LAUSR

Abstract Let P : ⋯ → C 2 → C 1 → P 1 be a Z p -cover of the projective line over a finite field of cardinality q and characteristic p which ramifies at exactly one rational point. We study the q -adic valuations of the reciprocal roots in C p of L -functions associated to characters of the Galois group of P . We show that for all covers P such that the genus of C n is a quadratic polynomial in p n for n large, the valuations of these reciprocal roots are uniformly distributed in the interval [ 0 , 1 ] . Furthermore, we show that for a large class of such covers P , the valuations of the reciprocal roots in fact form a finite union of arithmetic progressions.