[Paper Review] Syntomic cohomology and Beilinson's Tate conjecture for $K_2$

Masanori Asakura, Kanetomo Sato|arXiv (Cornell University)|Apr 23, 2009

Algebraic Geometry and Number Theory19 references3 citations

TL;DR

This paper establishes a syntomic cohomology bound on the étale cohomology of $K_2$ for elliptic surfaces over $p$-adic fields, proving that the image of syntomic cohomology maps into a space of formal Eisenstein series. This yields a new upper bound for the rank of the Galois-invariant étale cohomology, enabling verification of Beilinson’s Tate conjecture for $K_2$ in specific cases and proving finiteness of torsion 0-cycles on certain $K3$ surfaces over $\mathbb{Q}_p$. The key innovation lies in $p$-adic Hodge-theoretic computations on Tate curves and the use of Fontaine-Messing theory to relate syntomic and étale cohomology.

ABSTRACT

In this paper, we study an analogue of the Tate conjecture for $K_2$ of U, the complement of split multiplicative fibers in an elliptic surface. A main result is to give an upper bound of the rank of the Galois fixed part of the etale cohomology $H^2(\bar{U},Q_p(2))$. As an application, we give an elliptic K3 surface $X$ over a p-adic field for which the torsion part of the Chow group $CH_0(X)$ of 0-cycles is finite. This would be the first example of a surface $X$ over a p-adic field whose geometric genus is non-zero and for which the torsion part of $CH_0(X)$ is finite.

Motivation & Objective

• To establish a new upper bound for the rank of $H^2_{\text{ét}}(\overline{U}, \mathbb{Q}_p(2))^{G_K}$ using syntomic cohomology and formal Eisenstein series.
• To verify Beilinson’s Tate conjecture for $K_2$ in specific cases by constructing enough $K_2$-symbols to saturate the upper bound.
• To prove the finiteness of the torsion subgroup of the Chow group $\mathrm{CH}_0(X)_{\text{tors}}$ for a $K3$ surface $X$ over $\mathbb{Q}_p$ with good reduction.
• To extend Beilinson’s theorem on Eisenstein symbols from modular curves to arbitrary elliptic surfaces via $p$-adic Hodge theory.

Proposed method

• Use syntomic cohomology with log poles along split multiplicative fibers to define a map from $H^2_{\text{syn}}(\mathscr{X}(\mathscr{D}), \mathscr{S}_{\mathbb{Z}_p}(2))$ to global log 2-forms on $\mathscr{X}$.
• Define the space of formal Eisenstein series $\mathrm{E}(\mathscr{X}, \mathscr{D})_{\mathbb{Z}_p}$ as a subspace of $\Gamma(\mathscr{X}, \Omega^2_{\mathscr{X}/R}(\log \mathscr{D}))$.
• Prove that the image of the syntomic cohomology map lies inside $\mathrm{E}(\mathscr{X}, \mathscr{D})_{\mathbb{Z}_p}$ via detailed $p$-adic Hodge-theoretic computations on Tate curves.
• Leverage the Fontaine-Messing isomorphism to relate syntomic cohomology to étale cohomology, yielding the inequality $\dim_{\mathbb{Q}_p} c_{\text{ét}}(K_2(U)\otimes \mathbb{Q}_p) \leq \mathrm{rank}_{\mathbb{Z}_p} \mathrm{E}(\mathscr{X}, \mathscr{D})_{\mathbb{Z}_p}$.
• Use the surjectivity of the regulator map $K_1(X)^{(2)} \otimes \mathbb{Q}_\ell \to \mathrm{NS}(Y)_{\mathbb{Q}_\ell}$ and divisibility of $H^3(X, \mathbb{Q}_\ell/\mathbb{Z}_\ell(2))$ to prove finiteness of $\mathrm{CH}_0(X)_{\text{tors}}$.
• Apply the Hochschild-Serre spectral sequence and Deligne’s theorem on independence of Frobenius traces to show divisibility of Galois-invariant cohomology groups.

Experimental results

Research questions

• RQ1 Can syntomic cohomology provide a new upper bound for the rank of $H^2_{\text{ét}}(\overline{U}, \mathbb{Q}_p(2))^{G_K}$ in the context of elliptic surfaces over $\mathbb{Q}_p$?
• RQ2 Does the image of the syntomic cohomology map into the space of formal Eisenstein series $\mathrm{E}(\mathscr{X}, \mathscr{D})_{\mathbb{Z}_p}$, extending Beilinson’s theorem beyond modular curves?
• RQ3 Under what conditions is the étale Chern class map $c_{\text{ét}}: K_2(U) \otimes \mathbb{Q}_p \to H^2_{\text{ét}}(\overline{U}, \mathbb{Q}_p(2))^{G_K}$ surjective, verifying Beilinson’s Tate conjecture for $K_2$?
• RQ4 Is the torsion subgroup $\mathrm{CH}_0(X)_{\text{tors}}$ finite for a $K3$ surface $X$ defined over $\mathbb{Q}_p$ with good reduction and split multiplicative fibers?
• RQ5 Can the finiteness of $\mathrm{CH}_0(X)_{\text{tors}}$ be deduced from the surjectivity of the regulator map and divisibility of cohomology groups?

Key findings

• The image of the syntomic cohomology map $H^2_{\text{syn}}(\mathscr{X}(\mathscr{D}), \mathscr{S}_{\mathbb{Z}_p}(2)) \to \Gamma(\mathscr{X}, \Omega^2_{\mathscr{X}/R}(\log \mathscr{D}))$ is contained in the space of formal Eisenstein series $\mathrm{E}(\mathscr{X}, \mathscr{D})_{\mathbb{Z}_p}$, establishing a key geometric constraint.
• The inequality $\dim_{\mathbb{Q}_p} c_{\text{ét}}(K_2(U)\otimes \mathbb{Q}_p) \leq \mathrm{rank}_{\mathbb{Z}_p} \mathrm{E}(\mathscr{X}, \mathscr{D})_{\mathbb{Z}_p}$ provides a new upper bound for the rank of the Galois-invariant étale cohomology, strictly less than the number of split multiplicative fibers in some cases.
• Beilinson’s Tate conjecture for $K_2$ is verified in examples where the dimension of $c_{\text{ét}}(K_2(U)\otimes \mathbb{Q}_p)$ matches the upper bound, implying surjectivity of the étale Chern class map.
• The torsion subgroup $\mathrm{CH}_0(X)_{\text{tors}}$ is finite for a $K3$ surface $X$ over $\mathbb{Q}_p$ with good reduction and split multiplicative fibers, extending known finiteness results to $p$-adic fields.
• The regulator map $K_1(X)^{(2)} \otimes \mathbb{Q}_\ell \to \mathrm{NS}(Y)_{\mathbb{Q}_\ell}$ is surjective, and $H^3(X, \mathbb{Q}_\ell/\mathbb{Z}_\ell(2))$ is divisible for almost all $\ell$, which together imply the finiteness of $\mathrm{CH}_0(X)_{\text{tors}}$.
• The proof of finiteness relies on the independence of Frobenius traces (Deligne’s theorem) and the divisibility of Galois-invariant cohomology, showing that $H^1(\mathbb{Q}_p, H^2(\overline{X}, \mathbb{Q}_\ell/\mathbb{Z}_\ell(2))) \simeq N_\ell \otimes \mathbb{Q}_\ell/\mathbb{Z}_\ell$ is divisible for almost all $\ell$.