Higher regulators on quaternionic Shimura curves and values of L- functions. (English) Zbl 0609.14007

Ramakrishnan, Dinakar

Applications of algebraic K-theory to algebraic geometry and number theory, Proc. AMS-IMS-SIAM Joint Summer Res. Conf., Boulder/Colo. 1983, Part I, Contemp. Math. 55, 377-387 (1986).

[For the entire collection see Zbl 0588.00014.]
In this paper, the conjectures of Bloch and Beilinson are partially proved by applying the results of Beilinson concerning modular curves. These conjectures relate special values of the associated L-functions to special regulators obtained by comparing different \({\mathbb{Q}}\)-structures.
In this case, one has a regulator map \(r: H^ 0(Sh,{\mathcal K}_ 2)\otimes {\mathbb{Q}}\to H^ 1_ B(Sh_{{\mathbb{R}}},{\mathbb{R}}(1))\), \(Sh=Sh(H)= the\) Shimura variety associated to the multiplicative group H of a quaternionic algebra and \(H^ 1_ B(Sh_{{\mathbb{R}}},{\mathbb{R}}(1))\) the Gal(\({\mathbb{C}}/{\mathbb{R}})\)-invariants of \(H^ 1_ B(Sh^{an},{\mathbb{R}}(1))\). You can decompose the motive \({\bar {\mathbb{Q}}}(Sh)\) into its automorphic components \(\oplus_{V}Sh_ V\otimes V_ f.\quad Let\ell_ 0(V)=(d/ds)L(V_ f,s)|_{s=0}\), then the author proves that there exists a \(H({\mathbb{A}}_ f)\)-submodule \({\mathfrak G}\subset H^ 0(Sh,{\mathcal K}_ 2)\otimes {\mathbb{Q}}\), \({\mathfrak G}=\oplus {\mathfrak G}_ V\), such that r(\({\mathfrak G}_ V)=\ell_ 0(V)\cdot H^ 1_ B((Sh_ v)_{{\mathbb{R}}},{\mathbb{Q}}(1))\subset H^ 1_ B((Sh_ V)_{{\mathbb{R}}},{\mathbb{R}}(1))\). In order to prove this theorem analog to the theorem of Beilinson for modular curves the author uses the correspondence between an automorphic form V of H and an automorphic form W of \(GL_ 2\) given by Jacquet-Langlands theory. By the isogeny theorem for abelian varieties of Faltings one obtains an isogeny between the corresponding parts of the Jacobian varieties. By the results of Soulé one gets isomorphisms of the corresponding K-cohomology groups that commute with the regulator maps, such that you can define the module \({\mathfrak G}_ V\) to be the image of the module \(B_ W\) given by Beilinson’s theorem.

Reviewer: M.Heep

MSC:

14C35	Applications of methods of algebraic \(K\)-theory in algebraic geometry
14G10	Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
14H25	Arithmetic ground fields for curves
11R52	Quaternion and other division algebras: arithmetic, zeta functions
18F25	Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects)
14G25	Global ground fields in algebraic geometry
11F70	Representation-theoretic methods; automorphic representations over local and global fields

Keywords:

L-functions; Beilinson conjecture; Bloch conjecture; Shimura variety; quaternionic algebra; Jacquet-Langlands theory; isogeny between the corresponding parts of the Jacobian varieties; K-cohomology groups; regulator maps

Citations:

Zbl 0588.00014