https://hal.science/hal-03875683v7Bouali, JohannJohannBoualiLAGA - Laboratoire Analyse, Géométrie et Applications - UP8 - Université Paris 8 Vincennes-Saint-Denis - CNRS - Centre National de la Recherche Scientifique - Université Sorbonne Paris NordComplex vs etale Abel Jacobi map for higher Chow groups and algebraicity of the zero locus of etale normal functionsHAL CCSD2023[MATH] Mathematics [math]Bouali, Johann2023-02-20 10:59:312023-03-02 04:20:502023-03-01 11:23:01enPreprints, Working Papers, ...https://hal.science/hal-03875683v6application/pdf7We prove, using p-adic Hodge theory for open algebraic varieties, that for a smooth projective variety over k ⊂ C the complex abel jacobi map vanishes if the etale abel jacobi map vanishes. This implies that for a smooth projective morphism f : X → S of smooth complex algebraic varieties over k ⊂ C and Z ∈ Z d (X, n) f,∂=0 an algebraic cycle flat over S whose cohomology class vanishes on fibers, the zero locus of the etale normal function associated to Z is contained in the zero locus of the complex normal function associated to Z. From the work of Saito or Charles, we deduce that the zero locus of the complex normal function associated to Z is defined over the algebraic closure k of k if the zero locus of the etale normal function associated to Z is not empty. We also prove an algebraicity result for the zero locus of an etale normal function associated to an algebraic cycle.