Reduction of certain crystalline representations and local constancy in the weight space

TL;DR

This paper investigates how the mod p reduction of crystalline Galois representations varies with weight, providing explicit bounds on local constancy regions based on slope, advancing understanding of their structural behavior.

Contribution

It offers an explicit linear bound on the radius of local constancy in weight space for crystalline Galois representations, extending previous theoretical results.

Findings

Derived an explicit linear upper bound for local constancy radius

Confirmed local constancy of reductions for fixed Frobenius trace

Enhanced understanding of weight variation effects on Galois representations

Abstract

We study the mod $p$ reduction of crystalline local Galois representations of dimension 2 under certain conditions on its weight and slope. Berger showed that for a fixed non-zero trace of the Frobenius, the reduction process is locally constant for varying weights. By explicit computation we obtain an upper bound that is a linear function of the slope, for the radius of this local constancy around some special points in the weight space.