STNB

STNB2026 (39a edició)

Primes ramified in coefficient fields of modular forms

Ponents

Filip Gawron

Resum

Let $f\in S_k(\Gamma_0(N),\chi)^\text{new}$ be a newform of level $N$ and nebentypus $\chi$. Knowing $N$ and $\chi$, what can we say about the field $\mathbb{Q}_f$ generated by the Fourier coefficients of $f$? In particular,

What can we say about primes ramifying in $\mathbb{Q}_f$?

In the talk, I will show some numerical calculations in the case $k=2$ and trivial nebentypus. I will also discuss some existing results. Finally, I will comment on my recent work about ramification occurring when the level $N$ is non-optimal.

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