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1. Hodge classes and the Jacquet–Langlands correspondence

   https://doi.org/10.1017/fmp.2023.20  

   Ichino, Atsushi; Prasanna, Kartik (September 2023, Forum of Mathematics, Pi)

   We prove that the Jacquet–Langlands correspondence for cohomological automorphic forms on quaternionic Shimura varieties is realized by a Hodge class. Conditional on Kottwitz’s conjecture for Shimura varieties attached to unitary similitude groups, we also show that the image of this Hodge class in ℓ-adic cohomology is Galois invariant for all ℓ. 

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2. Perfectoidness via Sen theory and applications to Shimura varieties

   https://doi.org/10.1090/jams/1060  

   He, Tongmu (July 2025, Journal of the American Mathematical Society)

   Sen’s theorem on the ramification of a $p$-adic analytic Galois extension of $p$-adic local fields shows that its perfectoidness is equivalent to the non-vanishing of its arithmetic Sen operator. By developing $p$-adic Hodge theory for general valuation rings, we establish a geometric analogue of Sen’s criterion for any $p$-adic analytic Galois extension of $p$-adic varieties: its (Riemann-Zariski) stalkwise perfectoidness is necessary for the non-vanishing of the geometric Sen operators. As the latter is verified for general Shimura varieties by Pan and Rodríguez Camargo, we obtain the perfectoidness of every completed stalk of general Shimura varieties at infinite level at $p$. As an application, we prove that the integral completed cohomology groups vanish in higher degrees, verifying a conjecture of Calegari-Emerton for general Shimura varieties. 

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3. -ADIC -FUNCTIONS FOR UNITARY GROUPS

   https://doi.org/10.1017/fmp.2020.4  

   EISCHEN, ELLEN; HARRIS, MICHAEL; LI, JIANSHU; SKINNER, CHRISTOPHER (January 2020, Forum of Mathematics, Pi)

   This paper completes the construction of $$p$$ -adic $$L$$ -functions for unitary groups. More precisely, in Harris, Li and Skinner [‘ $$p$$ -adic $$L$$ -functions for unitary Shimura varieties. I. Construction of the Eisenstein measure’, Doc. Math. Extra Vol. (2006), 393–464 (electronic)], three of the authors proposed an approach to constructing such $$p$$ -adic $$L$$ -functions (Part I). Building on more recent results, including the first named author’s construction of Eisenstein measures and $$p$$ -adic differential operators [Eischen, ‘A $$p$$ -adic Eisenstein measure for unitary groups’, J. Reine Angew. Math. 699 (2015), 111–142; ‘ $$p$$ -adic differential operators on automorphic forms on unitary groups’, Ann. Inst. Fourier (Grenoble) 62 (1) (2012), 177–243], Part II of the present paper provides the calculations of local $$\unicode[STIX]{x1D701}$$ -integrals occurring in the Euler product (including at $$p$$ ). Part III of the present paper develops the formalism needed to pair Eisenstein measures with Hida families in the setting of the doubling method. 

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4. Chern classes of automorphic vector bundles, II

   Esnault, Hélène; Harris, Michael (September 2019, Épijournal de géométrie algébrique)

   We prove that the l-adic Chern classes of canonical extensions of automorphic vector bundles, over toroidal compactifications of Shimura varieties of Hodge type over \bar{Q}_p, descend to classes in the l-adic cohomology of the minimal compactifications. These are invariant under the Galois group of the p-adic field above which the variety and the bundle are defined. 

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5. MULTIPLICITY ONE AT FULL CONGRUENCE LEVEL

   https://doi.org/10.1017/S1474748020000225  

   Le, Daniel; Morra, Stefano; Schraen, Benjamin (May 2020, Journal of the Institute of Mathematics of Jussieu)

   null (Ed.)

   Let $$F$$ be a totally real field in which $$p$$ is unramified. Let $$\overline{r}:G_{F}\rightarrow \text{GL}_{2}(\overline{\mathbf{F}}_{p})$$ be a modular Galois representation that satisfies the Taylor–Wiles hypotheses and is tamely ramified and generic at a place $$v$$ above $$p$$ . Let $$\mathfrak{m}$$ be the corresponding Hecke eigensystem. We describe the $$\mathfrak{m}$$ -torsion in the $$\text{mod}\,p$$ cohomology of Shimura curves with full congruence level at $$v$$ as a $$\text{GL}_{2}(k_{v})$$ -representation. In particular, it only depends on $$\overline{r}|_{I_{F_{v}}}$$ and its Jordan–Hölder factors appear with multiplicity one. The main ingredients are a description of the submodule structure for generic $$\text{GL}_{2}(\mathbf{F}_{q})$$ -projective envelopes and the multiplicity one results of Emerton, Gee and Savitt [Lattices in the cohomology of Shimura curves, Invent. Math.   200 (1) (2015), 1–96]. 

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