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1. K-theoretic Tate–Poitou duality and the fiber of the cyclotomic trace

   https://doi.org/10.1007/s00222-020-00952-z  

   Blumberg, Andrew J.; Mandell, Michael A. (February 2020, Inventiones mathematicae)

   Let p∈Z be an odd prime. We prove a spectral version of Tate–Poitou duality for the algebraic K-theory spectra of number rings with p inverted. This identifies the homotopy type of the fiber of the cyclotomic trace K(OF)∧p→TC(OF)∧p after taking a suitably connective cover. As an application, we identify the homotopy type at odd primes of the homotopy fiber of the cyclotomic trace for the sphere spectrum in terms of the algebraic K-theory of Z. 

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2. Topological Cyclic Homology Via the Norm

   Angeltveit, Vigleik; Blumberg, Andrew J.; Gerhardt, Teena; Hill, Michael A.; Lawson, Tyler; Mandell, Michael A. (October 2018, Documenta mathematica)

   We describe a construction of the cyclotomic structure on topological Hochschild homology (THH) of a ring spectrum using the Hill-Hopkins-Ravenel multiplicative norm. Our analysis takes place entirely in the category of equivariant orthogonal spectra, avoiding use of the Bökstedt coherence machinery. We are also able to define two relative versions of topological cyclic homology (TC) and TR-theory: one starting with a ring C_n-spectrum and one starting with an algebra over a cyclotomic commutative ring spectrum A. We describe spectral sequences computing the relative theory over A in terms of TR over the sphere spectrum and vice versa. Furthermore, our construction permits a straightforward definition of the Adams operations on TR and TC. 

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3. The eigensplitting of the fiber of the cyclotomic trace for the sphere spectrum

   https://doi.org/10.1090/tran/8822  

   Blumberg, Andrew; Mandell, Michael (January 2022, Transactions of the American Mathematical Society)

   Let p ∈ Z p\in {\mathbb {Z}} be an odd prime. We show that the fiber sequence for the cyclotomic trace of the sphere spectrum S {\mathbb {S}} admits an “eigensplitting” that generalizes known splittings on K K -theory and T C TC . We identify the summands in the fiber as the covers of Z p {\mathbb {Z}}_{p} -Anderson duals of summands in the K ( 1 ) K(1) -localized algebraic K K -theory of Z {\mathbb {Z}} . Analogous results hold for the ring Z {\mathbb {Z}} where we prove that the K ( 1 ) K(1) -localized fiber sequence is self-dual for Z p {\mathbb {Z}}_{p} -Anderson duality, with the duality permuting the summands by i ↦ p − i i\mapsto p-i (indexed mod p − 1 p-1 ). We explain an intrinsic characterization of the summand we call Z Z in the splitting T C ( Z ) p ∧ ≃ j ∨ Σ j ′ ∨ Z TC({\mathbb {Z}})^{\wedge }_{p}\simeq j \vee \Sigma j’\vee Z in terms of units in the p p -cyclotomic tower of Q p {\mathbb {Q}}_{p} . 

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4. Reciprocity maps with restricted ramification

   Sharifi, Romyar T. (August 2022, Transactions of the American Mathematical Society)

   We compare two maps that arise in study of the cohomology of global fields with ramification restricted to a finite set S of primes. One of these maps, which we call an S-reciprocity map, interpolates the values of cup products in S-ramified cohomology. In the case of p-ramified cohomology of the pth cyclotomic field for an odd prime p, we use this to exhibit an intriguing relationship between particular values of the cup product on cyclotomic p-units. We then consider higher analogues of the S-reciprocity map and relate their cokernels to the graded quotients in augmentation filtrations of Iwasawa modules. 

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5. Abelian varieties with isogenous reductions

   https://doi.org/https://doi.org/10.5802/crmath.129  

   Khare, Chandreshekhar; Larsen, Michael: (October 2020, Comptes rendus)

   null (Ed.)

   Let $$A_1$$ and $$A_2$$ be abelian varieties over a number field $$K$$. We prove that if there exists a non-trivial morphism of abelian varieties between reductions of $$A_1$$ and $$A_2$$ at a sufficiently high percentage of primes, then there exists a non-trivial morphism $$A_1\to A_2$$ over $$\bar K$$. Along the way, we give an upper bound for the number of components of a reductive subgroup of $$GL_n$$ whose intersection with the union of $$Q$$-rational conjugacy classes of $$GL_n$$ is Zariski-dense. This can be regarded as a generalization of the Minkowski-Schur theorem on faithful representations of finite groups with rational characters. 

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