The brauer group of the product of two curves over a finite field

1985 ◽  
pp. 254-283
Author(s):  
Noriko Yui
Keyword(s):  
Finite Field ◽  
Brauer Group

COMPARING THE BRAUER GROUP TO THE TATE–SHAFAREVICH GROUP

2018 ◽  
Vol 19 (3) ◽  
pp. 965-970 ◽  
Author(s):  
Thomas H. Geisser
Keyword(s):  
Finite Field ◽  
Brauer Group ◽  
Generic Fiber

We give a formula relating the order of the Brauer group of a surface fibered over a curve over a finite field to the order of the Tate–Shafarevich group of the Jacobian of the generic fiber. The formula implies that the Brauer group of a smooth and proper surface over a finite field is a square if it is finite.

scholarly journals TATE’S CONJECTURE AND THE TATE–SHAFAREVICH GROUP OVER GLOBAL FUNCTION FIELDS

2019 ◽  
pp. 1-22 ◽  
Author(s):  
Thomas H. Geisser
Keyword(s):  
Finite Field ◽  
Function Fields ◽  
Brauer Group ◽  
Regular Variety ◽  
Regular Curve ◽  
Generic Fiber ◽  
Global Function ◽  
Relative Dimension ◽  
Global Function Fields ◽  
Albanese Variety

Let ${\mathcal{X}}$ be a regular variety, flat and proper over a complete regular curve over a finite field such that the generic fiber $X$ is smooth and geometrically connected. We prove that the Brauer group of ${\mathcal{X}}$ is finite if and only Tate’s conjecture for divisors on $X$ holds and the Tate–Shafarevich group of the Albanese variety of $X$ is finite, generalizing a theorem of Artin and Grothendieck for surfaces to arbitrary relative dimension. We also give a formula relating the orders of the group under the assumption that they are finite, generalizing the known formula for a surface.

scholarly journals Étale Steenrod operations and the Artin–Tate pairing

2020 ◽  
Vol 156 (7) ◽  
pp. 1476-1515
Author(s):  
Tony Feng
Keyword(s):  
Finite Field ◽  
Algebraic Topology ◽  
Affirmative Answer ◽  
Characteristic Classes ◽  
Brauer Group ◽  
Tate Pairing ◽  
Steenrod Operations ◽  
Etale Cohomology ◽  
Étale Cohomology ◽  
Topology Of Manifolds

We prove a 1966 conjecture of Tate concerning the Artin–Tate pairing on the Brauer group of a surface over a finite field, which is the analog of the Cassels–Tate pairing. Tate asked if this pairing is always alternating and we find an affirmative answer, which is somewhat surprising in view of the work of Poonen–Stoll on the Cassels–Tate pairing. Our method is based on studying a connection between the Artin–Tate pairing and (generalizations of) Steenrod operations in étale cohomology. Inspired by an analogy to the algebraic topology of manifolds, we develop tools allowing us to calculate the relevant étale Steenrod operations in terms of characteristic classes.

Beta dynamical systems over the formal fields

2014 ◽  
Vol 51 (4) ◽  
pp. 454-465
Author(s):  
Lu-Ming Shen ◽  
Huiping Jing
Keyword(s):  
Dynamical Systems ◽  
Finite Field ◽  
Haar Measure ◽  
Laurent Series ◽  
Formal Laurent Series ◽  
Almost All

Let \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathbb{F}_q ((X^{ - 1} ))$$ \end{document} denote the formal field of all formal Laurent series x = Σ n=ν∞anX−n in an indeterminate X, with coefficients an lying in a given finite field \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathbb{F}_q$$ \end{document}. For any \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\beta \in \mathbb{F}_q ((X^{ - 1} ))$$ \end{document} with deg β > 1, it is known that for almost all \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$x \in \mathbb{F}_q ((X^{ - 1} ))$$ \end{document} (with respect to the Haar measure), x is β-normal. In this paper, we show the inverse direction, i.e., for any x, for almost all \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\beta \in \mathbb{F}_q ((X^{ - 1} ))$$ \end{document}, x is β-normal.

Valued Field Graph and Some Related Parameters

2020 ◽  
pp. 389-395
Author(s):  
G. Suresh Singh ◽  
P. K. Prasobha
Keyword(s):  
Finite Field ◽  
Independence Number ◽  
Valued Field ◽  
Vertex Set ◽  
Adic Valuation

Let $K$ be any finite field. For any prime $p$, the $p$-adic valuation map is given by $\psi_{p}:K/\{0\} \to \R^+\bigcup\{0\}$ is given by $\psi_{p}(r) = n$ where $r = p^n \frac{a}{b}$, where $p,a,b$ are relatively prime. The field $K$ together with a valuation is called valued field. Also, any field $K$ has the trivial valuation determined by $\psi{(K)} = \{0,1\}$. Through out the paper K represents $\Z_q$. In this paper, we construct the graph corresponding to the valuation map called the valued field graph, denoted by $VFG_{p}(\Z_{q})$ whose vertex set is $\{v_0,v_1,v_2,\ldots, v_{q-1}\}$ where two vertices $v_i$ and $v_j$ are adjacent if $\psi_{p}(i) = j$ or $\psi_{p}(j) = i$. Here, we tried to characterize the valued field graph in $\Z_q$. Also we analyse various graph theoretical parameters such as diameter, independence number etc.

On the Security of Public-Key Algorithms Based on Chebyshev Polynomials over the Finite Field $Z_N$

2010 ◽  
Vol 59 (10) ◽  
pp. 1392-1401 ◽  
Author(s):  
Xiaofeng Liao ◽  
Fei Chen ◽  
Kwok-wo Wong
Keyword(s):  
Finite Field ◽  
Chebyshev Polynomials ◽  
Public Key

Image encryption based on the finite field cosine transform

2013 ◽  
Vol 28 (10) ◽  
pp. 1537-1547 ◽  
Author(s):  
J.B. Lima ◽  
E.A.O. Lima ◽  
F. Madeiro
Keyword(s):  
Finite Field ◽  
Image Encryption ◽  
Cosine Transform
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