Newton Polygons of L-functions Associated to Deligne Polynomials

TL;DR

This paper investigates the conditions under which Deligne polynomials are ordinary, providing a counterexample that challenges a previous conjecture relating to the combinatorial constant and prime congruences.

Contribution

It offers a counterexample disproving Le's conjecture about the generic ordinarity of Deligne polynomials based on prime congruences.

Findings

Counterexample to Le's conjecture established

The conjecture does not hold in general

Highlights the need for revised criteria for ordinarity

Abstract

A conjecture of Le says that the Deligne polytope $\Delta_d$ is generically ordinary if $p\equiv 1\ (\!\!\bmod\ D(\Delta_d))$, where $D(\Delta_d)$ is a combinatorial constant determined by $\Delta_d$. In this paper a counterexample is given to show that the conjecture is not true in general.