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1. Diagrams in the mod p cohomology of Shimura curves

   https://doi.org/10.1112/S0010437X21007375  

   Dotto, Andrea; Le, Daniel (August 2021, Compositio Mathematica)

   null (Ed.)

   Abstract We prove a local–global compatibility result in the mod $$p$$ Langlands program for $$\mathrm {GL}_2(\mathbf {Q}_{p^f})$$ . Namely, given a global residual representation $$\bar {r}$$ appearing in the mod $$p$$ cohomology of a Shimura curve that is sufficiently generic at $$p$$ and satisfies a Taylor–Wiles hypothesis, we prove that the diagram occurring in the corresponding Hecke eigenspace of mod $$p$$ completed cohomology is determined by the restrictions of $$\bar {r}$$ to decomposition groups at $$p$$ . If these restrictions are moreover semisimple, we show that the $$(\varphi ,\Gamma )$$ -modules attached to this diagram by Breuil give, under Fontaine's equivalence, the tensor inductions of the duals of the restrictions of $$\bar {r}$$ to decomposition groups at $$p$$ . 

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2. Rationality of twists of the Siegel modular variety of genus 2 and level 3

   https://doi.org/10.1090/proc/15854  

   Calegari, Frank; Chidambaram, Shiva (February 2022, Proceedings of the American Mathematical Society)

   Let  ρ ¯ : G Q → GSp 4 ⁡ ( F 3 ) \overline {\rho }: G_{\mathbf {Q}} \rightarrow \operatorname {GSp}_4(\mathbf {F}_3) be a continuous Galois representation with cyclotomic similitude character. Equivalently, consider ρ ¯ \overline {\rho } to be the Galois representation associated to the  3 3 -torsion of a principally polarized abelian surface  A / Q A/\mathbf {Q} . We prove that the moduli space  A 2 ( ρ ¯ ) \mathcal {A}_2(\overline {\rho }) of principally polarized abelian surfaces  B / Q B/\mathbf {Q} admitting a symplectic isomorphism  B [ 3 ] ≃ ρ ¯ B[3] \simeq \overline {\rho } of Galois representations is never rational over  Q \mathbf {Q} when  ρ ¯ \overline {\rho } is surjective, even though it is both rational over  C \mathbf {C} and unirational over  Q \mathbf {Q} via a map of degree  6 6 . 

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3. An Euler system for GU(2, 1)

   https://doi.org/10.1007/s00208-021-02224-4  

   Loeffler, David; Skinner, Christopher; Zerbes, Sarah Livia (April 2022, Mathematische Annalen)

   Abstract We construct an Euler system associated to regular algebraic, essentially conjugate self-dual cuspidal automorphic representations of $${{\,\mathrm{GL}\,}}_3$$ GL 3 over imaginary quadratic fields, using the cohomology of Shimura varieties for $${\text {GU}}(2, 1)$$ GU ( 2 , 1 ) . 

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4. Universal Tensor Categories Generated by Dual Pairs

   https://doi.org/10.1007/s10485-021-09640-2  

   Chirvasitu, Alexandru; Penkov, Ivan (October 2021, Applied Categorical Structures)

   Abstract Let $$V_*\otimes V\rightarrow {\mathbb {C}}$$ V ∗ ⊗ V → C be a non-degenerate pairing of countable-dimensional complex vector spaces V and $$V_*$$ V ∗ . The Mackey Lie algebra $${\mathfrak {g}}=\mathfrak {gl}^M(V,V_*)$$ g = gl M ( V , V ∗ ) corresponding to this pairing consists of all endomorphisms $$\varphi $$ φ of V for which the space $$V_*$$ V ∗ is stable under the dual endomorphism $$\varphi ^*: V^*\rightarrow V^*$$ φ ∗ : V ∗ → V ∗ . We study the tensor Grothendieck category $${\mathbb {T}}$$ T generated by the $${\mathfrak {g}}$$ g -modules V , $$V_*$$ V ∗ and their algebraic duals $$V^*$$ V ∗ and $$V^*_*$$ V ∗ ∗ . The category $${{\mathbb {T}}}$$ T is an analogue of categories considered in prior literature, the main difference being that the trivial module $${\mathbb {C}}$$ C is no longer injective in $${\mathbb {T}}$$ T . We describe the injective hull I of $${\mathbb {C}}$$ C in $${\mathbb {T}}$$ T , and show that the category $${\mathbb {T}}$$ T is Koszul. In addition, we prove that I is endowed with a natural structure of commutative algebra. We then define another category $$_I{\mathbb {T}}$$ I T of objects in $${\mathbb {T}}$$ T which are free as I -modules. Our main result is that the category $${}_I{\mathbb {T}}$$ I T is also Koszul, and moreover that $${}_I{\mathbb {T}}$$ I T is universal among abelian $${\mathbb {C}}$$ C -linear tensor categories generated by two objects X , Y with fixed subobjects $$X'\hookrightarrow X$$ X ′ ↪ X , $$Y'\hookrightarrow Y$$ Y ′ ↪ Y and a pairing $$X\otimes Y\rightarrow {\mathbf{1 }}$$ X ⊗ Y → 1 where 1 is the monoidal unit. We conclude the paper by discussing the orthogonal and symplectic analogues of the categories $${\mathbb {T}}$$ T and $${}_I{\mathbb {T}}$$ I T . 

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5. SERRE WEIGHTS AND BREUIL’S LATTICE CONJECTURE IN DIMENSION THREE

   https://doi.org/10.1017/fmp.2020.1  

   LE, DANIEL; LE HUNG, BAO V.; LEVIN, BRANDON; MORRA, STEFANO (January 2020, Forum of Mathematics, Pi)

   We prove in generic situations that the lattice in a tame type induced by the completed cohomology of a $U(3)$ -arithmetic manifold is purely local, that is, only depends on the Galois representation at places above $$p$$ . This is a generalization to $$\text{GL}_{3}$$ of the lattice conjecture of Breuil. In the process, we also prove the geometric Breuil–Mézard conjecture for (tamely) potentially crystalline deformation rings with Hodge–Tate weights $(2,1,0)$ as well as the Serre weight conjectures of Herzig [‘The weight in a Serre-type conjecture for tame $$n$$ -dimensional Galois representations’, Duke Math. J.   149 (1) (2009), 37–116] over an unramified field extending the results of Le et al. [‘Potentially crystalline deformation 3985 rings and Serre weight conjectures: shapes and shadows’, Invent. Math.   212 (1) (2018), 1–107]. We also prove results in modular representation theory about lattices in Deligne–Lusztig representations for the group $$\text{GL}_{3}(\mathbb{F}_{q})$$ . 

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