Ordinary primes in Hilbert modular varieties

TL;DR

This paper investigates the distribution of ordinary primes in Hilbert modular varieties, providing new methods to produce such primes and formulating conjectures for higher weights, with some verified cases.

Contribution

It introduces techniques combining Sato-Tate equidistribution and Galois theory to produce ordinary primes for Hilbert modular forms, and formulates an ordinariness conjecture for higher weights.

Findings

More ordinary primes can be produced using Sato-Tate and Galois theory.

Conditional results depend on stronger Sato-Tate assumptions.

Verified the ordinariness conjecture for certain abelian Galois representations.

Abstract

A well-known conjecture, often attributed to Serre, asserts that any motive over any number field has infinitely many ordinary reductions (in the sense that the Newton polygon coincides with the Hodge polygon). In the case of Hilbert modular cuspforms $f$ of parallel weight $(2, \cdots , 2)$, we show how to produce more ordinary primes by using the Sato-Tate equidistribution and combining it with the Galois theory of the Hecke field. Under the assumption of stronger forms of Sato-Tate equidistribution, we get stronger (but conditional) results. In the case of higher weights, we formulate the ordinariness conjecture for submotives of the intersection cohomology of proper algebraic varieties with motivic coefficients, and verify it for the motives whose $l$-adic Galois realisations are abelian on a finite index subgroup. We get some results for Hilbert cuspforms of weight $(3, \cdots , 3)$, weaker than those for $(2, \cdots , 2)$.