Newton Polygons of Hecke Operators

TL;DR

This paper computationally verifies a truncated version of the Buzzard-Calegari conjecture concerning the Newton polygon of the Hecke operator T_2 for large weights, using explicit formulas and calculations.

Contribution

Develops a formula for p-adic valuations of exponential sums and verifies the conjecture for vertices up to n=15 in the Newton polygon.

Findings

Verified the conjecture for vertices at n ≤ 15.

Established a method to compute p-adic valuations of traces of Hecke operators.

Confirmed the Newton polygon of the Buzzard-Calegari polynomial matches that of T_2 up to certain vertices.

Abstract

In this computational paper we verify a truncated version of the Buzzard-Calegari conjecture on the Newton polygon of the Hecke operator $T_2$ for all large enough weights. We first develop a formula for computing $p$-adic valuations of exponential sums, which we then implement to compute $2$-adic valuations of traces of Hecke operators acting on spaces of cusp forms. Finally, we verify that if Newton polygon of the Buzzard-Calegari polynomial has a vertex at $n\leq 15$, then it agrees with the Newton polygon of $T_2$ up to $n$.