P-adic L-functions and the coniveau filtration on Chow groups

Abstract

Abstract We give evidence for the refined version of the Beilinson–Bloch conjecture involving coniveau filtrations, by studying several infinite families of CM motives (indexed by the integers r ≥ 1 {r\geq 1} ) that are irreducible of Hodge type ( 2 ⁢ r + 1 , 0 ) + ( 0 , 2 ⁢ r + 1 ) {(2r+1,0)+(0,2r+1)} and whose L-functions vanish at the center. In each case, we construct a corresponding algebraic cycle that is homologically trivial but nontrivial in the top graded piece for the coniveau filtration. Consequently, we obtain new explicit examples of cycles on varieties over number fields that are nontorsion in the Griffiths group. The proof of the main result relies crucially on p-adic Hodge theoretic methods, and in particular on the relation between images of homologically trivial cycles under the p-adic Abel–Jacobi map and the values of p-adic L-functions.