On the p-adic Beilinson Conjecture for Number Fields

Abstract

Abstract:We formulate a conjectural p-adic analogue of Borel's theorem relating regulators for higher K-groups of number fields to special values of the corresponding ζ-functions, using syntomic regulators and p-adic Lfunctions. We also formulate a corresponding conjecture for Artin motives, and state a conjecture about the precise relation between the p-adic and classical situations. Parts of the conjectures are proved when the number field (or Artin motive) is Abelian over the rationals, and all conjectures are verified numerically in various other cases.

Étale cohomology, cofinite generation, and p-adic L-functions

“…For the same k, e and χ, (1) implies (2), and, as discussed before the statement of the conjecture, (2) is equivalent to (3). As also implied by the discussion there, if we fix e, then (3) for all k, χ and E implies (4), and the converse is clear. …”

“…(For the existence and uniqueness of such functions we refer to the overview statement in [4,Theorem 2.9], and for the radius of convergence to [18, p.82].) In particular, L p (s, χ, k) is not identically zero because χ is even, so that the right-hand side of (3.5) is non-zero for m < 0.…”

“…For example, the equivalent statement for Galois cohomology of the first part of Theorem 1. 4(1) below follows already by combining [32, Theorems 2.1 and 3.1] with the Corollary on page 260 of [33]; a succinct general overview is given in Appendix A.1 of [28]. (For the relation between those cohomology groups and the groups we consider, we refer to Remark 1.5 or Remark 2.6.)…”

“…However, the uniform bound across all e and E in the second part of Theorem 1. 4(1) is new. Theorem 1.…”

“…Theorem 1. 4. Let k be a number field, p a prime number, E a finite extension of Q p , and η : G k → E an Artin character realizable over E. Let M (E, η) be an O E -lattice for η and let W (E, η) = M (E, η)⊗ Zp Q p /Z p .…”