Regulators, algebraic cycles, and values of L-functions.

*(English)*Zbl 0694.14002
Algebraic \(K\)-theory and algebraic number theory, Proc. Semin., Honolulu/Hawaii 1987, Contemp. Math. 83, 183-310 (1989).

[For the entire collection see Zbl 0655.00010.]

In this expository paper a general survey of conjectures and results on L-functions and related subjects such as regulators and algebraic cycles is given. These conjectures are the starting point for a general philosophy, set forth by A. Beilinson, that advocates the universality of algebraic K-theory as a good cohomology theory with “generalized cycle maps” (regulators) to any other suitable cohomology theory, e.g. Deligne-Beilinson cohomology. The K-groups ocurring in the formalism should be Yoneda extensions in a conjectured category of so-called mixed motives. These ideas are presented in the five introductory pages and they turn up every now and then in the main text consisting of nine sections, each followed by one or more interesting pages of “notes and comments” where some details and vistas on specific topics are revealed.

Starting from Pell’s equation one is naturally led to introduce the notion of fundamental units in real quadratic number fields, and more generally, one introduces the regulator of a general number field. Its importance becomes evident in the statement of Dedekind’s class number formula, which in this context, says that the first non-zero coefficient of the Taylor series expansion at \(s=0\) of the zeta-function \(\zeta_ F(s)\) of the number field F is (up to a non-zero rational multiple) equal to the regulator. In this form Borel generalized the class number formula to other values of the argument s of \(\zeta_ F(s)\). The right setting becomes, on the one hand, the algebraic K-groups of the ring of integers of F and, on the other hand, the (de Rham invariant part of) Deligne- Beilinson cohomology of the complex embeddings of F, the relation between both being provided by regulator maps (one for each K-group), whose volumes are called the regulators \(R_ m\). These \(R_ m\) give (up to a non-zero rational multiple) the values of the first non-zero coefficient of the Taylor series expansion of F at the point \(s=1-m\), \(m\geq 2\). - The next step is the introduction of Artin L-functions for representations of the (relative or absolute) Galois group of number fields. One also has Artin L-functions for arbitrary (smooth, projective) varieties X defined over number fields, where the \(\ell\)-adic cohomology of X gives a representation of the (absolute) Galois group of the number field. For number fields there is a conjecture due to Stark, Tate and Gross relating the values of the L-function at negative integers to the values of regulators, again coming from the K-groups of the ring of integers of the number field.

For a curve X defined over a number field F one may consider the associated complex curve (Riemann surface) and construct explicitly a regulator map from K-theory to Deligne-Beilinson cohomology using the construction of the Heisenberg line bundle on \({\mathbb{C}}^*\times {\mathbb{C}}^*\) with connection. This leads to a conjecture, due to Beilinson and Bloch, again relating the value of the first non-zero coefficient of the Taylor series of the L-function of X to the regulator. The K-group \(K_ 2(X)\) has to be replaced by \(K_ 2({\mathcal X})\) of a regular model \({\mathcal X}\) of X defined over the ring of integers of F. \({\mathcal X}\) may be compared to the Néron model of an elliptic curve (or, more generally, an abelian variety). Such \({\mathcal X}\) exists by results of Abhyankar.

In § 5 algebraic cycles of smooth projective varieties enter the scene. For a variety X over \({\mathbb{C}}\) the classical Hodge conjecture is reminded and for X defined over a finitely generated field Tate’s conjecture I (the \(\ell\)-adic counterpart of the Hodge conjecture) is formulated. Also, Tate’s conjecture II on the order of the pole of the L- function in terms of algebraic cycles is stated.

In the next section Beilinson’s general conjectures on the relation between K-groups and Deligne cohomology and on the values of L-functions at special values of the argument are formulated, thus extending the conjectures and examples of the foregoing sections. Compatibility with Deligne’s conjecture on critical points of motives over number fields is indicated and a list of cases where (part of) the conjectures are verified is included. Almost all known cases are of a modular nature.

Section 7 treats in some detail polylogarithms and their role in hyperbolic geometry, the description of higher cyclotomic regulators and the theory of variations of mixed Hodge structures on the projective line minus three points. Also the interpretation of the (odd) higher K-groups of a number field F as extensions (of Tate objects) in a conjectured category of mixed motives over F is briefly discussed.

In § 8 variants of the classical Hodge conjecture are discussed. The generalized Hodge group is defined and the surjectivity of a generalized cycle map from K-theory to this Hodge group is conjectured for varieties defined over \({\bar {\mathbb{Q}}}\). This is a corrected version of a conjecture of Beilinson, the correction being due to U. Jannsen. Also, the general Hodge \({\mathcal D}\)-conjecture (for varieties over \({\bar {\mathbb{Q}}})\) is discussed which asserts that the image of motivic cohomology (i.e. a suitable K-group) is dense in the corresponding Deligne-Beilinson cohomology. The truth of this conjecture would imply as a special case the classical Hodge conjecture.

The final section treats the Birch and Swinnerton-Dyer conjectures and related topics. A generalized height pairing for arithmetic varieties, due to Beilinson and, independently, Gillet and Soulé, is defined. A conjecture generalizing the Birch and Swinnerton-Dyer conjectures and expressing the value of the first non-zero coefficient of the Taylor series of the L-function of the variety at a suitable integer value of the argument is stated in terms of the product of the determinant of the height pairing and the Deligne period. Again this conjecture is due to Beilinson in its greatest generality.

The paper closes with an impressive bibliography of almost 300 references.

In this expository paper a general survey of conjectures and results on L-functions and related subjects such as regulators and algebraic cycles is given. These conjectures are the starting point for a general philosophy, set forth by A. Beilinson, that advocates the universality of algebraic K-theory as a good cohomology theory with “generalized cycle maps” (regulators) to any other suitable cohomology theory, e.g. Deligne-Beilinson cohomology. The K-groups ocurring in the formalism should be Yoneda extensions in a conjectured category of so-called mixed motives. These ideas are presented in the five introductory pages and they turn up every now and then in the main text consisting of nine sections, each followed by one or more interesting pages of “notes and comments” where some details and vistas on specific topics are revealed.

Starting from Pell’s equation one is naturally led to introduce the notion of fundamental units in real quadratic number fields, and more generally, one introduces the regulator of a general number field. Its importance becomes evident in the statement of Dedekind’s class number formula, which in this context, says that the first non-zero coefficient of the Taylor series expansion at \(s=0\) of the zeta-function \(\zeta_ F(s)\) of the number field F is (up to a non-zero rational multiple) equal to the regulator. In this form Borel generalized the class number formula to other values of the argument s of \(\zeta_ F(s)\). The right setting becomes, on the one hand, the algebraic K-groups of the ring of integers of F and, on the other hand, the (de Rham invariant part of) Deligne- Beilinson cohomology of the complex embeddings of F, the relation between both being provided by regulator maps (one for each K-group), whose volumes are called the regulators \(R_ m\). These \(R_ m\) give (up to a non-zero rational multiple) the values of the first non-zero coefficient of the Taylor series expansion of F at the point \(s=1-m\), \(m\geq 2\). - The next step is the introduction of Artin L-functions for representations of the (relative or absolute) Galois group of number fields. One also has Artin L-functions for arbitrary (smooth, projective) varieties X defined over number fields, where the \(\ell\)-adic cohomology of X gives a representation of the (absolute) Galois group of the number field. For number fields there is a conjecture due to Stark, Tate and Gross relating the values of the L-function at negative integers to the values of regulators, again coming from the K-groups of the ring of integers of the number field.

For a curve X defined over a number field F one may consider the associated complex curve (Riemann surface) and construct explicitly a regulator map from K-theory to Deligne-Beilinson cohomology using the construction of the Heisenberg line bundle on \({\mathbb{C}}^*\times {\mathbb{C}}^*\) with connection. This leads to a conjecture, due to Beilinson and Bloch, again relating the value of the first non-zero coefficient of the Taylor series of the L-function of X to the regulator. The K-group \(K_ 2(X)\) has to be replaced by \(K_ 2({\mathcal X})\) of a regular model \({\mathcal X}\) of X defined over the ring of integers of F. \({\mathcal X}\) may be compared to the Néron model of an elliptic curve (or, more generally, an abelian variety). Such \({\mathcal X}\) exists by results of Abhyankar.

In § 5 algebraic cycles of smooth projective varieties enter the scene. For a variety X over \({\mathbb{C}}\) the classical Hodge conjecture is reminded and for X defined over a finitely generated field Tate’s conjecture I (the \(\ell\)-adic counterpart of the Hodge conjecture) is formulated. Also, Tate’s conjecture II on the order of the pole of the L- function in terms of algebraic cycles is stated.

In the next section Beilinson’s general conjectures on the relation between K-groups and Deligne cohomology and on the values of L-functions at special values of the argument are formulated, thus extending the conjectures and examples of the foregoing sections. Compatibility with Deligne’s conjecture on critical points of motives over number fields is indicated and a list of cases where (part of) the conjectures are verified is included. Almost all known cases are of a modular nature.

Section 7 treats in some detail polylogarithms and their role in hyperbolic geometry, the description of higher cyclotomic regulators and the theory of variations of mixed Hodge structures on the projective line minus three points. Also the interpretation of the (odd) higher K-groups of a number field F as extensions (of Tate objects) in a conjectured category of mixed motives over F is briefly discussed.

In § 8 variants of the classical Hodge conjecture are discussed. The generalized Hodge group is defined and the surjectivity of a generalized cycle map from K-theory to this Hodge group is conjectured for varieties defined over \({\bar {\mathbb{Q}}}\). This is a corrected version of a conjecture of Beilinson, the correction being due to U. Jannsen. Also, the general Hodge \({\mathcal D}\)-conjecture (for varieties over \({\bar {\mathbb{Q}}})\) is discussed which asserts that the image of motivic cohomology (i.e. a suitable K-group) is dense in the corresponding Deligne-Beilinson cohomology. The truth of this conjecture would imply as a special case the classical Hodge conjecture.

The final section treats the Birch and Swinnerton-Dyer conjectures and related topics. A generalized height pairing for arithmetic varieties, due to Beilinson and, independently, Gillet and Soulé, is defined. A conjecture generalizing the Birch and Swinnerton-Dyer conjectures and expressing the value of the first non-zero coefficient of the Taylor series of the L-function of the variety at a suitable integer value of the argument is stated in terms of the product of the determinant of the height pairing and the Deligne period. Again this conjecture is due to Beilinson in its greatest generality.

The paper closes with an impressive bibliography of almost 300 references.

Reviewer: W.Hulsbergen

##### MSC:

14C35 | Applications of methods of algebraic \(K\)-theory in algebraic geometry |

14A20 | Generalizations (algebraic spaces, stacks) |

14G10 | Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) |

14C30 | Transcendental methods, Hodge theory (algebro-geometric aspects) |

18F25 | Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects) |

14Fxx | (Co)homology theory in algebraic geometry |

11R42 | Zeta functions and \(L\)-functions of number fields |