Limiting measures of supersingularities

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 $p+1\nmid k-1$.