On the number of Galois orbits of newforms

• Autores: Dieulefait, Luis Victor; Pacetti, Ariel Martín; Tsaknias, Panagiotis
• Año de publicación: 2021
• Idioma: inglés
• Tipo de recurso: artículo
• Materia: GALOIS ORBITS; MAEDA'S CONJECTURE
• Institución: Consejo Nacional de Investigaciones Científicas y Técnicas

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.
Fil: Dieulefait, Luis Victor. Universidad de Barcelona; España
Fil: Pacetti, Ariel Martín. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomia y Física. Sección Matemática; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Córdoba. Centro de Investigación y Estudios de Matemática. Universidad Nacional de Córdoba. Centro de Investigación y Estudios de Matemática; Argentina
Fil: Tsaknias, Panagiotis. No especifíca;