DESCENT FOR l-ADIC POLYLOGARITHMS

Let L be a finite Galois extension of a number field K. Let G := Gal(L/K). Let z(1),...,z(N) is an element of L* \ {1} and let m(1),...,m(n) is an element of Q(l). Let us assume that the linear combination of l-adic polylogarithms c(n) := Sigma(N)(i=1) m(i)l(n) (z(i))gamma(i) (constructed in some given way) is a cocycle on G(L) and that the formal sum Sigma(N)(i=1) m(i)[z(i)] is G-invariant. Then we show that c(n) determines a unique cocycle s(n) on G(K). We also prove a weak version of Zagier conjecture for l-adic dilogarithm. Finally we show that if c(2) is "motivic" (m(1),...,m(N) is an element of Q) then s(2) is also "motivic"