On the Riemann hypothesis for the characteristic \(p\) zeta function
Let \(\mathbb{F}_q\) be the finite field with \(q=p^{n_0}\) elements. Let \({\mathcal C}\) be a smooth, projective, geometrically connected curve over \(\mathbb{F}_q\) and let \(\infty\in {\mathcal C}\) be a fixed closed point. As is standard, we let \({\mathbf A}\) be the affine ring of those functions regular outside of \(\infty\); so \(\text{Spec} ({\mathbf A}) = {\mathcal C} \backslash \infty\), and we let \({\mathbf k}\) be the quotient field of \({\mathbf A}\). If \(a\in {\mathbf A}\) is an invertible function, then \(a\) must also have no poles at \(\infty\) (as the degree of a principal divisor must be 0) and thus \(a\in \mathbb{F}^*_q\). Moreover it is very easy to see that the class group of \({\mathbf A}\) must be finite. It is precisely such \({\mathbf A}\) that are the base (or ``bottom'') rings of the theory of Drinfeld modules. It is also with such rings \({\mathbf A}\) that we may define characteristic \(p\) \(L\)-functions. The procedure is as follows: let \(\pi\) be a uniformizer in the complete field \({\mathbf K} : = {\mathbf k}_\infty\); we call an element \(x\in {\mathbf K}^*\) ``positive'' if and only if it can be written in the form \(x=\pi^d \langle x \rangle\) where \(\langle x \rangle\) is a 1-unit in \({\mathbf K}\). It is trivial to see that there is a constant \(t\) such that \(I^t\) is principal and generated by a positive element \(i(I)\) for all \({\mathbf A}\)-fractional ideals \(I\). Let \(s=(x,y)\in S_\infty: = \overline {\mathbf K}^* \times \mathbb{Z}_p\), then we set \(I^s: =x^{\deg_{\mathbb{F}_q}I} \cdot (\langle i(I) \rangle^y)^{1 /t}\) where we take the unique 1-unit \(t\)-th root. One sees directly that this has the usual good properties of ``\(n^s\)''. Now let \({\mathcal X}\) be a scheme of finite type over \(\text{Spec} ({\mathbf A})\) and let \(x\in{\mathcal X}\) be a closed point. Let \(x\) lie over \(\wp_x\in \text{Spec} ({\mathbf A})\) and let \(d(x)\) be the relative degree of the residue field at \(x\) over \({\mathbf A}/ \wp_x\). Suppose e.g., that \(\psi\) is a Drinfeld module over \({\mathcal X}\) (i.e., a family of Drinfeld modules) and let \(\psi^x\) be the fiber at \(x\); so \(\psi^x\) is a Drinfeld module over a finite field and has a ``good'' characteristic polynomial \(f_x(u) = 1+u \dots\) of the Frobenius morphism. Thus following arithmetic geometry, we define an \(L\)-series \(L(\psi,s): = \prod_{x \text{closed}} f_x(\wp_x^{-d(x)s})^{-1}\). In the paper, ``\(L\)-series of \(t\)-motives and Drinfeld modules'' [The arithmetic of function fields, de Gruyter, 313-402 (1992; Zbl 0806.11028)] the reviewer made a conjecture about the meromorphy of \(L(\psi,s)\). When \({\mathbf A} = \mathbb{F}_q[T]\), this conjecture was proved by \textit{D. Wan} and \textit{Y. Taguchi} via the use of the Dwork trace formula [see: \(L\)-functions of \(\varphi\)-sheaves and Drinfeld modules, J. Am. Math. Soc. 9, No. 3, 755-781 (1996)]. In the spring of 1994, the reviewer had a number of conversations with Y. Taguchi and D. Wan about this proof. Before the use of the Dwork trace formula, the standard technique in the theory of such \(L\)-series was certain estimates obtained via the multinomial theorem. These estimates insured that the order of the coefficients of these \(L\)-series grew at a quadratic rate which was enough to establish the analytic continuation of such series. However, for the most basic such \(L\)-series, \(\zeta_{\mathbb{F}_q[T]} (s)\), (i.e., take \({\mathcal X}= \text{Spec} (\mathbb{F}_q[T])\) and \(\psi\) equal to the Carlitz module) there exist some very old results of Carlitz that imply that this growth is sometimes exponential [see Subsection 8.24 of the reviewer's book, Basic Structures of Function Field Arithmetic, Springer (1996)]. Upon hearing of this exponential growth, the author was inspired to calculate explicitly the Newton polygons involved which led to the paper being reviewed. It is well known that, via a simple change of variable, the Riemann zeta function (with the \(\Gamma\)-factors etc.) can be expressed as an entire real power series with the property that the Riemann hypothesis is equivalent to the roots of this power series also lying in \(\mathbb{R} = \mathbb{Q}_\infty\). The author shows in the present paper that this formalism is exactly reflected in the theory of \(\zeta_{\mathbb{F}_p[T]} (s)\). More precisely, he computes the Newton polygons associated to this function and shows that these Newton polygons are simple (i.e., there exists at most one root of a given absolute value). As such, the roots are obviously simple and, moreover, they must be in \({\mathbf K} = {\mathbf k}_\infty\). The author's proof is based on a clever use of techniques from linear programming. Another and somewhat simpler proof has been given by D. Thakur and his student \textit{J. Diaz-Vargas} [Riemann hypothesis for \(\mathbb{F}_q[T]\), J. Number Theory 59, 313-318 (1996)]. Diaz-Vargas' proof is based on combinatorial formulas for polynomial sums due to L. Carlitz. In a recent preprint J. Sheats claims to have extended the Diaz-Vargas-Carlitz method to all prime powers \(q\). The techniques in this paper are combinatorial in nature and involve a detailed study of certain compositions of positive integers. In this sense the proof is formal and analogous to, though much more difficult than, the use of the geometric series to show that the classical zeta function of \(\mathbb{P}^1_{\mathbb{F}_q}\) (in the sense of E. Artin, A. Weil, etc.) is a rational function. It thus remains to ponder whether there is also a conceptual proof of the characteristic \(p\) results involving the theory of Drinfeld modules.
scientific article; zbMATH DE number 883845