Counting newforms with prescribed ramified supercuspidal components
Andrew Knightly, Kimball Martin
TL;DR
The paper derives explicit dimension formulas for spaces of newforms S_k^{new}(N) with prescribed ramified supercuspidal local components at a set T of primes. By blending a simple trace formula with local Galois-orbit analysis, the authors show that dim S_k^{new}(N;π_T) equals a main term plus an Atkin–Lehner–trace correction that depends only on the ramified data E_p and local root numbers ε_p, and is explicit in cases with M=1 or T prime. The key insight is that the global dimension is governed by the Galois orbits of ramified supercuspidals and the trace of W_T on the full newspace, revealing root-number biases and invariance under local Galois conjugacy. The results connect local representation theory (ramified dihedral supercuspidals) with global automorphic dimensions, and extend prior work by treating ramified odd-conductor cases and outlining extensions to depth-zero components. This provides exact counts and structural understanding of newforms with prescribed ramified components, with consequences for root-number bias and Galois decomposition in spaces of cusp forms.
Abstract
We give a formula for the number of newforms in $S_k^{\mathrm{new}}(N)$ that have prescribed ramified supercuspidal components $π_p$ 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 $π_p$, and the root number of each $π_p$. The formula is completely explicit when $T$ consists of either a single prime or all prime factors of $N$.