p-adic level raising on the eigenvariety for U(3)
Ruishen Zhao
TL;DR
This work establishes a p-adic level-raising mechanism for automorphic forms on the definite unitary group $U(3)/\mathbb{Q}$, showing that a non-very-Eisenstein old point with $T_l=l(l^3+1)$ attains a position in the new component of the l-adic eigenvariety, i.e., an intersection of old and new loci. Central to the argument are an abelian Ihara lemma for $U(n)$ and duality arguments on Hecke modules that control abelian and non-abelian forms, enabling a precise analysis of the level-raising map and its dual. The results generalize classical mod $p$ level-raising and Newton’s p-adic level-raising, and yield non-classical intersection points on eigenvarieties, with discussions on explicit constructions and potential p-adic Langlands functoriality. The paper also develops local analogues and points toward broader applicability to higher ranks and other definite unitary groups, highlighting the role of degenerate Satake parameters and endoscopy in shaping the local-global picture.
Abstract
We prove level raising results for $p$-adic automorphic forms on definite unitary groups $U(3)/\mathbb{Q}$ and deduce some intersection points on the eigenvariety. Let $l$ be an inert prime where the level subgroups varies, if there is a non-very-Eisenstein point $φ$ on the old component (generically parametrizing forms old at $l$) satisfying $T_{l}(φ)=l(l^3+1)$, then this point also lies in the new component (generically parametrizing forms new at $l$). This provides a $p$-adic analogue of Bellaïche and Graftieaux's mod $p$ level raising for classical automorphic forms on $U(3)$, and also generalizes James Newton's $p$-adic level raising results for definite quaternion algebras. Key ingredients include abelian Ihara lemma (proved for any definite unitary group $U(n)$) and some duality arguments about certain Hecke modules. Finally we also discuss some methods to construct such points explicitly and further development.