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arXiv · 1805.10361

On the number of Galois orbits of newforms

Luis Dieulefait·Ariel Pacetti·Panagiotis Tsaknias
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Abstract

Counting the number of Galois orbits of newforms in $S_k(Γ_0(N))$ 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 $Γ_0(N)$ 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.

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Luis Dieulefait, Ariel Pacetti, Panagiotis Tsaknias. 2018-05-25. On the number of Galois orbits of newforms. https://arxiv.org/abs/1805.10361

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