On the number of Galois orbits of newforms - Dieulefait, Luis Victor (2021)
Counting the number of Galois orbits of newforms in Sk(Γ0(N) and giving some arithmetic sense to this number is an interesting open problem. The case N D 1 corresponds to Maeda's conjecture (still an open problem) and the expected number of orbits in this case is 1, for any k ≥ 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 Γ00(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.