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PreprintCC BY 4.0HAL

THE DE RHAM AND THE SYNTOMIC LOGARITHM

  • Matthias Flach
  • Achim Krause
  • Baptiste Morin
Open at sourcePDFCite

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 (Q_p(n)) of Fontaine and Perrin-Riou for all local fields K/Q_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.

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Cite

BibTeXRIS
BibTeX
@misc{flach2026rham,
  title = {{THE DE RHAM AND THE SYNTOMIC LOGARITHM}},
  author = {Matthias Flach and Achim Krause and Baptiste Morin},
  howpublished = {HAL},
  year = {2026},
  month = jul,
  eprint = {2603.21471},
  archivePrefix = {arXiv},
  url = {https://hal.science/hal-05803702v1},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - THE DE RHAM AND THE SYNTOMIC LOGARITHM
AU  - Matthias Flach
AU  - Achim Krause
AU  - Baptiste Morin
PY  - 2026
DA  - 2026/07/12
PB  - HAL
UR  - https://hal.science/hal-05803702v1
AB  - 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 (Q_p(n)) of Fontaine and Perrin-Riou for all local fields K/Q_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.
ER  -