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Register a new userOn the p-adic Beilinson conjecture for number fields
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We formulate a conjectural p-adic analogue of Borels theorem relating regulators for higher K-groups of number fields to special values of the corresponding zeta-functions, using syntomic regulators and p-adic L-functions. We also formulate a corresponding conjecture for Artin motives, and state a conjecture about the precise relation between the p-adic and classical situations. Parts of he conjectures are proved when the number field (or Artin motive) is Abelian over the rationals, and all conjectures are verified numerically in some other cases.
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Using topological cyclic homology, we give a refinement of Beilinsons $p$-adic Goodwillie isomorphism between relative continuous $K$-theory and cyclic homology. As a result, we generalize results of Bloch-Esnault-Kerz and Beilinson on the $p$-adic deformations of $K$-theory classes. Furthermore, we prove structural results for the Bhatt-Morrow-Scholze filtration on $TC$ and identify the graded pieces with the syntomic cohomology of Fontaine-Messing.
For a global function field K of positive characteristic p, we show that Artin conjecture for L-functions of geometric p-adic Galois representations of K is true in a non-trivial p-adic disk but is false in the full p-adic plane. In particular, we prove the non-rationality of the geometric unit root L-functions.
In this paper, we completely prove a standard conjecture on the local converse theorem for generic representations of GLn(F), where F is a non-archimedean local field.
We construct a Chern character map from the K-theory of the reduced C^* algebra of the p-adic GL(n) with values in the periodic cyclic homology of the Schwartz algebra of this group. We prove that this map is an isomorphism after tensoring with C by comparing an explicit formula, stated in the algebraic case by Cuntz and Quillen, with the classical Chern character. This Chern character is a crucial ingredient in the proof of the Baum-Connes conjecture for the p-adic GL(n) due to Baum, Higson and Plymen.
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