Let X , be a geometrically connected smooth and proper curve over a local or global field K . Following GROTHENDIECK [3] there is a canonical exact sequence for the (etale) fundamental group z,(X,) of X,. (Here and in the following we will omit the base points.)
where I? denotes an algebraic closure of K and G, = G ( K / K ) is the absolute Galois group of K. The structure of the geometric fundamental group nl(XK) is completely known for the part prime to the characteristic of K (see [3] X Cor. 3.10). In particular for a prime 1 # char(K) the maximal pro-1 factor group n1(XE)(l) of zl(XK) is an I-POINCARE group of dimension 2, i.e. the cohomology groups of this group have the well known duality properties.
In this paper we are interested in duality properties of the maximal pro-1 factor of the arithmetic fundamental group nl(X,). It is easy to see that the sequence 1 +nl(XR)(1)-'n,(XK)(l) -+G,(O+ 1 remains exact for 1 # char(K) if the 1-torsion points of the JAcoBIan of X, are K-rational.
As a consequence we obtain Theorem 1. Let K be a localfield and let 1 be a prime number different to char(K).
Assume that X , has genus g 2 1 and that the 1-torsion points of the JAcoBIan of X , are K-rational. m e n nl(XK)(l) is a POINCARB group of dimension 4 with dualizing module Q1/Z,(2) = 1 5 p ; 2 , i.e. for all i 2 0 and allfinite 1-primary nl(XK)(l)modules M the cup product Hi(nl(X,)(l), M ) x H4-' (nl(XK) (I), Ql/zl(2))) % H 4 ( ~A & ) ( O , Ql/W)) ~Q l / z l induces a perfect pairing offinite groups.