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We prove, using $p$-adic Hodge theory for open algebraic varieties, that fora smooth projective variety over a subfield $k\subset\mathbb C$ which is offinite type over $\mathbb Q$, the complex abel jacobi map vanishes if the etaleabel jacobi map vanishes. This implies that for a smooth projective morphism$f:X\to S$ of smooth complex algebraic varieties over $k\subset\mathbb C$ whichis of finite type over $\mathbb Q$ and $Z\in\mathcal Z^d(X,n)^{f,\partial=0}$an algebraic cycle flat over $S$ whose cohomology class vanishes on fibers, thezero locus of the etale normal function associated to $Z$ is contained in thezero locus of the complex normal function associated to $Z$. From the work ofSaito or Charles, we deduce that the zero locus of the complex normal functionassociated to $Z$ is defined over the algebraic closure $\bar k$ of $k$ if thezero locus of the etale normal function associated to $Z$ is not empty. We alsoprove an algebraicity result for the zero locus of an etale normal functionassociated to an algebraic cycle over a field of finite type over $\mathbb Q$.By the way, for a smooth morphism $f:X\to S$ of smooth algebraic varieties overa field of finite type over $\mathbb Q$, we embed the locus of Hodge-Tateclasses of $f$ inside the locus of Hodge classes of $f$.

Complex vs etale Abel Jacobi map and algebraicity of the zero locus of etale normal functions