Variations on a theme of rationality of cycles

2013 ◽  
Vol 11 (6) ◽  
Author(s):  
Nikita Karpenko
Keyword(s):  
Function Field ◽  
Algebraic Cycles ◽  
Base Field ◽  
Arbitrary Characteristic ◽  
Steenrod Operations ◽  
Cobordism Theory ◽  
Characteristic 2 ◽  
Algebraic Cobordism ◽  
Variations On A Theme

AbstractWe prove certain weak versions of some celebrated results due to Alexander Vishik comparing rationality of algebraic cycles over the function field of a quadric and over the base field. The original proofs use Vishik’s symmetric operations in the algebraic cobordism theory and work only in characteristic 0. Our proofs use the modulo 2 Steenrod operations in the Chow theory and work in any characteristic ≠ 2. Our weak versions are still sufficient for existing applications. In particular, Vishik’s construction of fields of u-invariant 2r + 1, for r ≥ 3, is extended to arbitrary characteristic ≠ 2.

scholarly journals Around rationality of cycles

2013 ◽  
Vol 11 (6) ◽  
Author(s):  
Raphaël Fino
Keyword(s):  
Function Field ◽  
Algebraic Cycles ◽  
Base Field ◽  
Steenrod Operations ◽  
Cobordism Theory ◽  
Characteristic 2 ◽  
Algebraic Cobordism

AbstractWe prove certain results comparing rationality of algebraic cycles over the function field of a quadric and over the base field. These results have already been obtained by Alexander Vishik in the case of characteristic 0, which allowed him to work with algebraic cobordism theory. Our proofs use the modulo 2 Steenrod operations in the Chow theory and work in any characteristic ≠ 2.

scholarly journals Algebraic Surfaces of General Type with Small c21 in Positive Characteristic

2008 ◽  
Vol 191 ◽  
pp. 111-134 ◽  
Author(s):  
Christian Liedtke
Keyword(s):  
Characteristic Zero ◽  
General Type ◽  
Positive Characteristic ◽  
Algebraic Surfaces ◽  
Surfaces Of General Type ◽  
Arbitrary Characteristic ◽  
Numerical Invariants ◽  
Characteristic 2

AbstractWe establish Noether’s inequality for surfaces of general type in positive characteristic. Then we extend Enriques’ and Horikawa’s classification of surfaces on the Noether line, the so-called Horikawa surfaces. We construct examples for all possible numerical invariants and in arbitrary characteristic, where we need foliations and deformation techniques to handle characteristic 2. Finally, we show that Horikawa surfaces lift to characteristic zero.

scholarly journals The autocorrelation of the Möbius function and Chowla's conjecture for the rational function field in characteristic 2

2015 ◽  
Vol 373 (2040) ◽  
pp. 20140311 ◽  
Author(s):  
Dan Carmon
Keyword(s):  
Finite Field ◽  
Rational Function ◽  
Function Field ◽  
Möbius Function ◽  
Rational Function Field ◽  
Mobius Function ◽  
Characteristic 2

We prove a function field version of Chowla's conjecture on the autocorrelation of the Möbius function in the limit of a large finite field of characteristic 2, extending previous work in odd characteristic.

scholarly journals DUALITY FOR COHOMOLOGY OF CURVES WITH COEFFICIENTS IN ABELIAN VARIETIES

2018 ◽  
Vol 240 ◽  
pp. 42-149 ◽  
Author(s):  
TAKASHI SUZUKI
Keyword(s):  
Function Field ◽  
Positive Characteristic ◽  
Abelian Varieties ◽  
Base Field ◽  
Tate Pairing ◽  
Global Function ◽  
Crystalline Cohomology ◽  
Local Duality ◽  
Height Pairing ◽  
Finite Base

In this paper, we formulate and prove a duality for cohomology of curves over perfect fields of positive characteristic with coefficients in Néron models of abelian varieties. This is a global function field version of the author’s previous work on local duality and Grothendieck’s duality conjecture. It generalizes the perfectness of the Cassels–Tate pairing in the finite base field case. The proof uses the local duality mentioned above, Artin–Milne’s global finite flat duality, the nondegeneracy of the height pairing and finiteness of crystalline cohomology. All these ingredients are organized under the formalism of the rational étale site developed earlier.

The Zeta Function of a Pair of Quadratic Forms

2001 ◽  
Vol 44 (2) ◽  
pp. 242-256
Author(s):  
Laura Mann Schueller
Keyword(s):  
Finite Field ◽  
Zeta Function ◽  
Function Field ◽  
Quadratic Forms ◽  
Arbitrary Characteristic ◽  
Number Of Zeros ◽  
Even Order ◽  
Hyperelliptic Function

AbstractThe zeta function of a nonsingular pair of quadratic forms defined over a finite field, k, of arbitrary characteristic is calculated. A. Weil made this computation when char k ≠ 2. When the pair has even order, a relationship between the number of zeros of the pair and the number of places of degree one in an appropriate hyperelliptic function field is established.

scholarly journals ESSENTIAL DIMENSION OF GENERIC SYMBOLS IN CHARACTERISTIC

2017 ◽  
Vol 5 ◽  
Author(s):  
KELLY MCKINNIE
Keyword(s):  
Lower Bound ◽  
Essential Dimension ◽  
Base Field ◽  
Witt Group ◽  
New Techniques ◽  
Characteristic 2 ◽  
Symbol Algebra ◽  
Bounded Below ◽  
Independent Parameters

In this article the $p$-essential dimension of generic symbols over fields of characteristic $p$ is studied. In particular, the $p$-essential dimension of the length $\ell$ generic $p$-symbol of degree $n+1$ is bounded below by $n+\ell$ when the base field is algebraically closed of characteristic $p$. The proof uses new techniques for working with residues in Milne–Kato $p$-cohomology and builds on work of Babic and Chernousov in the Witt group in characteristic 2. Two corollaries on $p$-symbol algebras (i.e, degree 2 symbols) result from this work. The generic $p$-symbol algebra of length $\ell$ is shown to have $p$-essential dimension equal to $\ell +1$ as a $p$-torsion Brauer class. The second is a lower bound of $\ell +1$ on the $p$-essential dimension of the functor $\operatorname{Alg}_{p^{\ell },p}$. Roughly speaking this says that you will need at least $\ell +1$ independent parameters to be able to specify any given algebra of degree $p^{\ell }$ and exponent $p$ over a field of characteristic $p$ and improves on the previously established lower bound of 3.

COSTANDARD MODULES OVER SCHUR SUPERALGEBRAS IN CHARACTERISTIC p

2008 ◽  
Vol 07 (02) ◽  
pp. 147-166 ◽  
Author(s):  
ROBERTO LA SCALA ◽  
ALEXANDER ZUBKOV
Keyword(s):  
Null Space ◽  
General Linear ◽  
Base Field ◽  
Characteristic P ◽  
Schur Algebra ◽  
Arbitrary Characteristic ◽  
Linear Basis ◽  
General Linear Supergroup ◽  
Schur Superalgebra

In this paper we consider the problem of describing the costandard modules ∇(λ) of a Schur superalgebra S(m|n,r) over a base field K of arbitrary characteristic. Precisely, if G = GL(m|n) is a general linear supergroup and Dist (G) its distribution superalgebra we compute the images of the Kostant ℤ-form under the epimorphism Dist (G) → S(m|n,r). Then, we describe ∇(λ) as the null-space of some set of superderivations and we obtain an isomorphism ∇(λ) ≈ ∇(λ+|0) ⊗ ∇(0|λ-) assuming that λ = (λ+|λ-) and λm = 0. If char (K) = p we give a Frobenius isomorphism ∇(0|pμ) ≈ ∇(μ)p where ∇(μ) is a costandard module of the ordinary Schur algebra S(n,r). Finally we provide a characteristic free linear basis for ∇(λ|0) which is parametrized by a set of superstandard tableaux.

scholarly journals Pseudo-elliptic integrals, units, and torsion

2005 ◽  
Vol 79 (3) ◽  
pp. 335-347 ◽  
Author(s):  
Francesco Pappalardi ◽  
Alfred J. Van Der Poorten
Keyword(s):  
Function Field ◽  
Function Fields ◽  
Quadratic Function ◽  
Elliptic Integrals ◽  
Square Root ◽  
Base Field ◽  
Quadratic Function Field

AbstractWe remark on pseudo-elliptic integrals and on exceptional function fields, namely function fields defined over an infinite base field but nonetheless containing non-trivial units. Our emphasis is on some elementary criteria that must be satisfied by a squarefree polynomialD(x)whose square root generates a quadratic function field with non-trivial unit. We detail the genus I case.

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