Syntomic cohomology and Beilinson's Tate conjecture for \(K_{2}\) (Q2841803)

The paper under review provides new geometric examples where one can prove deep conjectures about \(K\)-theory and regulators of arithmetic schemes. The main result (denoted (1) below) is the proof of new cases of Belinsons's generalization of the Tate conjecture to higher \(K\)-theory (or higher Chow groups). Recall that Jannsen found counter-examples to the original formulation [\textit{U. Jannsen}, Mixed motives and algebraic \(K\)-theory. (Almost unchanged version of the author's habilitation at Univ. Regensburg 1988). Berlin etc.: Springer-Verlag (1990; Zbl 0691.14001)].NEWLINENEWLINEThe geometric situation studied in the paper is the following. Let \(k\) be a base field, and consider a map \(\pi : X \to C\) over \(k\), from a smooth projective surface to a smooth projective curve, whose generic fiber is an elliptic curve. Let \(U\) be an open subscheme of \(X\) whose complement is a union of fibers of \(\pi\) (those of ``multiplicative typ'').NEWLINENEWLINEA motivating example is given by the universal family of elliptic curves \(U\) over a modular curve, and its canonical compactification \(X\). Regulators in this case were extensively studied by \textit{A. A. Beilinson} [Contemp. Math. 55, 1--34 (1986; Zbl 0609.14006)]. In several points the present paper can be seen as a generalization of Beilinson's work, although the techniques in the proofs are different (the use of \(p\)-adic Hodge theory here is essential).NEWLINENEWLINEThe authors construct new examples where the following can be proved :NEWLINENEWLINE(1) (\(k\) a number field) The étale Chern class map NEWLINE\[NEWLINEc_{\mathrm{\'et}} : K_2 (U ) \otimes \mathbb Q_p \to H^2_{\mathrm{\'et}} (U_{\overline{k}} , \mathbb Q_p (2))^{\mathrm{Gal}(\overline{k}/k)}NEWLINE\]NEWLINE is surjective (Section 6.1).NEWLINENEWLINE(2) (\(k\) a \(p\)-adic local field) The \(p\)-adic regulator NEWLINE\[NEWLINE \rho : K_1 (X)^{(2)} \otimes \mathbb Q_p \to H^!_g (k, H^1_{\mathrm{\'et}} (X_{\overline{k}} , \mathbb Q_p (2)))NEWLINE\]NEWLINE is surjective (Theorem 7.0.3).NEWLINENEWLINE(3) (\(k\) a \(p\)-adic local field) The torsion subgroup \(\mathrm{CH}^2 (X)_{\mathrm{tors}}\) of the Chow group \(\mathrm{CH}^2 (X)\) is finite (Section 7.3).NEWLINENEWLINEThe key ingredient is the introduction of the ``space of formal Eisenstein series'' (Section 5.2), NEWLINE\[NEWLINEE(\mathcal {X , D})_{\mathbb Z_p} \subset \Gamma(\mathcal{X }, \Omega^2_{\mathcal{X }/R} (\log \mathcal D)).NEWLINE\]NEWLINE (Here \(k\) is a \(p\)-adic local field, \(R\) its ring of integers, \(\mathcal X\) and \(\mathcal D\) are suitable models of \(X\) and \(X - U\) over \(R\).) This generalizes a construction due to Beilinson in the case of modular curves (see [loc. cit.]).NEWLINENEWLINEIn 5.3 it is shown that the \(\mathbb Z_p\)-rank of \(E(\mathcal {X , D})_{\mathbb Z_p}\) bounds the dimension of the Galois invariant classes appearing in (1). Hence, if one can construct enough algebraic classes (reaching the bound), then one can deduce the surjectivity in (1).