Modular analogs of character formulas and minimal lifts of modular forms

Abstract

If $f$ is a mod-$3$ eigenform of weight 2 and level ${\Gamma }_0({\ell }^2)$ for a prime $\ell$ such that $\ell \equiv -1(\mathrm{mod}3)$, and $\ell$ is a vexing prime for $f$, we show that there is no obstruction to finding a minimal lift of $f$, but that there is an obstruction to finding a nonminimal lift. The key new ingredient that we prove is a modular analog of the standard character formula for a cuspidal representation of ${\mathrm{GL}}_2({\mathbb{F}}_{\ell })$, an enhancement that allows us to easily compute the group cohomology of a $3$-adic lattice in such a representation. In fact, we provide a general framework for proving such modular analogs for a broader class of representations using results of Broué and Puig in modular representation theory. We show that this class includes certain Deligne--Lusztig representations and representations coming from higher-depth supercuspidal representations of ${\mathrm{GL}}_2$.

Parts of this paper are from arXiv:2508.06657v1, which has now been split into two papers. This version is revised based on referee comments