MOTIVES OVER TOTALLY-REAL FIELDS AND P-ADIC L-FUNCTIONS

Special values of certain L functions of the type L(M, s) are studied where M is a motive over a totally real field F with coefficients in another field T, and L(M, s) = Pi (p) Lp (M, Np--s) is an Euler product P running through maximal ideals of the maximal order O-F of F and Lp (M, X)(-1) = (1 - alpha((1))(p)X).(1 - alpha((2))(p)X)....(1 - alpha(d)(p)X) = 1 + A(1)(p)X + ... + Ad(p)X(d) being a polynomial with coefficients in T. Using the Newton and the Hedge polygons of M one formulate a conjectural criterium for the existence of a p-adic analytic continuation of the special values. This conjecture is verified in a number of cases related to Hilbert modular forms.