On the p-adic Beilinson conjecture for number fields

TL;DR

This paper proposes a conjectural p-adic analogue of Borel's theorem linking regulators and zeta-values for number fields, introduces related conjectures for Artin motives, and verifies some cases both theoretically and numerically.

Contribution

It formulates new conjectures connecting p-adic regulators and L-functions, extending Borel's theorem to p-adic settings, with partial proofs and numerical verifications.

Findings

Conjectures are proved for Abelian cases over rationals.

Numerical verification of conjectures in specific instances.

Extension of classical regulator relations to p-adic context.

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 zeta-functions, using syntomic regulators and p-adic L-functions. 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 he conjectures are proved when the number field (or Artin motive) is Abelian over the rationals, and all conjectures are verified numerically in some other cases.