Goss polynomials provide a substitute of trigonometric functions and their identities for the arithmetic of function fields. We study the Goss polynomials G k (X) for the lattice A = F q [T ] and obtain, in the case when q is prime, an explicit description of the Newton polygon N P (G k (X)) of the k-th Goss polynomial in terms of the q-adic expansion of k − 1. In the case of an arbitrary q, we have similar results on N P (G k (X)) for special classes of k, and we formulate a general conjecture about its shape. The proofs use rigid-analytic techniques and the arithmetic of power sums of elements of A.
MSC 2010 Primary 11F52 Secondary 11G09, 11J93, 11T55Introduction. Throughout, F = F q will denote a finite field with q elements, where q is a power of the natural prime p, and A = F[T ] the polynomial ring over A in an indeterminate T . It is a well-established fact that the arithmetic of A and its quotient field K := F(T ) is largely similar to that of their number theoretical counterparts Z and Q. Both Z and A are euclidean rings, discrete in the completions R (resp. K ∞ := F((T −1 ))) of Q at the archimedean valuation (resp. of K at the place at infinity) with compact quotients R/Z and K ∞ /A. The finite abelian extensions of Q and A, described in both cases by classical abelian class field theory, may be explicitly constructed through the adjunction of roots of unity or torsion points of the Carlitz module, respectively. Comparable similarities hold for the non-abelian class field theories of Q and K, presumably governed by the predictions of the Langlands conjectures, and for topics like elliptic curves and (semi-) abelian varieties over Q, which to some extent correspond to Drinfeld modules and their generalizations over K. Likewise, there is a strong analogy between classical (elliptic) modular forms/modular curves and Drinfeld modular forms/curves.