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Yigeng Zhao - One of the best experts on this subject based on the ideXlab platform.

  • HIGHER IDELES AND Class Field Theory
    Nagoya Mathematical Journal, 2018
    Co-Authors: Moritz Kerz, Yigeng Zhao
    Abstract:

    We use higher ideles and duality theorems to develop a universal approach to higher dimensional Class Field Theory.

  • duality for relative logarithmic de rham witt sheaves and wildly ramified Class Field Theory over finite Fields
    Compositio Mathematica, 2018
    Co-Authors: Uwe Jannsen, Shuji Saito, Yigeng Zhao
    Abstract:

    In order to study $p$ -adic etale cohomology of an open subvariety $U$ of a smooth proper variety $X$ over a perfect Field of characteristic $p>0$ , we introduce new $p$ -primary torsion sheaves. It is a modification of the logarithmic de Rham–Witt sheaves of $X$ depending on effective divisors $D$ supported in $X-U$ . Then we establish a perfect duality between cohomology groups of the logarithmic de Rham–Witt cohomology of $U$ and an inverse limit of those of the mentioned modified sheaves. Over a finite Field, the duality can be used to study wildly ramified Class Field Theory for the open subvariety $U$ .

  • Duality for relative logarithmic de Rham–Witt sheaves and wildly ramified Class Field Theory over finite Fields
    Compositio Mathematica, 2018
    Co-Authors: Uwe Jannsen, Shuji Saito, Yigeng Zhao
    Abstract:

    In order to study $p$ -adic etale cohomology of an open subvariety $U$ of a smooth proper variety $X$ over a perfect Field of characteristic $p>0$ , we introduce new $p$ -primary torsion sheaves. It is a modification of the logarithmic de Rham–Witt sheaves of $X$ depending on effective divisors $D$ supported in $X-U$ . Then we establish a perfect duality between cohomology groups of the logarithmic de Rham–Witt cohomology of $U$ and an inverse limit of those of the mentioned modified sheaves. Over a finite Field, the duality can be used to study wildly ramified Class Field Theory for the open subvariety $U$ .

  • duality for relative logarithmic de rham witt sheaves and wildly ramified Class Field Theory over finite Fields
    arXiv: Algebraic Geometry, 2016
    Co-Authors: Uwe Jannsen, Shuji Saito, Yigeng Zhao
    Abstract:

    With the aim of studying $p$-adic \'etale cohomology of an open subvariety $U$ of a smooth proper variety $X$ over a perfect Field of characteristic $p>0$, we introduce new $p$-primary torsion sheaves, which are modifications of the logarithmic de Rham-Witt sheaves depending on divisors $D$ in the complement of the open subvariety. Then we establish a perfect duality between the logarithmic de Rham-Witt cohomology of $U$ and an inverse limit of the mentioned modified sheaves. Over a finite Field, the duality can be used to study wild ramification Class Field Theory for the open subvariety $U$.

Uwe Jannsen - One of the best experts on this subject based on the ideXlab platform.

  • duality for relative logarithmic de rham witt sheaves and wildly ramified Class Field Theory over finite Fields
    Compositio Mathematica, 2018
    Co-Authors: Uwe Jannsen, Shuji Saito, Yigeng Zhao
    Abstract:

    In order to study $p$ -adic etale cohomology of an open subvariety $U$ of a smooth proper variety $X$ over a perfect Field of characteristic $p>0$ , we introduce new $p$ -primary torsion sheaves. It is a modification of the logarithmic de Rham–Witt sheaves of $X$ depending on effective divisors $D$ supported in $X-U$ . Then we establish a perfect duality between cohomology groups of the logarithmic de Rham–Witt cohomology of $U$ and an inverse limit of those of the mentioned modified sheaves. Over a finite Field, the duality can be used to study wildly ramified Class Field Theory for the open subvariety $U$ .

  • Duality for relative logarithmic de Rham–Witt sheaves and wildly ramified Class Field Theory over finite Fields
    Compositio Mathematica, 2018
    Co-Authors: Uwe Jannsen, Shuji Saito, Yigeng Zhao
    Abstract:

    In order to study $p$ -adic etale cohomology of an open subvariety $U$ of a smooth proper variety $X$ over a perfect Field of characteristic $p>0$ , we introduce new $p$ -primary torsion sheaves. It is a modification of the logarithmic de Rham–Witt sheaves of $X$ depending on effective divisors $D$ supported in $X-U$ . Then we establish a perfect duality between cohomology groups of the logarithmic de Rham–Witt cohomology of $U$ and an inverse limit of those of the mentioned modified sheaves. Over a finite Field, the duality can be used to study wildly ramified Class Field Theory for the open subvariety $U$ .

  • duality for relative logarithmic de rham witt sheaves and wildly ramified Class Field Theory over finite Fields
    arXiv: Algebraic Geometry, 2016
    Co-Authors: Uwe Jannsen, Shuji Saito, Yigeng Zhao
    Abstract:

    With the aim of studying $p$-adic \'etale cohomology of an open subvariety $U$ of a smooth proper variety $X$ over a perfect Field of characteristic $p>0$, we introduce new $p$-primary torsion sheaves, which are modifications of the logarithmic de Rham-Witt sheaves depending on divisors $D$ in the complement of the open subvariety. Then we establish a perfect duality between the logarithmic de Rham-Witt cohomology of $U$ and an inverse limit of the mentioned modified sheaves. Over a finite Field, the duality can be used to study wild ramification Class Field Theory for the open subvariety $U$.

  • Bertini theorems and Lefschetz pencils over discrete valuation rings, with applications to higher Class Field Theory
    Journal of Algebraic Geometry, 2012
    Co-Authors: Uwe Jannsen, Shuji Saito
    Abstract:

    Good hyperplane sections, whose existence is assured by Bertini’s theorem, and good families of hyperplane sections, so-called Lefschetz pencils, are well-known constructions and powerwful tools in Classical geometry, i.e., for varieties over a Field. But for arithmetic questions one is naturally led to the consideration of models over Dedekind rings and, for local questions, to schemes over discrete valuation rings. It is the aim of this note to provide extensions of the mentioned constructions to this situation. We point out some new phenomena, and give an application to the Class Field Theory of varieties over local Fields with good reduction. See also [JS] for more arithmetic applications. Let A be a discrete valuation ring with fraction Field K, maximal ideal m and residue Field F = A/m. Let η = Spec(K) and s = Spec(F ) be the generic and closed point of Spec(A), respectively. For any scheme X over A we let Xη = X ×A K and Xs = X ×A F be its generic and special fibre, respectively.

  • Kato homology of arithmetic schemes and higher Class Field Theory over local Fields.
    Documenta Mathematica, 2003
    Co-Authors: Uwe Jannsen, Shuji Saito
    Abstract:

    For arithmetical schemes X, K. Kato introduced certain complexes C r,s (X) of Gersten-Bloch-Ogus type whose components in- volve Galois cohomology groups of all the residue Fields of X. For specific (r,s), he stated some conjectures on their homology generalizing the fun- damental isomorphisms and exact sequences for Brauer groups of local and global Fields. We prove some of these conjectures in small degrees and give applications to the Class Field Theory of smooth projecive varieties over local Fields, and finiteness questions for some motivic cohomology groups over local and global Fields.

Shuji Saito - One of the best experts on this subject based on the ideXlab platform.

  • duality for relative logarithmic de rham witt sheaves and wildly ramified Class Field Theory over finite Fields
    Compositio Mathematica, 2018
    Co-Authors: Uwe Jannsen, Shuji Saito, Yigeng Zhao
    Abstract:

    In order to study $p$ -adic etale cohomology of an open subvariety $U$ of a smooth proper variety $X$ over a perfect Field of characteristic $p>0$ , we introduce new $p$ -primary torsion sheaves. It is a modification of the logarithmic de Rham–Witt sheaves of $X$ depending on effective divisors $D$ supported in $X-U$ . Then we establish a perfect duality between cohomology groups of the logarithmic de Rham–Witt cohomology of $U$ and an inverse limit of those of the mentioned modified sheaves. Over a finite Field, the duality can be used to study wildly ramified Class Field Theory for the open subvariety $U$ .

  • Duality for relative logarithmic de Rham–Witt sheaves and wildly ramified Class Field Theory over finite Fields
    Compositio Mathematica, 2018
    Co-Authors: Uwe Jannsen, Shuji Saito, Yigeng Zhao
    Abstract:

    In order to study $p$ -adic etale cohomology of an open subvariety $U$ of a smooth proper variety $X$ over a perfect Field of characteristic $p>0$ , we introduce new $p$ -primary torsion sheaves. It is a modification of the logarithmic de Rham–Witt sheaves of $X$ depending on effective divisors $D$ supported in $X-U$ . Then we establish a perfect duality between cohomology groups of the logarithmic de Rham–Witt cohomology of $U$ and an inverse limit of those of the mentioned modified sheaves. Over a finite Field, the duality can be used to study wildly ramified Class Field Theory for the open subvariety $U$ .

  • duality for relative logarithmic de rham witt sheaves and wildly ramified Class Field Theory over finite Fields
    arXiv: Algebraic Geometry, 2016
    Co-Authors: Uwe Jannsen, Shuji Saito, Yigeng Zhao
    Abstract:

    With the aim of studying $p$-adic \'etale cohomology of an open subvariety $U$ of a smooth proper variety $X$ over a perfect Field of characteristic $p>0$, we introduce new $p$-primary torsion sheaves, which are modifications of the logarithmic de Rham-Witt sheaves depending on divisors $D$ in the complement of the open subvariety. Then we establish a perfect duality between the logarithmic de Rham-Witt cohomology of $U$ and an inverse limit of the mentioned modified sheaves. Over a finite Field, the duality can be used to study wild ramification Class Field Theory for the open subvariety $U$.

  • Chow group of $0$-cycles with modulus and higher-dimensional Class Field Theory
    Duke Mathematical Journal, 2016
    Co-Authors: Moritz Kerz, Shuji Saito
    Abstract:

    One of the main results of this article is a proof of the rank-one case of an existence conjecture on lisse (Q) over bar (l)-sheaves on a smooth variety U over a finite Field due to Deligne and Drinfeld. The problem is translated into the language of higher-dimensional Class Field Theory over finite Fields, which describes the abelian fundamental group of U by Chow groups of 0-cycles with moduli. A key ingredient is the construction of a cycle-theoretic avatar of a refined Artin conductor in ramification Theory originally studied by Kazuya Kato.

  • Bertini theorems and Lefschetz pencils over discrete valuation rings, with applications to higher Class Field Theory
    Journal of Algebraic Geometry, 2012
    Co-Authors: Uwe Jannsen, Shuji Saito
    Abstract:

    Good hyperplane sections, whose existence is assured by Bertini’s theorem, and good families of hyperplane sections, so-called Lefschetz pencils, are well-known constructions and powerwful tools in Classical geometry, i.e., for varieties over a Field. But for arithmetic questions one is naturally led to the consideration of models over Dedekind rings and, for local questions, to schemes over discrete valuation rings. It is the aim of this note to provide extensions of the mentioned constructions to this situation. We point out some new phenomena, and give an application to the Class Field Theory of varieties over local Fields with good reduction. See also [JS] for more arithmetic applications. Let A be a discrete valuation ring with fraction Field K, maximal ideal m and residue Field F = A/m. Let η = Spec(K) and s = Spec(F ) be the generic and closed point of Spec(A), respectively. For any scheme X over A we let Xη = X ×A K and Xs = X ×A F be its generic and special fibre, respectively.

Alexander Schmidt - One of the best experts on this subject based on the ideXlab platform.

  • tame Class Field Theory for singular varieties over finite Fields
    Journal of the European Mathematical Society, 2017
    Co-Authors: Thomas Geisser, Alexander Schmidt
    Abstract:

    In the 1980’s, Kato and Saito (based on ideas of Bloch) generalized the Class Field Theory for smooth, projective curves over finite Fields to smooth, projective varieties of arbitrary dimension [KaSa]: The map from the free abelian group generated by the closed points which sends a generator x ∈ X to the image of the Frobenius of k(x) under π 1 (k(x))→ π 1 (X) factors through the Chow group of zero cycles, and induces an isomorphism

  • Covering data and higher dimensional global Class Field Theory
    Journal of Number Theory, 2009
    Co-Authors: Moritz Kerz, Alexander Schmidt
    Abstract:

    For a connected regular scheme X, flat and of finite type over Spec(Z), we construct a reciprocity homomorphism rX : CX ! p ab (X), which is surjective and whose kernel is the connected compo- nent of the identity. The (topological) group CX is explicitly given and built solely out of data attached to points and curves on X. A similar but weaker statement holds for smooth varieties over finite Fields. Our results are based on earlier work of G. Wiesend. To the memory of Gotz Wiesend 1 The aim of global Class Field Theory is the description of abelian extensions of arithmetic schemes (i.e. regular schemes X of finite type over Spec(Z)) in terms of arithmetic invariants attached to X. The solution of this problem in the case dim X = 1 was one of the major achievements of number Theory in the first part of the previous century. In the 1980s, mainly due to K. Kato and S. Saito (8), a generalization to higher dimensional schemes has been found. The description of the abelian extensions is given in terms of a generalized idele Class group, whose rather involved definition is based on Milnor K-sheaves. In the course of the last years, G. Wiesend developed a new approach to higher dimensional Class Field Theory which only uses data attached to points and curves on the scheme. The central and new idea was to consider data which describe not necessarily abelian Galois coverings of all curves on the scheme, together with some compatibility condition. Then one investigates the question whether these data are given by a single Galois covering of the scheme. The essential advantage of this nonabelian approach is that one can use the topological finite generation of the tame fundamental groups of smooth curves over separably closed Fields as an additional input. The restriction to abelian coverings is made at a later stage. One obtains an explicitly given Class group CX together with a reciprocity ho- momorphism rX : CX! p ab

  • tame Class Field Theory for arithmetic schemes
    Inventiones Mathematicae, 2005
    Co-Authors: Alexander Schmidt
    Abstract:

    Takagi's Class Field Theory gave a decription of the abelian extensions of a number Field $K$ in terms of ideal groups in $K$. In the 1980s, {\it K. Kato} and {\it S. Saito} [``Global Class Field Theory of arithmetic schemes". Applications of algebraic K-Theory to algebraic geometry and number Theory, Proc. AMS-IMS-SIAM Joint Summer Res. Conf., Boulder/Colo. 1983, Part I, Contemp. Math. 55, 255--331 (1986; Zbl 0614.14001)] were able to generalize Class Field Theory to higher dimensional Fields, and to describe their abelian extensions using a generalized idele Class group whose definition is quite involved. In the case of unramified extensions, however, the Class Fields can be described geometrically using Chow groups. For Fields of positive characteristic, a similarly geometric description for tamely ramified extensions was obtained by the author and {\it M.~Spiess} [J. Reine Angew. Math. 527, 13--36 (2000; Zbl 0961.14013)]. In this article, an analogous result is proved for the case of mixed characteristic.

  • Tame Class Field Theory for arithmetic schemes
    Inventiones mathematicae, 2004
    Co-Authors: Alexander Schmidt
    Abstract:

    We extend the unramified Class Field Theory for arithmetic schemes of K. Kato and S. Saito to the tame case. Let $X$ be a regular proper arithmetic scheme and let $D$ be a divisor on $X$ whose vertical irreducible components are normal schemes. Theorem: There exists a natural reciprocity isomorphism \[ \rec_{X,D}: \CH_0(X,D) \liso \tilde \pi_1^t(X,D)^\ab\. \] Both groups are finite. This paper corrects and generalizes my paper "Relative K-Theory and Class Field Theory for arithmetic surfaces" (math.NT/0204330)

  • Relative K-Theory and Class Field Theory for arithmetic surfaces
    arXiv: Number Theory, 2002
    Co-Authors: Alexander Schmidt
    Abstract:

    In this paper we extend the unramified Class Field Theory for arithmetic surfaces of K. Kato and S. Saito to the relative case. Let X be a regular proper arithmetic surface and let Y be the support of divisor on X. Let CH_0(X,Y) denote the relative Chow group of zero cycles and let \tilde \pi_1^t(X,Y)^ {ab} denote the abelianized modified tame fundamental group of (X,Y) (which Classifies finite etale abelian covings of X-Y which are tamely ramified along Y and in which every real point splits completely). THEOREM: There exists a natural reciprocity isomorphism rec: CH_0(X,Y) --> \tilde \pi_1^t(X,Y)^{ab}. Both groups are finite.

Samir Siksek - One of the best experts on this subject based on the ideXlab platform.

  • Class Field Theory, Diophantine analysis and the asymptotic Fermat's Last Theorem
    Advances in Mathematics, 2020
    Co-Authors: Nuno Freitas, Alain Kraus, Samir Siksek
    Abstract:

    Abstract Recent results of Freitas, Kraus, Şengun and Siksek, give sufficient criteria for the asymptotic Fermat's Last Theorem to hold over a specific number Field. Those works in turn build on many deep theorems in arithmetic geometry. In this paper we combine the aforementioned results with techniques from Class Field Theory, the Theory of p-groups and p-extensions, Diophantine approximation and linear forms in logarithms, to establish the asymptotic Fermat's Last Theorem for many infinite families of number Fields, and for thousands of number Fields of small degree. For example, we prove the effective asymptotic Fermat's Last Theorem for the infinite family of Fields Q ( ζ 2 r ) + where r ≥ 2 .

  • Class Field Theory, Diophantine analysis and the asymptotic Fermat's Last Theorem
    arXiv: Number Theory, 2019
    Co-Authors: Nuno Freitas, Alain Kraus, Samir Siksek
    Abstract:

    Recent results of Freitas, Kraus, Sengun and Siksek, give sufficient criteria for the asymptotic Fermat's Last Theorem to hold over a specific number Field. Those works in turn build on many deep theorems in arithmetic geometry. In this paper we combine the aforementioned results with techniques from Class Field Theory, the Theory of p-groups and p-extensions, Diophantine approximation and linear forms in logarithms, to establish the asymptotic Fermat's Last Theorem for many infinite families of number Fields, and for thousands of number Fields of small degree. For example, we prove the effective asymptotic Fermat's Last Theorem for the infinite family of Fields $\mathbb{Q}(\zeta_{2^r})^+$.