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arXiv · 2610.12605

Properness of the Cuspidal Hilbert Eigenvariety

Yi Zhu
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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.

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Yi Zhu. 2026-10-08. Properness of the Cuspidal Hilbert Eigenvariety. https://arxiv.org/abs/2610.12605

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