The de Rham and the syntomic logarithm
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914413753139200 |
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| author | Flach, Matthias Krause, Achim Morin, Baptiste |
| author_facet | Flach, Matthias Krause, Achim Morin, Baptiste |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_21471 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The de Rham and the syntomic logarithm Flach, Matthias Krause, Achim Morin, Baptiste Number Theory 14F30 (Primary), 11G40 (Secondary) 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. |
| title | The de Rham and the syntomic logarithm |
| topic | Number Theory 14F30 (Primary), 11G40 (Secondary) |
| url | https://arxiv.org/abs/2603.21471 |