On the number of Galois orbits of newforms

Abstract: A conjecture of Maeda states that all newforms of level $1$ and weight $k \ge 16$ are Galois conjugate. This striking observation has very deep implications in the study of Galois representations and until now stays as a completely open problem. A natural question is to understand what happens when we move from level $1$ to modular forms of level $\Gamma_0(N)$, with $N > 1$. In this talk we will present a list of invariants of Galois orbits of newforms of level $N$, giving a lower bound for the number of Galois orbits (when $k$ is large enough). It is a natural question to understand whether such a lower bound is attached (a question related to Maeda's conjecture).

number theoryrepresentation theory

Audience: researchers in the topic

Number Theory and Representations in Valparaiso