arXiv subjects

Yuanyang Jiang

Publications and source records attributed to Yuanyang Jiang.

3 recordsLinked to original sources

arXiv2026

Classicality for Hilbert modular forms

Let $F$ be a totally real number field. We prove that a character of the spherical Hecke algebra appearing in the completed cohomology of Hilbert modular varieties is modular if the associated Galois representation is absolutely irreducible, and de Rham of regular parallel weights. As an application, we prove some new cases of the Langlands-Clozel-Fontaine-Mazur conjecture of $\mathrm{GL}_2$ over totally real fields. For the proof, we generalize the method in [Pan26], calculate geometric partial Fontaine operators, and study the cohomology of the associated Koszul-type partial de Rham complexes. The key step is the establishment of a locally analytic Jacquet-Langlands transfer, whose proof consists of several novel ingredients including a comparison of Igusa stacks for different quaternionic Shimura data constructed by [DvHKZ26], and the Grothendieck-Messing theory for locally analytic infinite level Shimura varieties established in [Jiang26a].

math.NT↗

arXiv2026

Differential operators on locally analytic Shimura varieties

We investigate infinite-level Shimura varieties within the framework of analytic stacks of Clausen-Scholze, developing their smooth, completed, locally analytic, and de Rham realizations. We formulate a Grothendieck-Messing-Hodge-Tate period map, and establish a Grothendieck-Messing theory for locally analytic infinite-level Shimura varieties. This theory, combined with a reformulation of Riemann-Hilbert correspondence, implies that the locally analytic infinite-level Shimura variety can be fully reconstructed purely from its perfectoid counterpart and its $\mathbb{B}_{\mathrm{dR}}^{+}$-thickening. Building upon this geometric structure, we systematically construct differential operators generalizing those of Pan, and we introduce a Bernstein-Gelfand-Gelfand-Fontaine complex based on dual BGG complexes, conjecturing its automorphic properties. These constructions will be used to establish a locally analytic Jacquet-Langlands correspondence in a companion paper ([Jia26a]).

math.NT↗

arXiv2023

Theta Operator Equals Fontaine Operator on Modular Curves

Inspired by [Pan22], we give a new proof that for an overconvergent modular eigenform $f$ of weight $1+k$ with $k\in\mathbb{Z}_{\ge1}$, assuming that its associated global Galois representation $\rho_{f}$ is irreducible, then $f$ is classical if and only if $\rho_{f}$ is de Rham at $p$. For the proof, we prove that theta operator $\theta^{k}$ coincides with Fontaine operator in a suitable sense.

math.NT↗