On the Zeroes of Goss Polynomials

Goss polynomials provide a substitute of trigonometric functions and their identities for the arithmetic of function fields. We study the Goss polynomials Gk(X) for the lattice A = Fq[T ] and obtain, in the case when q is prime, an explicit description of the Newton polygon NP (Gk(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 NP (Gk(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. Introduction. Throughout, F = Fq 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. In both cases, the arithmetic behind modular forms is encoded in their series expansions around cusps and in the action of Hecke operators. The study of these questions for Drinfeld modular forms requires substitutes for certain classical, notably trigonometric, functions and their identities, which are routinely used in elliptic modular forms theory. The required substitute is provided by the Goss polynomials Gk,Λ = Gk,Λ(X) of F-lattices Λ in C∞, the completed algebraic closure of K∞, and in particular, the polynomials Gk := Gk,A for the F-lattice Λ = A. It turns out that the series (Gk)k≥1 of these is crucial for the understanding of modular forms and modular curves for the group GL(2, A) and its congruence subgroups, and for many other topics in the arithmetic of A and K, see, e.g., [3], [4] and [9]. 2000 Mathematics Subject Classification. 11G09, 11J93, 11T55. 1 2 ERNST-ULRICH GEKELER The most important question is about the size and arithmetic nature of their zeroes. The behavior of Gk is rather erratic, depending in a complicated fashion on the p-adic expansion of k − 1 and the vanishing/nonvanishing of certain multinomial coefficients (mod p), and a general answer, though desirable, is not in sight. Nevertheless, besides some incomplete results for general q, we succeed to give an explicit description of the Newton polygon of Gk over the valued field K∞ (equivalent with the description of the size of its zeroes), in the important special case where q = p is prime: see Theorem 6.12, which is our principal result. Its proof uses non-archimedean contour integration and the arithmetic of power sums of elements of A. In a weakend form, this result may be (hypothetically) generalized to arbitrary finite fields F = Fq, which is the contents of Conjecture 3.10. It is compatible with extensive numerical calculations and may perhaps be approached by means different from those in this paper. The research leading to the present result was partially carried out when I was on sabbatical leave at the Centre de Recerca Matematica CRM at Bellaterra, Spain. With pleasure I acknowledge the support of that institution, and I heartily thank its staff for their hospitality. Notation. A = F[T ], where F = Fq, #(F) = q = power of the prime p K = F(T ), endowed with the valuation v∞ at infinity and the absolute value | . | normalized such that v∞(T ) = −1 and |T | = q K∞ = F((T−1)), the completion of K w.r.t. | . |, with ring of integers O∞ and maximal ideal m∞ C∞ = the completed algebraic closure of K∞, with its canonical extensions of | . | and v∞, denoted by the same symbols B(z, r) the open ball {w ∈ C∞ | |w−z| < r} around z ∈ C∞ with radius r ∈ |C∗ ∞| and corresponding closed ball B(z, r) = {w ∈ C∞ | |w − z| ≤ r}. Review of Goss polynomials (see [4]). An F-lattice (lattice for short) in C∞ is a discrete F-subspace Λ of C∞. Discreteness means that Λ intersected with each ball B(0, q) in C∞ is finite. Hence