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Rigid syntomic cohomology and \(p\)-adic polylogarithms. (English) Zbl 1006.19002

Special values of \(p\)-adic \(L\)-functions of projective smooth varieties over a number field \(K\) are believed to be related with values of the syntomic regulator. Beilinson defined a simplicial scheme and constructed a motivic polylogarithm \(\Pi_M\) in a certain quotient \(H_M^{(n)}\) of a cohomology of the scheme such that if one fixes a nontrivial torsion element \(\omega\) in \(K^\times\), then there is a homomorphism \(i_\omega^*: H_M^{(n)}\to \prod_{j=1}^n K_{2j-1}({\mathcal O}_K)\otimes \mathbb Q\) which sends \(\Pi_M\) to the Beilinson elements \((C_{2j-1}(\omega))_j\). M. Somekawa [\(K\)-Theory 17, No. 3, 256-294 (1999; Zbl 0978.19004)] constructed regulator maps from motivic cohomologies to his logarithmic syntomic cohomology. Using the latter he defined a \(p\)-adic version \(H_{\text{syn}}^{(n)}\) of the group \(H_M^{(n)}\) with map \(\text{reg}: H_M^{(n)}\to H_{\text{syn}}^{(n)}\), and explicitly calculated the \(p\)-adic polylogarithm \([\eta]=\text{reg}(\Pi_M)\). This gave an alternative proof of a result of M. Gros [Invent. Math. 115, No. 1, 61-79 (1994; Zbl 0799.14010)] on the image of the Beilinson element with respect to the syntomic regulator.
This paper develops elements of \(p\)-adic mixed sheaves theory which in particular provides a sheaf theoretic interpretation of the work of Somekawa. Let \(X\) be a smooth scheme over \({\mathcal O}_K\) of finite type, let \(\overline{X}\) be a smooth projective compactification of \(X\) over \({\mathcal O}_K\) such that \(\overline{X}\setminus X\) is a relative strict normal crossing divisor relative to \({\mathcal O}_K\), let \(\overline{\mathbb{X}}\) be the formal completion of \(\overline{X}\) with respect to the special fibre. Suppose that the absolute Frobenius map of the special fibre can be lifted to \(\phi: \overline{\mathbb{X}}\to \overline{\mathbb{X}}\). The triple \({\mathbb X}=(X,\overline{X},\phi)\) is called a syntomic datum over \({\mathcal O}_K\). After reviewing results from rigid cohomology the author defines in the first section a category of syntomic coefficients \(S(\mathbb X)\) and its full subcategory of admissible syntomic coefficients; these are \(p\)-adic analogs of variation of mixed Hodge structures.
In the second section an absolute syntomic cohomology \(H^i_{\text{syn}}({\mathbb X},M)\) for an admissible object \(M\) is defined; in particular, \(H^i_{\text{syn}}({\mathbb X}, K(j))\) is isomorphic to the rigid syntomic cohomology defined by A. Besser [Isr. J. Math. 120, Pt. B, 291-334 (2000; Zbl 1001.19003)]. Theorem 1 in section 4 proves that if \(K(0)\) is admissible then there is a natural isomorphism between \(\text{Ext}_{S(\mathbb X)}^1(K(0), M)\) and \(H_{\text{syn}}^1({\mathbb X},M)\), this result is in analogy to the classical result on the first absolute Hodge cohomology as extension classes.
In the fifth section the author defines a logarithmic sheaf \(\text{Log}=(\text{Log}^{(n)})\) as an interated extension of Tate objects in \(S(\mathbb G_m)\). The splitting property is proved: if \(i_\omega: \text{Spec} {\mathcal O}_K\to \text{Spec} {\mathcal O}_K[t,t^{-1}]\), \(t\mapsto \omega\), then \(i_\omega^* \text{Log}^{(n)}=\prod_{0\leq j\leq n} K(j)\) in \(S({\mathcal O}_K)\).
In the last section, following the method of A. Huber and J. Wildeshaus [Doc. Math., J. DMV 3, 27-133, 297-299 (1998; Zbl 0906.19004)] the author constructs a polylogarithmic element \(\text{pol}=( \text{pol}^{(n)})\in H^1_{\text{syn}}({\mathbb G_m}\setminus\{1\},\text{Log})\). Its explicit description as a coherent module with connection, Hodge filtration and Frobenius is given in Theorem 2. It implies that the specialization of the polylogarithm to roots of unity can be expressed by the \(p\)-adic polylogarithm function, which is compatible with the previous results of Gros-Kurihara and Somekawa. The author anticipates that there is a canonical isomorphism between \(H_{\text{syn}}^1({\mathbb G_m}\setminus\{1\}\), \(\text{Log}^{(n)})\) and \(H_{\text{syn}}^{(n)}\) which sends \(\text{pol}^{(n)}\) to Somekawa’s \([\eta]\).

MSC:

19F27 Étale cohomology, higher regulators, zeta and \(L\)-functions (\(K\)-theoretic aspects)
14F43 Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies)
11S40 Zeta functions and \(L\)-functions
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