Counting locally supercuspidal newforms

Abstract

The trace formula is a versatile tool for computing sums of spectral data across families of automorphic forms. Using specialized test functions, one can treat small families with refined spectral properties. This has proven fruitful in analytic applications. We detail such methodology here, with the aim of counting newforms in certain small families. The result (Theorem 7.1) is a general formula for the number of holomorphic newforms of weight $k$ and level $N$ whose local representation type at each $p|N$ is a fixed supercuspidal representation $\sigma_p$ of $\operatorname{GL}_2(\mathbf{Q}_p)$. This is given in terms of local elliptic orbital integrals attached to matrix coefficients of the $\sigma_p$. We evaluate the formula explicitly in the case where each $\sigma_p$ has conductor $\le p^3$. The technical heart of the paper is the explicit calculation of elliptic orbital integrals attached to such $\sigma_p$. We also compute the traces of Hecke operators on the span of these newforms. Some applications are given to biases among root numbers of newforms.