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Classical Motivic Polylogarithm According to Beilinson and Deligne

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Classical Motivic Polylogarithm According to Beilinson and Deligne Doc Math J DMV Classical Motivic Polylogarithm According to Beilinson and Deligne Annette Huber Jorg Wildeshaus Received February Revised September Communicated by Peter Schneider Abstract The main purp ose of this pap er is the construction in motivic cohomology of the cyclotomic or classical polylogarithm on the pro jective line minus three p oints and the identication of its image under the regula tor to absolute Deligne or l adic cohomology By sp ecialization to ro ots of unity one obtains a compatibility statement on cyclotomic elements in mo tivic and absolute cohomology of ab elian number elds As shown in BlK this compatibility completes the pro of of the Tamagawa number conjecture on sp ecial values of the Riemann zeta function The main constructions and ideas are contained in Beilinsons and Delignes unpublished preprint Motivic Polylogarithm and Zagier Conjecture BD We work out the details of the pro of setting up the foundational material which was missing from the original source the pap er contains an app endix on absolute Ho dge cohomology with co ecients and its inter pretation in terms of Saitos Ho dge mo dules The second app endix treats K theory and regulators for simplicial schemes Mathematics Sub ject Classication Primary F Secondary R R R D F Keywords Polylogarithm motivic and absolute cohomology regulators cy clotomic elements Introduction The aim of this work is to present the construction of the class of the cyclotomic or classical p olylogarithm in motivic cohomology It maps to the elements in Deligne and l adic cohomology dened and studied in Beilinsons Polylogarithm and cyclotomic elements B The latter elements can b e seen as b eing represented by a pro variation of Ho dge structure or a prol adic sheaf on the pro jective line minus three p oints 1 See correction on page of this volume Documenta Mathematica Annette Huber Jorg Wildeshaus Our main interest lies in the sp ecialization of these sheaves to ro ots of unity they represent the cyclotomic oneextensions of Tate twists already studied by Soule Sou Deligne D and Beilinson B Let us b e more precise denote by the set of primitive dth ro ots of unity in d Q Q T T d We get an alternative pro of of the following theorem of d d Beilinsons Corollary Assume n and denote by r the regulator map D M n C i R H Sp ec Q Q n d M Q C d There is a map of sets H Sp ec Q Q n n d d M such that M n C i R r D n d Q C d P k maps a ro ot of unity to Li n n k k Now x a dth primitive ro ot of unity in Q This choice allows to identify Sp ec Q Q n with a Q subspace of continuous etale cohomology H d l l cont 1 Gal Q Q l r n l A 1 1 Q Q Q lim r Z l l l l l r Note that there is a distinguished ro ot of unity T in Q As was observed already d in B the study of the cyclotomic p olylogarithm gives a pro of of BlK Conjecture cf Sou Theoreme for the case n Gr Theoreme IV for the lo cal version if l d Corollary Let b e the map constructed in Under the ab ove n inclusion the l adic regulator r H Sp ec Q Q n H Sp ec Q Q n l d d l M cont b maps T to n X d n A n d n r l b r This result implies in particular that Soules cyclotomic elements in the group K F Z for an ab elian number eld F and a prime l are induced by el n Z l ements in K theory itself Corollary Furthermore the case d of forms a central ingredient of the pro of of the Tamagawa number conjecture mo dulo p owers of for o dd Tate twists Q n n BlK x Finally as shown in KNF Theorem the general case of implies the mo died version of the Lichtenbaum conjecture for ab elian number elds Documenta Mathematica Classical Motivic Polylogarithm The main ideas necessary for b oth the construction of the motivic p olylogarithm and the identication of the realization classes together with a sketch of pro of are contained in the unpublished preprint Motivic Polylogarithm and Zagier Conjecture BD and its predecessors B BDp Our aim in this pap er is to work out the details of the pro ofs To do this we have to set up a lot of foundational material which was missing from the original sources K theory of simplicial schemes regulators to absolute Ho dge and l adic cohomology of simplicial schemes and an interpretation of the latter as Ext groups of Ho dge mo dules and l adic sheaves resp ectively This material is contained in the two app endices which we regard as our main contribution to the sub ject We hop e they prove to b e useful in other contexts than that treated in the main text Other parts of BD deal with the weak version of the Zagier conjecture We do not treat this since a complete pro of has b een given by de Jeu Jeu although by somewhat dierent means from those used in BD We see two main groups of pap ers related to p olylogarithms The rst deals with mixed sheaves ie variations of Ho dge structure or l adic mixed lisse sheaves Maybe the nucleus of these pap ers is Delignes observation that the analytic and top ological prop erties of the dilogarithm Li viewed as a multivalued holomorphic function on P C nf g can b e co ded by saying that Li is an entry of the p erio d matrix of a certain rank three variation of Q TateHodge structure on P n f g C We refer to Rm section for a nice survey of the construction of a pro variation on P n f g containing all Li The etale analogue is constructed in k C Beilinsons Polylogarithm and Cyclotomic elements B where he dened pro ob jects in the categories of l adic sheaves on P nf g In b oth settings the bres at ro ots of unity dierent from coincide with the cyclotomic extensions mentioned ab ove The hop e and indeed the motivation underlying these pap ers is that once a satisfactory formalism of motivic sheaves is developed the denition of p olylogarithms should basically carry over We would thus obtain p olylogarithmic classes in Ext groups of motives these groups b eing supp osedly closely connected to K theory of which everything already dened on the level of realizations would turn out to b e the resp ective regulator Nowhere is this hop e do cumented more manifestly than in Beilinsons and Delignes Interpretation motivique de la conjecture de Zagier reliant p olylogarithmes et regulateurs BD if there is such a motivic formalism then the weak version of Zagiers conjecture necessarily holds not only the values at ro ots of unity of higher logarithms but also appropriate linear combinations of arbitrary values must lie in the image of the regulator For the time b eing and in each case separately honest work is needed to p erform the K theoretic constructions and calculate their images under the regulators The second class of pap ers is concerned with precisely that task In analogy with the ab ove one should rst mention Blo chs Application of the dilogarithm function in algebraic K theory and algebraic geometry Bl Beilinsons Higher regulators and values of Lfunctions B provided the K theoretic construction of cyclotomic elements together with the computation of their images in Deligne cohomology lo c cit Theorem Neu E Documenta Mathematica Annette Huber Jorg Wildeshaus As for Zagiers conjecture we mention Goncharovs Polylogarithms and Motivic Galois Groups Go where Zagiers conjecture including the surjectivity statement is proved for K of a number eld and de Jeus Zagiers Conjecture and Wedge Complexes in Algebraic K theory Jeu which contains the pro of of the weak version of Zagiers conjecture indep endently of motivic considerations for K of n a number eld and arbitrary n Typically the ob jects of interest in this class of pap ers are complexes co cy cles and symbols ie ob jects which do not constantly aord a geometric or sheaf theoretic interpretation It is by no means easy to see say how a concrete element in some Deligne cohomology group can b e interpreted as an extension of variations of RHo dge structure These and similar diculties present themselves to the reader willing to translate from one class to the other The authors like to think of the present article as an attempt to bridge the gap b etween the two disciplines In a sense the coarse structure of the article follows the ab ove scheme sections are entirely sheaftheoretic Anything we say there is therefore a priori restricted to the level of realizations ie nonmotivic In sections K theory enters The app endices provide the foundations necessary to connect the two p oints of view Given that quite a lot has b een said ab out the l adic and Ho dge theoretic incar nations of the classical p olylogarithm B BD WiIV the reader may wonder why sheaf theoretic considerations still take up one third of this work Indeed the construction of the motivic p olylog could b e achieved much more eas ily if a satisfactory formalism of mixed motivic sheaves were available The necessity to replace a simple geometric situation by a rather complicated one in order to replace complicated co ecients like Log by Tate twists should b e seen as the main source of diculty in any attempt to the construction of motivic versions of p olylogarithms We now turn to the description of the ner structure of the main text sections In section we normalize the sheaf theoretic notations used throughout the whole article Section gives a quick axiomatic description of the logarithmic sheaf Log and the small p olylogarithmic extension pol The universal prop erty is needed only to connect the general denition of the logarithmic sheaf as a solution of a representability problem to the
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