Chow group of 0-cycles with modulus and higher-dimensional class field theory

Abstract

One of the main results of this paper is a proof of the rank one case of an existence conjecture on lisse Q 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 zero cycles with moduli.A key ingredient is the construction of a cycle theoretic avatar of refined Artin conductor in ramification theory originally studied by Kazuya Kato.where D runs through all effective Cartier divisors on X with |D| ⊂ X \ U and endow it with the inverse limit topology where C(X, D) is endowed with the discrete

Cube invariance of higher Chow groups with modulus

“…This result is closely related to [3,Theorem 2.7], which states that if X is proper over k, then the "unipotent part" (=non-homotopcally invariant part) of CH 0 (X ) = CH dim X (X , 0) is p-primary torsion when char(k) = p > 0 and that it is divisible when char(k) = 0. The group CH 0 (X ) = CH 0 (X|D) is called the Chow group of zero cycles with modulus, and it is used by M. Kerz and S. Saito in [11] to give a cycle-theoretic description of theétale fundamental group of the interior X \ |D|.…”

Ramification theory and formal orbifolds in arbitrary dimension

“…Note that this is an increasing filtration with f il log 0 W s (K) = W s (O K ). A modification of this filtration by Matsuda was used in [KS2]. Let V : W s (K) → W s+1 (K) be the function sends (a s−1 , .…”

“…One way to approach this problem is to realize that π ab 1 (X ) is a quotient of π ab 1 (X, D) for D = Sw(X ). So by Kerz-Saito's class field theory chow groups of zero cycles on X should be an appropriate quotient of chow groups of X with modulus as defined in [KS2]. And one could try to provide a geometric or hopefully a motivic interpretation of the this quotient group directly in terms of X .…”

Deformation Theory of the Chow Group of Zero Cycles

“…In order to prove Proposition 5.1, we need to prove key Lemma 5.4. Its proof is inspired by the techniques of[17] from which we cite the following lemma:…”