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Milnor K-theory, F-isocrystals and Syntomic Regulators

Masanori Asakura, Kazuaki Miyatani

Abstract

We introduce a category of filtered F-isocrystals and construct a symbol maps on Milnor K-theory which is compatible with the syntomic symbol maps to the log syntomic cohomology. These are fundamental materials in our applications on syntomic regulators which we work in other papers.

Milnor K-theory, F-isocrystals and Syntomic Regulators

Abstract

We introduce a category of filtered F-isocrystals and construct a symbol maps on Milnor K-theory which is compatible with the syntomic symbol maps to the log syntomic cohomology. These are fundamental materials in our applications on syntomic regulators which we work in other papers.

Paper Structure

This paper contains 19 sections, 32 theorems, 269 equations.

Table of Contents

  1. Introduction
  2. Filtered $F$-isocrystals and Milnor $K$-theory
  3. The category of Filtered $F$-isocrystals
  4. The complex ${\mathscr S}(M)$
  5. Log objects
  6. Polylog objects
  7. Relative cohomologies.
  8. Extensions associated to Milnor symbols
  9. Explicit formula
  10. Comparison with the Syntomic Symbol maps
  11. Log syntomic cohomology
  12. Syntomic symbol maps
  13. Comparison of Symbol maps for $U/S$ with Syntomic symbol maps
  14. $p$-adic regulators of $K_2$ of curves
  15. Symbol map on $K_2$ of a projective smooth family of curves
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let $p\geq 5$ be a prime. Let $W=W(\overline{{\mathbb F}}_p)$ be the Witt ring, and $K:={\mathrm{Frac}}(W)$. Let $a\in W$ satisfy $a\not\equiv 0,1$ mod $p$. Let $E_a$ be the elliptic curve over $W$ defined by a Weierstrass equation $y^2=x^3+(3x+4-4a)^2$. Let where we note that the divisors $\mathrm{div}(h_i)$ have supports in $3$-torsion points. Then there are overconvergent functions $\varepsilo

Theorems & Definitions (51)

  • Theorem 1.1: Corollary \ref{['ell-cor']}
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Lemma 2.5
  • proof
  • Definition 2.6
  • Lemma 2.7
  • proof
  • ...and 41 more