Beilinson’s Conjectures
نویسندگان
چکیده
We give a survey of Beilinson’s conjectures about special values of Lfunctions, with emphasis on the underlying philosophy of mixed motives and motivic cohomology. Introduction. In his seminal paper [1], A.A. Beilinson formulated far reaching conjectures about values of motivic L-functions at integers, and produced a compelling body of evidence in their favour by making ingenious calculations in several special cases. The main gist of [1] was a construction of “higher regulators”, expected to explain these L-values in the same spirit as the (slightly modified) classical Dirichlet regulator r : O∗ F ⊕ Z −→ R12 does for the zeta function of a number field F at s = 0 (resp. s = 1). In this case, ζF (s) satisfies a functional equation relating its values at s and 1− s. It has a simple pole at s = 1 and a zero of order r1 + r2 − 1 at s = 0. Its leading Taylor coefficient at s = 0 is equal to lim s−→0 ζF (s)s−(r1+r2−1) = − ]Pic(OF ) ·R ](O∗ F )tors , (0.1) where R denotes the covolume of the lattice Im(r) in R12 . The quest for higher regulators, extending (0.1) to other values of ζF (s), has been initiated by Lichtenbaum [47]. He observed that for m > 1, the order dm of vanishing of ζF (s) at s = 1 −m is equal to the dimension of the higher K-group K2m−1(F )⊗Q. This led to a conjecture that the leading coefficient lim s−→1−m ζF (s)(s+m− 1)−dm should be equal, up to a rational factor, to the covolume of Im(rm) for a certain map rm : K2m−1(F ) −→ Rm 1991 Mathematics Subject Classification. Primary 19F27, 11G40, 14A20; Secondary 14C35, 14G10. 1Miller Fellow Typeset by AMS-TEX 1
منابع مشابه
Algebraic K-theory and Special Values of L-functions: Beilinson’s Conjectures. (talk Notes)
1. Classical motivation 2 1.1. Some classical identities 2 1.2. Riemann’s zeta function 2 1.3. Dedekind zeta functions 3 1.4. Higher regulators 4 2. Motivic L-functions 4 2.1. Realizations of motives 4 2.2. L-functions 6 3. Beilinson’s conjectures on special values of L-functions 7 3.1. Elementary reduction 7 3.2. The regulator map 8 3.3. The conjectures 8 3.4. Known cases 9 4. Motivic cohomolo...
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