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**Realizations of polylogarithms.**
*(English)*
Zbl 0877.11001

Lecture Notes in Mathematics. 1650. Berlin: Springer. xi, 343 p. (1997).

Beilinson’s conjecture on special values of \(L\)-functions generalize the classical analytic class number formula, and it relates integral motivic cohomology groups and Deligne cohomology groups of smooth projective schemes defined over \(\mathbb Q\). The weak version of Beilinson’s conjecture involves the existence of a certain subspace of the integral motivic cohomology group of a given scheme, and it requires the construction of explicit elements in motivic cohomology. In the cases where the approach of constructing explicit elements proved to be successful, the underlying geometric objects are special cases of pure or mixed Shimura varieties. It is expected in these cases that the collection of the images under the regulator of these explicit elements is interpolated by a single object called the polylogarithmic extension. This monograph attempts to provide a unified explanation for such constructions. It provides a sheaf-theoretic foundation of the theory of polylogarithms and generalizes the approaches of Beilinson, Deligne and Levin to the context of Shimura varieties.

This monograph consists of five parts. The first two parts are mostly expository describing the generic sheaf and the canonical construction on a comprehensive level. Part I discusses the construction and properties of the mixed structure on the completed group ring of the topological fundamental group of a scheme, which is smooth over a number field. Special emphasis is given to the universal property satisfied by this mixed structure. The construction can be performed in a relative setting when there is a morphism \(\pi: X \to Y\) of schemes satisfying a certain regularity condition. In this case the generic sheaf on \(X\) can be defined. In Part II, such a construction is specialized to the case, where the morphism \(\pi\) is the natural projection of a mixed Shimura variety to the underlying pure Shimura variety. The main tool used for this purpose is the identification of the generic sheaf, called the logarithmic sheaf, with the canonical construction of a certain pro-representation. The central part of this book is Part III, which generalizes the definition of polylogarithmic extensions in the context of mixed Shimura varieties and discusses characteristic features of such an extension such as the splitting principle, rigidity, and norm compatibility. The treatment of the special case of the morphism \(\pi: G_{m, \mathbb Q} \to \text{Spec} (\mathbb Q)\) is contained in Part IV, where both the Hodge and \(\ell\)-adic versions are described, recovering results of Beilinson and Deligne. Part V discusses the case of the universal elliptic curve \(\pi: \mathcal E \to M\) over a modular curve \(M\) defined over \(\mathbb Q\). It also discusses the base change to CM-points of \(M\).

{Authors remark: This monograph is based on the thesis with the same title reviewed in Zbl 0824.14018}

This monograph consists of five parts. The first two parts are mostly expository describing the generic sheaf and the canonical construction on a comprehensive level. Part I discusses the construction and properties of the mixed structure on the completed group ring of the topological fundamental group of a scheme, which is smooth over a number field. Special emphasis is given to the universal property satisfied by this mixed structure. The construction can be performed in a relative setting when there is a morphism \(\pi: X \to Y\) of schemes satisfying a certain regularity condition. In this case the generic sheaf on \(X\) can be defined. In Part II, such a construction is specialized to the case, where the morphism \(\pi\) is the natural projection of a mixed Shimura variety to the underlying pure Shimura variety. The main tool used for this purpose is the identification of the generic sheaf, called the logarithmic sheaf, with the canonical construction of a certain pro-representation. The central part of this book is Part III, which generalizes the definition of polylogarithmic extensions in the context of mixed Shimura varieties and discusses characteristic features of such an extension such as the splitting principle, rigidity, and norm compatibility. The treatment of the special case of the morphism \(\pi: G_{m, \mathbb Q} \to \text{Spec} (\mathbb Q)\) is contained in Part IV, where both the Hodge and \(\ell\)-adic versions are described, recovering results of Beilinson and Deligne. Part V discusses the case of the universal elliptic curve \(\pi: \mathcal E \to M\) over a modular curve \(M\) defined over \(\mathbb Q\). It also discusses the base change to CM-points of \(M\).

{Authors remark: This monograph is based on the thesis with the same title reviewed in Zbl 0824.14018}

Reviewer: Min Ho Lee (Cedar Falls)

### MSC:

11-02 | Research exposition (monographs, survey articles) pertaining to number theory |

11G18 | Arithmetic aspects of modular and Shimura varieties |

11F67 | Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols |

11G09 | Drinfel’d modules; higher-dimensional motives, etc. |

14G35 | Modular and Shimura varieties |

19F27 | Étale cohomology, higher regulators, zeta and \(L\)-functions (\(K\)-theoretic aspects) |

32G20 | Period matrices, variation of Hodge structure; degenerations |