On the number of Galois orbits of newforms

TL;DR

This paper investigates the number of Galois orbits of newforms in modular form spaces, providing local invariants, lower bounds, and numerical evidence, and explores connections to Maeda's conjecture.

Contribution

It introduces local invariants for Galois orbits of newforms, establishes lower bounds for their count, and suggests a potential generalization of Maeda's conjecture.

Findings

Lower bounds for the number of Galois orbits for large weights

Numerical evidence supports the bounds as exact in many cases

Potential link between Galois orbit counts and Maeda's conjecture

Abstract

Counting the number of Galois orbits of newforms in $S_k(\Gamma_0(N))$ and giving some arithmetic sense to this number is an interesting open problem. The case $N=1$ corresponds to Maeda's conjecture (still an open problem) and the expected number of orbits in this case is 1, for any $k \ge 16$. In this article we give local invariants of Galois orbits of newforms for general $N$ and count their number. Using an existence result of newforms with prescribed local invariants we prove a lower bound for the number of non-CM Galois orbits of newforms for $\Gamma_0(N)$ for large enough weight $k$ (under some technical assumptions on $N$). Numerical evidence suggests that in most cases this lower bound is indeed an equality, thus we leave as a Question the possibility that a generalization of Maeda's conjecture could follow from our work. We finish the paper with some natural generalizations of the problem and show some of the implications that a generalization of Maeda's conjecture has.