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Syntomic formalism with coefficients

Fabrizio Andreatta, Massimo Bertolini, Marco Adamo Seveso, Rodolfo Venerucci

TL;DR

The paper develops a comprehensive syntomic formalism with coefficients to study p-adic étale Abel–Jacobi maps and explicit reciprocity laws. It constructs analytic, crystalline, convergent, and log-rigid syntomic complexes, and proves deep links to Hyodo–Kato and de Rham theories via comparison isomorphisms, traces, and period sheaves. It further introduces polynomially deformed syntomic complexes, Kunneth and cup-product structures, Gysin morphisms, and dualities, and establishes finite-dimensionality and compatibility results under semistable reduction and good reduction assumptions. These frameworks enable robust comparisons with étale cohomology and yield a versatile toolkit for semistable p-adic representations and reciprocity laws, with potential applications to GSp4 and related arithmetic questions.

Abstract

This paper provides the technical tools needed in ongoing work of the authors to compute p-adic étale Abel-Jacobi maps in order to obtain explicit reciprocity laws for GSp4. In particular, we define and study syntomic polynomial cohomology for filtered Frobenius log-isocrystals over proper and semistable schemes over the ring of integers of a local field, with smooth generic fiber, endowed with horizontal divisors. We introduce syntomic polynomial cohomology with support, we define Kunneth morphisms, trace maps and cup products, Gysin maps with respect to divisors and we study some of their properties. We establish the relation with Hyodo-Kato cohomology of the special fiber and de Rham cohomology of the generic fiber. We also introduce overconvergent variants with and without support by restricting to open smooth formal subschemes. Most of all, in case that the filtered log-isocrystal is associated to a p-adic local system on the generic fiber, we establish comparison morphisms between étale and syntomic cohomology and compatibilities with Hochschild-Serre morphisms and between Gysin morphisms.

Syntomic formalism with coefficients

TL;DR

The paper develops a comprehensive syntomic formalism with coefficients to study p-adic étale Abel–Jacobi maps and explicit reciprocity laws. It constructs analytic, crystalline, convergent, and log-rigid syntomic complexes, and proves deep links to Hyodo–Kato and de Rham theories via comparison isomorphisms, traces, and period sheaves. It further introduces polynomially deformed syntomic complexes, Kunneth and cup-product structures, Gysin morphisms, and dualities, and establishes finite-dimensionality and compatibility results under semistable reduction and good reduction assumptions. These frameworks enable robust comparisons with étale cohomology and yield a versatile toolkit for semistable p-adic representations and reciprocity laws, with potential applications to GSp4 and related arithmetic questions.

Abstract

This paper provides the technical tools needed in ongoing work of the authors to compute p-adic étale Abel-Jacobi maps in order to obtain explicit reciprocity laws for GSp4. In particular, we define and study syntomic polynomial cohomology for filtered Frobenius log-isocrystals over proper and semistable schemes over the ring of integers of a local field, with smooth generic fiber, endowed with horizontal divisors. We introduce syntomic polynomial cohomology with support, we define Kunneth morphisms, trace maps and cup products, Gysin maps with respect to divisors and we study some of their properties. We establish the relation with Hyodo-Kato cohomology of the special fiber and de Rham cohomology of the generic fiber. We also introduce overconvergent variants with and without support by restricting to open smooth formal subschemes. Most of all, in case that the filtered log-isocrystal is associated to a p-adic local system on the generic fiber, we establish comparison morphisms between étale and syntomic cohomology and compatibilities with Hochschild-Serre morphisms and between Gysin morphisms.

Paper Structure

This paper contains 42 sections, 46 theorems, 258 equations.

Table of Contents

  1. The geometric setup
  2. Notation
  3. Setting
  4. Frobenius log-isocrystals
  5. The syntomic complex
  6. Explicit complexes
  7. Chech covers
  8. De Rham complexes
  9. Crystalline complexes
  10. Convergent complexes
  11. Log-rigid complexes
  12. Relation with Hyodo-Kato cohomology I
  13. Syntomic complexes with support conditions
  14. Alternative descriptions
  15. Relation with Hyodo-Kato cohomology with support
  16. ...and 27 more sections

Key Result

Proposition 2.2

The Frobenius crystal $\bigl(\mathfrak{E},\Phi_{\mathfrak{E}}\bigr)$ defines a convergent Frobenius isocrystal $\bigl(\mathfrak{E}^{\rm conv},\Phi_{\mathfrak{E}^{\rm conv}}\bigr)$ on the log-convergent site of $\bigl(\mathcal{X}_k,\alpha_{\mathcal{X}_k}\bigr)$ for the Zariski topology, relative to t

Theorems & Definitions (111)

  • Remark 2.1
  • Proposition 2.2
  • proof
  • Definition 1
  • Definition 2
  • Proposition 3.1
  • proof
  • Remark 3.2
  • Remark 4.1
  • Lemma 4.2
  • ...and 101 more