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Properness of the Cuspidal Hilbert Eigenvariety
Abstract
We prove that for any prime p and tame level N, the weight map from the cuspidal Hilbert eigenvariety constructed by Andreatta-Iovita-Pilloni for a totally real field F to the weight space is proper in the sense of Buzzard-Calegari, under a non-vanishing hypothesis on certain Sen polynomials of Galois representations at the primes above p. In particular, this yields the properness of the eigenvariety away from non-positive integral Sen weights. Our approach extends the p-adic Hodge-theoretic argument of Diao-Liu.
Related
- Left properness of Moore flows
- Cuspidal Class Number Formulae for Intermediate Modular Curves of Prime Level
- Cuspidal sheaves on the stack of L-parameters
- An analogue of Γ-function for irreducible cuspidal representations of the group GL(2) over a two-dimensional local field
- EKR Results for Cuspidal Representations of GL2(q)
BibTeX
@misc{zhu2026properness,
title = {{Properness of the Cuspidal Hilbert Eigenvariety}},
author = {Yi Zhu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.12605},
eprint = {2610.12605},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.12605},
}
RIS
TY - UNPB
TI - Properness of the Cuspidal Hilbert Eigenvariety
AU - Yi Zhu
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.12605
UR - https://arxiv.org/abs/2610.12605
AB - We prove that for any prime p and tame level N, the weight map from the cuspidal Hilbert eigenvariety constructed by Andreatta-Iovita-Pilloni for a totally real field F to the weight space is proper in the sense of Buzzard-Calegari, under a non-vanishing hypothesis on certain Sen polynomials of Galois representations at the primes above p. In particular, this yields the properness of the eigenvariety away from non-positive integral Sen weights. Our approach extends the p-adic Hodge-theoretic argument of Diao-Liu.
ER -