The p-part of higher-dimensional local class field theory

Establish the p-primary reciprocity isomorphism for smooth projective schemes of dimension greater than two over local fields, thereby resolving the remaining p-coefficient cases in higher-dimensional local class field theory.

Background

The paper studies reciprocity maps from higher Chow groups to the abelianized étale fundamental group for smooth projective schemes over local fields. It proves the prime-to-p part of the reciprocity theorem and reviews the known p-primary result in dimension two.

For schemes of dimension greater than two, the p-primary component remains unresolved. This concerns understanding whether the p-primary quotient of CH{d+1}(X_K,1) maps isomorphically to the corresponding p-primary quotient of the abelianized étale fundamental group of the generic fiber.

References

This study has been continued in the 21st century in particular by Forr´e, Jannsen, Kerz, Saito, Schmidt and Wiesend and, at least in the smooth projective case, the only remaining open problem is to understand the p-part for schemes of dimension > 2 over local fields (see for example [4, 8, 15, 30, 33]).

— Higher dimensional local class field theory  (2609.34437 - Lüders, 28 Sep 2026) in Section 1, Introduction