TAILIEUCHUNG - Đề tài " Deligne’s integrality theorem in unequal characteristic and rational points over finite fields "

A Pierre Deligne, a l ’occasion de son 60-i`me anniversaire, ` e en t´moignage de profonde admiration e Abstract If V is a smooth projective variety defined over a local field K with fi¯ nite residue field, so that its ´tale cohomology over the algebraic closure K is e supported in codimension 1, then the mod p reduction of a projective regular model carries a rational point. As a consequence, if the Chow group of 0-cycles of V over a large algebraically closed field is trivial, then the mod p reduction of a projective regular model carries a rational. | Annals of Mathematics Deligne s integrality theorem in unequal characteristic and rational points over finite fields By H el ene Esnault Annals of Mathematics 164 2006 715 730 Deligne s integrality theorem in unequal characteristic and rational points over finite fields By Helene Esnault A Pierre Deligne a l occasion de son 60-ieme anniversaire en témoignage de profonde admiration Abstract If V is a smooth projective variety defined over a local field K with finite residue field so that its etale cohomology over the algebraic closure K is supported in codimension 1 then the mod p reduction of a projective regular model carries a rational point. As a consequence if the Chow group of 0-cycles of V over a large algebraically closed field is trivial then the mod p reduction of a projective regular model carries a rational point. 1. Introduction If Y is a smooth geometrically irreducible projective variety over a finite field k we singled out in 10 a motivic condition forcing the existence of a rational point. Indeed if the Chow group of 0-cycles of Y fulfills base change CH0 Y Xk k Y Z Q Q then the number of rational points of Y is congruent to 1 modulo k . In general it is hard to control the Chow group of 0-cycles but if Y is rationally connected for example if Y is a Fano variety then the base change condition is fulfilled and thus rationally connected varieties over a finite field have a rational point. Recall the Leitfaden of the proof. By S. Bloch s decomposition of the diagonal acting on cohomology as a correspondence 2 Appendix to Lecture 1 the base change condition implies that etale cohomology Hm Y Q is supported in codimension 1 for all m 1 that is that etale cohomology for m 1 lives in coniveau 1. Here t is a prime number not dividing k . On the other hand by Deligne s integrality theorem 6 Cor. the coniveau condition implies that the eigenvalues of the geometric Frobenius acting on Hm Y O are divisible by k as algebraic Partially supported by the .

• Thạc sĩ - Tiến sĩ - Cao học
• finite fields
• subjects of mathematics
• science reports
• studied mathematics
• natural sciences
• scientific research
• Finite Fields in Vector Spaces
• Vector Finite Fields
• The Elliptic Cryptography
• Finite Vector Spaces
• Different Finite Fields
• Applications to Cryptography
• Efficient Software Implementation
• The Finite Fields
• The Basis Representation
• Addition and Subtraction
• Elliptic Curve Cryptography
• Network security
• Lecture notes on Computer and network security
• Arithmetic theoretical
• Modern cryptography
• Normal polynomial
• Self reciprocal
• Computationally simple
• Explicit construction
• Electromagnetic shieldings
• Magetic fields
• Underground power cables
• Finite element approach
• Magnetic vector potential formulations
• Data security
• Lecture Data security and encryption
• Mã hóa dữ liệu
• Bảo mật dữ liệu
• Number theory
• Cryptographic hash functions
• Fields theoretical underpinnings
• Eddy current
• Mangetic fields
• Finite element method
• Subproblem method
• Magnetic circuits
• Vietnam Journal of Mechanics
• Numerical simulation of the crack 2D
• The finite element incorporated the discontinuity
• The extended finite element method
• Dimensional asymptotic crack tip displacement fields
• tài liệu báo cáo nghiên cứu khoa học
• cách trình bày báo cáo
• kiến thức toán học
• báo cáo toán học
• công trình toán học
• báo cáo khoa học
• báo cáo hóa học
• công trình nghiên cứu về hóa học
• tài liệu về hóa học
• Heron Problem
• Sum Matrices
• Some Inequalities
• Series Approach
• Diameter Distance
• Riemann Maps
• Bertini theorems
• Annals of Mathematics
• Public Key Cryptography
• Diffie Helman
• Encryption techniques
• Caesar cipher
• Monoalphabetic ciphers
• Block ciphers
• Data encryption standard
• Feistel cipher structure
• Computer security
• Polynomial arithmetic theoretical underpinnings
• Theoretical underpinnings
• A course in number theory
• Elementary number theory
• Quadratic residues
• The Euclidean algorithm
• Enciphering matrices
• Data security and encryption
• Symmetric ciphers
• Asymmetric ciphers
• Cryptographic data integrity algorithms
• Mutual trust