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1. The stable Galois correspondence for real closed fields

   https://doi.org/10.1090/conm/707/14250  

   Heller, J.; Ormsby, K. (January 2018, Contemporary mathematics - American Mathematical Society)

   In previous work [7], the authors constructed and studied a lift of the Galois correspondence to stable homotopy categories. In particular, if L/k is a finite Galois extension of fields with Galois group G, there is a functor c∗L/k : SHG → SHk from the G-equivariant stable homotopy category to the stable motivic homotopy category over k such that c∗L/k(G/H+) = Spec(LH)+. The main theorem of [7] says that when k is a real closed field and L = k[i], the restriction of c∗L/k to the η-complete subcategory is full and faithful. Here we “uncomplete” this theorem so that it applies to c∗L/k itself. Our main tools are Bachmann’s theorem on the (2,η)- periodic stable motivic homotopy category and an isomorphism range for the map πRS → πC2 S induced by C2-equivariant Betti realization. 

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2. A supplement to Chebotarev’s density theorem

   https://doi.org/10.1007/s11425-022-2141-1  

   Harcos, Gergely; Soundararajan, Kannan (December 2023, Science China Mathematics)

   Abstract LetL/Kbe a Galois extension of number fields with Galois groupG. We show that if the density of prime ideals inKthat split totally inLtends to 1/∣G∣ with a power saving error term, then the density of prime ideals inKwhose Frobenius is a given conjugacy classC⊂Gtends to ∣C∣/∣G∣ with the same power saving error term. We deduce this by relating the poles of the corresponding Dirichlet series to the zeros ofζ_L(s)/ζ_K(s). 

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3. Class groups and local indecomposability for non-CM forms

   Castella, Francesc; Wang-Erickson, Carl; Hida, Haruzo (July 2019, Journal of the European Mathematical Society)

   In the late 1990s, R. Coleman and R. Greenberg (independently) asked for a global property characterizing those p-ordinary cuspidal eigenforms whose associated Galois representation becomes decomposable upon restriction to a decomposition group at p. It is expected that such p-ordinary eigenforms are precisely those with complex multiplication. In this paper, we study Coleman-Greenberg's question using Galois deformation theory. In particular, for p-ordinary eigenforms which are congruent to one with complex multiplication, we prove that the conjectured answer follows from the p-indivisibility of a certain class group. 

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4. 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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5. The Galois action on the lower central series of the fundamental group of the Fermat curve

   https://doi.org/10.1007/s11856-023-2571-z  

   Davis, Rachel; Pries, Rachel; Wickelgren, Kirsten (November 2023, Israel Journal of Mathematics)

   Abstract. Information about the absolute Galois group G K of a number field K is encoded in how it acts on the ´etale fundamental group π of a curve X defined over K. In the case that K = Q ( ζ n ) is the cyclotomic field and X is the Fermat curve of degree n ≥ 3, Anderson determined the action of G K on the ´etale homology with coefficients in Z/nZ. The ´etale homology is the first quotient in the lower central series of the ´etale fundamental group. In this paper, we determine the Galois module structure of the graded Lie algebra for π. As a consequence, this determines the action of G K on all degrees of the associated graded quotient of the lower central series of the ´etale fundamental group of the Fermat curve of degree n, with coefficients in Z/nZ. 

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