The de Rham and the syntomic logarithm

TL;DR

This paper introduces the de Rham logarithm as an integral refinement of the inverse Bloch-Kato exponential map, utilizing syntomic logarithms and prismatic cohomology to prove conjectures and compute correction factors in number theory.

Contribution

It defines the de Rham logarithm and employs syntomic logarithms and prismatic cohomology to prove conjectures and compute correction factors in arithmetic geometry.

Findings

Proved Conjecture C_{EP}(Q_p(n)) for all local fields K/Q_p.

Established a version of the Beilinson fibre square for derived formal schemes.

Computed the correction factor C(X,n) related to the Bloch-Kato Tamagawa number conjecture.

Abstract

We define and study an integral refinement of the inverse of the Bloch-Kato exponential map which we call the de Rham logarithm. Our main tool to analyze the de Rham logarithm is the syntomic logarithm, a certain limit construction based on the theory of filtered prismatic cohomology initiated by Antieau, Krause and Nikolaus. We use the syntomic logarithm to prove a version of the Beilinson fibre square for all quasicompact, quasiseparated derived formal schemes. We also use our techniques to prove Conjecture $C_{EP}(\bq_p(n))$ of Fontaine and Perrin-Riou for all local fields $K/\bq_p$ and to compute the correction factor $C(X,n)$ introduced by Flach and Morin in their reformulation of the Bloch-Kato Tamagawa number conjecture for the Zeta function of a smooth projective scheme $X$ over a number ring.