  We establish rigorously the regularization of the p-adic open string
amplitudes, with Chan-Paton rules and a constant B-field, introduced by Ghoshal
and Kawano. In this study we use techniques of multivariate local zeta
functions depending on multiplicative characters and a phase factor which
involves an antisymmetric bilinear form. These local zeta functions are new
mathematical objects. We attach to each amplitude a multivariate local zeta
function depending on the kinematics parameters, the B-field and the Chan-Paton
factors. We show that these integrals admit meromorphic continuations in the
kinematic parameters, this result allows us to regularize the Goshal-Kawano
amplitudes, the regularized amplitudes do not have ultraviolet divergences. Due
to the need of a certain symmetry, the theory works only for prime numbers
which are congruent to 3 modulo 4. We also discuss the limit p tends to 1 in
the noncommutative effective field theory and in the Ghoshal-Kawano amplitudes.
We show that in the case of four points, the limit p tends to 1 of the
regularized Ghoshal-Kawano amplitudes coincides with the Feynman amplitudes
attached to the limit p tends to 1 of the noncommutative Gerasimov-Shatashvili
Lagrangian.
