementioned representations over the non-subtle components of weight space, both for integer and non-integer slopes.
A conjecture of Breuil, Buzzard, and Emerton says that the slopes of certain reducible $p$-adic Galois representations must be integers. In previous work we showed this conjecture for representations that lie over certain non-subtle components of weight space. This article is a continuation of that work in which we also show the conjecture for the subtle components for slopes less than $\frac{p-1}{2}$.
Crystabelline representations are representations of the absolute Galois group $G_{\mathbb{Q}_p}$ over $\mathbb{Q}_p$ that become crystalline on $G_{F}$ for some abelian extension $F/\mathbb{Q}_p$. Their relation to modular forms is that the representation associated with a finite slope newform of level divisible by $p^2$ is crystabelline. In this article we study the connection between the slopes of two-dimensional crystabelline representations and the reducibility of their modulo $p$ reductions. This question is inspired by a theorem by Buzzard and Kilford which implies that the slopes on the boundary of the $2$-adic eigencurve of tame level $1$ are integers (and in arithmetic progression); an analogous theorem by Roe which says that the same is true for the $3$-adic eigencurve; Coleman's halo conjecture and the ghost conjecture which give predictions about the slopes on the $p$-adic eigencurve of general tame level; and Hodge theoretic conjectures by Breuil, Buzzard, Emerton, and Gee which indicate that there is a connection between all of these and the slopes of locally reducible two-dimensional crystabelline representations. We prove that the reductions of certain two-dimensional crystabelline representations with slopes in $(0,\frac{p-1}{2})\backslash \mathbb{Z}$ are usually irreducible, with the exception of a small region where the slopes are half-integers and reducible representations do occur.
The question of computing the reductions modulo $p$ of two-dimensional crystalline $p$-adic Galois representations has been studied extensively, and partial progress has been made for representations that have small weights, very small slopes, or very large slopes. It was conjectured by Breuil, Buzzard, and Emerton that these reductions are irreducible if they have even weight and non-integer slope. We prove some instances of this conjecture for slopes up to $\frac{p-1}{2}$.
We use the p-adic local Langlands correspondence for GL_2(Q_p) to find the reduction modulo p of certain two-dimensional crystalline Galois representations. In particular, we resolve a conjecture of Breuil, Buzzard, and Emerton in the case when the slope is strictly between one and three, and prove partial results towards this conjecture for arbitrary slopes. Moreover, we partially classify the reduction modulo p of these representations when the slope is equal to one.