Remarks on special values of \(L\)-functions.

*(English)*Zbl 0817.14007
\(L\)-functions and arithmetic, Proc. Symp., Durham/UK 1989, Lond. Math. Soc. Lect. Note Ser. 153, 373-392 (1991).

[For the entire collection see Zbl 0718.00005.]

This article does not represent precisely a talk given at the symposium, but is complementary to C. Deninger and the author [same collection, Lond. Math. Soc. Lect. Note 153, 173-209 (1991; Zbl 0729.14002)]. Its purpose is to explain a setting in which the various conjectures on special values of \(L\)-functions admit a unified formulation. At critical points, Deligne’s conjecture relates the value of an \(L\)-function of a certain period, and at non-critical points, the conjectures of Beilinson give an interpretation in terms of regulators. Finally, at the point of symmetry of the functional equation, there is the conjecture of Birch and Swinnerton-Dyer, generalized by Bloch and Beilinson, in which the determinant of the height pairing on cycles appears.

Both the periods and the regulators are constructed globally, and their definitions are in some sense archimedean. The height pairing, on the other hand, is defined as a sum of local terms. The author’s aim is to show how all of these objects – periods, regulators, and heights – may be interpreted as ‘periods of mixed motives’.

That such a reformulation is possible in the case of regulators in clearly indicated in a letter of Deligne to SoulĂ© [1985]. Perhaps the only novel feature of the author’s account is to regard the mixed motives as primary objects, rather than the Ext groups. It is appropriate to mention in this connection work of Anderson and of Harder, in which certain particular mixed motives arising in the study of the cohomology of Shimura varieties are investigated. These motives fit directly (and without assuming a grand conjectural framework) into our setting, although their connection with the \(K\)-theoretical formulation of Beilinson’s conjectures remains obscure.

This article does not represent precisely a talk given at the symposium, but is complementary to C. Deninger and the author [same collection, Lond. Math. Soc. Lect. Note 153, 173-209 (1991; Zbl 0729.14002)]. Its purpose is to explain a setting in which the various conjectures on special values of \(L\)-functions admit a unified formulation. At critical points, Deligne’s conjecture relates the value of an \(L\)-function of a certain period, and at non-critical points, the conjectures of Beilinson give an interpretation in terms of regulators. Finally, at the point of symmetry of the functional equation, there is the conjecture of Birch and Swinnerton-Dyer, generalized by Bloch and Beilinson, in which the determinant of the height pairing on cycles appears.

Both the periods and the regulators are constructed globally, and their definitions are in some sense archimedean. The height pairing, on the other hand, is defined as a sum of local terms. The author’s aim is to show how all of these objects – periods, regulators, and heights – may be interpreted as ‘periods of mixed motives’.

That such a reformulation is possible in the case of regulators in clearly indicated in a letter of Deligne to SoulĂ© [1985]. Perhaps the only novel feature of the author’s account is to regard the mixed motives as primary objects, rather than the Ext groups. It is appropriate to mention in this connection work of Anderson and of Harder, in which certain particular mixed motives arising in the study of the cohomology of Shimura varieties are investigated. These motives fit directly (and without assuming a grand conjectural framework) into our setting, although their connection with the \(K\)-theoretical formulation of Beilinson’s conjectures remains obscure.

##### MSC:

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

11G40 | \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture |

14G40 | Arithmetic varieties and schemes; Arakelov theory; heights |

14A20 | Generalizations (algebraic spaces, stacks) |

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