Jacobi sum, differential equation modulo prime and logarithmic growth.

Abstract: We study the function for , and . We regard as a p-adic analytic function. We prove that satisfies a Frobenius equation and see that Jacobi sum appears in this equation. We consider the relation between the dimension of space of bounded solutions of and the dimension of the space of solutions of L in . We prove that for almost all primes and that for almost all primes if and only if for almost all primes. Moreover in this case is an algebraic function. [Copyright &y& Elsevier]