Counting newforms with prescribed ramified supercuspidal components

TL;DR

This paper derives an explicit formula for counting newforms with specific ramified supercuspidal components at certain primes, linking the count to trace formulas and local data such as quadratic extensions and root numbers.

Contribution

It provides a new explicit formula for the number of newforms with prescribed ramified supercuspidal components, depending only on local data and the trace of Atkin--Lehner operators.

Findings

Explicit formula for counting newforms with prescribed ramified supercuspidal components.

Dependence of the count on local quadratic extensions and root numbers.

Complete explicitness when the set of primes is a single prime or all prime factors of the level.

Abstract

We give a formula for the number of newforms in $S_k^{\mathrm{new}}(N)$ that have prescribed ramified supercuspidal components $\pi_p$ at a set $T$ of primes dividing $N$. This dimension is given in terms of the trace of the Atkin--Lehner operator at $T$ on $S_k^{\mathrm{new}}(N)$. It depends only upon the weight, the level, the ramified quadratic extensions $E_p/{\mathbb Q}_p$ attached to the $\pi_p$, and the root number of each $\pi_p$. The formula is completely explicit when $T$ consists of either a single prime or all prime factors of $N$.