Limiting measures of supersingularities

TL;DR

This paper improves bounds on the reduction of crystalline Galois representations with large slopes, advancing understanding of supersingularities and supporting conjectures about their measure distribution.

Contribution

It asymptotically refines bounds on the reduction of crystalline Galois representations, approaching the predicted optimal bound under specific assumptions.

Findings

Improved the bound to loor((k-1)/(p+1)) + \u00bflog_p(k-1)b1.

Partial validation of Gouveab4s conjecture on measures of supersingularities.

Potential extension of methods to cases p=2,3 and when p+1 does not divide k-1.

Abstract

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 $(\varphi,\Gamma)$-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 Gouv\^ea: 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 $\Gamma_0(N)$-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 $p+1\nmid k-1$.