Framed mixed Tate motives and the value of zeta-functions of number fields at 2 and 3 (Q2726222)

This treatise is the author's doctoral dissertation, which has been written under the scientific supervision by G. Harder at the University of Bonn, Germany, and vindicated in that very place a year ago.NEWLINENEWLINENEWLINEThe subject of study in this work is the zeta function of an arbitrary algebraic number field. More precisely, the author's main goal is to elaborate an algebro-geometric method that leads to an effective description of the values of the zeta function \(\zeta_k(s)\) at \(s=2\) and \(s=3\). The basic approach to this program relies on the philosophy that the study of certain varieties over the respective number field, together with their Hodge structures and their Galois representations, should transpire some realizable information about the values of the zeta function. In particular, the mixed motives arising from the various cohomology theories for a variety over a number field and, among them, the so-called mixed Tate motives \(\mathbb{Q}(-n)\) for positive integers \(n\) are suspected to be linked with the values of the zeta function of the respective number field. This is based upon several strong indications and conjectures, in this context, which are due to previous results by \textit{A. Beilinson} and \textit{P. Deligne} [in: ``Motives'', Proc. Summer Res. Conf. Motives, Univ. Washington 1991, Proc. Symp. Pure Math. 55, Pt. 2, 97-121 (1994; Zbl 0799.19004)], \textit{J.-M. Fontaine} and \textit{B. Perrin-Riou} [ibid. Pt. 1, 599-706 (1994; Zbl 0821.14013)], \textit{A. J. Scholl} [in: ``\(L\)-functions and arithmetic'', Proc. Symp., Durham 1989, Lond. Math. Soc. Lect. Note Ser. 153, 373-392 (1991; Zbl 0817.14007)], and others.NEWLINENEWLINENEWLINEMore precisely, it is conjectured that the motivic extension group \(\text{Ext}^1_{MM_k} (\mathbb{Q}(-n), \mathbb{Q}(0))\) can be realized as a lattice in some \(\mathbb{R}^m\), the volume of which \(\pmod{\mathbb{Q}^*}\) is related to the value \(\zeta_k(n)\). Then, due to an approach suggested by \textit{G. Harder} [``Modular construction of mixed motives. II'', (preprint, ftp://ftp.math.uni-bonn.de/people/harder/Eisenstein/MixMot.ps)], it looks promising to study extensions of the form \(0\to\mathbb{Z}(0)\to X\to\mathbb{Z}(-n)\to 0\) arising from the cohomologies of varieties, and to relate them to the value \(\zeta_k(n)\).NEWLINENEWLINENEWLINEIn the work under review, the author makes this idea more precise, in two special cases, and carries out the conjectural program for these two cases in a very subtle way.NEWLINENEWLINENEWLINEThe two special cases investigated in this thesis are the following ones:NEWLINENEWLINENEWLINE(i) \(n=2\) and \(k\) is an imaginary quadratic number field;NEWLINENEWLINENEWLINE(ii) \(n=3\) and \(k\) is the rational number field \(\mathbb{Q}\) itself. NEWLINENEWLINENEWLINEIn both cases, the extensions in \(\text{Ext}^1_{MHS_k} (\mathbb{Z}(-n), \mathbb{Z}(0))\), i.e., with respect to motivic mixed Hodge structures, are well-known, and the author tackles Harder's conjecture by trying to construct successive extensions of Tate motives via analyzing configurations of projective spaces (embedded in a larger projective space) and their resulting relative cohomology. This ambitious and difficult undertaking is conducted by introducing and studying the parametrized family of so-called polylogarithmic motives, within this particular set-up, which constitute the basic new toolkit developed here. NEWLINENEWLINENEWLINEThis treatise consists of six chapters.NEWLINENEWLINENEWLINEChapter 1 discusses the general theory of mixed Tate motives, including their various cohomological realizations, framings, and their conjectured relations to the values of the zeta function.NEWLINENEWLINENEWLINEChapter 2 deals with motives that arise from relative cohomology, together with their realizations and their mixed Hodge structures. NEWLINENEWLINENEWLINEChapter 3 is devoted to the study of the so-called Kummer motives, which arise from extensions of \(\mathbb{Z}(-1)\) by \(\mathbb{Z}(0)\) as framed mixed Tate motives. NEWLINENEWLINENEWLINEChapter 4 provides the construction of the afore-mentioned polylogarithmic motives in the special cases of \(n=2\) and \(n=3\), including the computation of their mixed Hodge structures.NEWLINENEWLINENEWLINEChapter 5 describes then explicitly those extensions of \(\mathbb{Z}(-2)\) and \(\mathbb{Z}(-3)\) by \(\mathbb{Z}(0)\) which are generated by polylogarithmic motives. Also, the author determines all those extensions of \(\mathbb{Z}(-2)\) by \(\mathbb{Z}(0)\) that can be constructed exclusively by Kummer motives, without using polylogarithmic motives. The explicit numerical results strengthen the conjecture on the relation between \(\zeta_k(2)\) and the extensions of \(\mathbb{Z}(-2)\) by \(\mathbb{Z}(0)\) according to G. Harder.NEWLINENEWLINENEWLINEChapter 6 deals with the more complicated case of \(n=3\), where the occurring difficulties are carefully analyzed. However, some extensions of \(\mathbb{Z}(-3)\) by \(\mathbb{Z}(0)\) can be constructed in the case of the rational number field, and those are shown to behave as conjectured.NEWLINENEWLINENEWLINEAlltogether, this work represents a very valuable contribution to the difficult problem of determining the values of the zeta function of a number field for positive integer arguments. The author's approach to carrying out G. Harder's proposed method to come to grips with the values \(\zeta_k(n)\) of the zeta function for small positive integers \(n\) is highly original, subtle, and very enlightening in view of further investigations along this path.