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1. Big images of two-dimensional pseudorepresentations

   https://doi.org/10.1007/s00208-021-02345-w  

   Conti, Andrea; Lang, Jaclyn; Medvedovsky, Anna (February 2022, Mathematische Annalen)

   Bellaïche has recently applied Pink-Lie theory to prove that, under mild conditions, the image of a continuous 2-dimensional pseudorepresentation ρ of a profinite group on a local pro- p domain A contains a nontrivial congruence subgroup of SL2(B) for a certain subring B of A. We enlarge Bellaïche’s ring and give this new B a conceptual interpretation both in terms of conjugate self-twists of ρ, symmetries that constrain its image, and in terms of the adjoint trace ring of ρ, which we show is both more natural and the optimal ring for these questions in general. Finally, we use our purely algebraic result to recover and extend a variety of arithmetic big-image results for GL2-Galois representations arising from elliptic, Hilbert, and Bianchi modular forms and p-adic Hida or Coleman families of elliptic and Hilbert modular forms. 

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2. Galois automorphisms on Harish-Chandra series and Navarro’s self-normalizing Sylow $$2$$-subgroup conjecture

   https://doi.org/10.1090/tran/7590  

   Schaeffer Fry, A. A. (July 2019, Transactions of the American Mathematical Society)

   Navarro has conjectured a necessary and sufficient condition for a finite group G to have a self-normalizing Sylow 2-subgroup, which is given in terms of the ordinary irreducible characters of G. In a previous article, Schaeffer Fry has reduced the proof of this conjecture to showing that certain related statements hold for simple groups. In this article, we describe the action of Galois automorphisms on the Howlett-Lehrer parametrization of Harish-Chandra induced characters. We use this to complete the proof of the conjecture by showing that the remaining simple groups satisfy the required conditions. 

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3. Entire Theta Operators at Unramified Primes

   https://doi.org/10.1093/imrn/rnab190  

   Eischen, E; Mantovan, E (July 2021, International Mathematics Research Notices)

   null (Ed.)

   Abstract Starting with the work of Serre, Katz, and Swinnerton-Dyer, theta operators have played a key role in the study of $$p$$-adic and $$\textrm{mod}\; p$$ modular forms and Galois representations. This paper achieves two main results for theta operators on automorphic forms on PEL-type Shimura varieties: (1) the analytic continuation at unramified primes $$p$$ to the whole Shimura variety of the $$\textrm{mod}\; p$$ reduction of $$p$$-adic Maass–Shimura operators a priori defined only over the $$\mu $$-ordinary locus, and (2) the construction of new $$\textrm{mod}\; p$$ theta operators that do not arise as the $$\textrm{mod}\; p$$ reduction of Maass–Shimura operators. While the main accomplishments of this paper concern the geometry of Shimura varieties and consequences for differential operators, we conclude with applications to Galois representations. Our approach involves a careful analysis of the behavior of Shimura varieties and enables us to obtain more general results than allowed by prior techniques, including for arbitrary signature, vector weights, and unramified primes in CM fields of arbitrary degree. 

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4. Reductions of abelian surfaces over global function fields

   https://doi.org/10.1112/S0010437X22007473  

   Maulik, Davesh; Shankar, Ananth N.; Tang, Yunqing (April 2022, Compositio Mathematica)

   Let $$A$$ be a non-isotrivial ordinary abelian surface over a global function field of characteristic $p>0$ with good reduction everywhere. Suppose that $$A$$ does not have real multiplication by any real quadratic field with discriminant a multiple of $$p$$ . We prove that there are infinitely many places modulo which $$A$$ is isogenous to the product of two elliptic curves. 

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5. The Simplicial Coalgebra of Chains Under Three Different Notions of Weak Equivalence

   https://doi.org/10.1093/imrn/rnae031  

   Raptis, George; Rivera, Manuel (July 2024, International Mathematics Research Notices)

   Abstract We study the simplicial coalgebra of chains on a simplicial set with respect to three notions of weak equivalence. To this end, we construct three model structures on the category of reduced simplicial sets for any commutative ring $$R$$. The weak equivalences are given by: (1) an $$R$$-linearized version of categorical equivalences, (2) maps inducing an isomorphism on fundamental groups and an $$R$$-homology equivalence between universal covers, and (3) $$R$$-homology equivalences. Analogously, for any field $${\mathbb{F}}$$, we construct three model structures on the category of connected simplicial cocommutative $${\mathbb{F}}$$-coalgebras. The weak equivalences in this context are (1′) maps inducing a quasi-isomorphism of dg algebras after applying the cobar functor, (2′) maps inducing a quasi-isomorphism of dg algebras after applying a localized version of the cobar functor, and (3′) quasi-isomorphisms. Building on a previous work of Goerss in the context of (3)–(3′), we prove that, when $${\mathbb{F}}$$ is algebraically closed, the simplicial $${\mathbb{F}}$$-coalgebra of chains defines a homotopically full and faithful left Quillen functor for each one of these pairs of model categories. More generally, when $${\mathbb{F}}$$ is a perfect field, we compare the three pairs of model categories in terms of suitable notions of homotopy fixed points with respect to the absolute Galois group of $${\mathbb{F}}$$. 

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