2023/10/25 by Andrew Knightly, Knightly, Andrew · 1 citation
Mathematics · #11F11 #11F25 #11F70 #11F72 #Advanced Algebra and Geometry #Analytic Number Theory Research #FOS: Mathematics #Finite Group Theory Research #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2310.17047
openalex publication_date 2023/10/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 σp of GL2(Qp). This is given in terms of local elliptic orbital integrals attached to matrix coefficients of the σp. We evaluate the formula explicitly in the case where each σp has conductor ≤ p3. The technical heart of the paper is the explicit calculation of elliptic orbital integrals attached to such σp. 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.