Riemann hypothesis for \(\mathbb{F}_ p[T]\)
Let \(\mathbb{F}_r\), \(r= p^m\), be the finite field with \(r\) elements. Let \(N\) be a positive integer and let \(k\) be a nonnegative integer. Let \((n_0, \dots, n_k)\) be a \((k+1)\)-tuple of nonnegative integers. We write \(N=n_0 \oplus n_1 \oplus \cdots \oplus n_k\) if and only if the following are true: 1. \(N= \sum^k_{j=0} n_j\); 2. There is no carry over of \(p\)-adic digits in this sum; and 3. For \(0\leq j \leq k-1\) we have \(n_j>0\) and \(n_j \equiv 0 \pmod {r-1}\). Now let \({\mathbf A}: = \mathbb{F}_r [T]\) be the polynomial ring over \(\mathbb{F}_r\). Let \[ S_k(N): = \sum_{\substack{ a \in {\mathbf A}, a \text{monic}\\ \deg (a) = k}} a^N. \] In ``Finite sums and interpolation formulas over \(GF [p^n,x]\)'' [Duke Math. J. 15, 1001-1012 (1948; Zbl 0032.00303)] \textit{L. Carlitz} stated that \(S_k(N) \neq 0\) if and only if there exists a \((k+1)\)-tuple \((n_0, \dots, n_k)\) with \(N=n_0 \oplus n_1 \oplus \cdots \oplus n_k\); indeed, it is a straightforward exercise with Lucas' formula to show that this condition is a necessary one. Carlitz did not however give a proof of the sufficiency and only gave an inadequate sketch of the sufficiency of a result implying that the degree of \(S_k (N) \) arises from a certain unique \((k+1)\)-tuple satisfying the above conditions. In any case, this result of Carlitz, which seemed so innocuous for almost 50 years, is now turing out to be of fundamental importance for the arithmetic of polynomials. It is certainly yet another case of the amazing influence of Carlitz's old work on current research. The importance of Carlitz's statement is that it allows one to prove a certain ``Riemann hypothesis'' which naturally occurs in function field arithmetic. To see how this might come about we proceed as follows: first of all, we introduce a new variable \(x\) and set \[ z(x,-N): = \sum^\infty_{k=0} S_k (N)x^{-k}. \] It is easy to see from the binomial theorem, or Carlitz's result above, that the above formal power series in \(x^{-1}\) is actually a polynomial. We now want to fit these polynomials together into a continuous function of two variables; in order to do so, we pick a uniformizer \(\pi\in \mathbb{F}_r ((1/T))\) of the form \(\pi= 1/T+\) higher terms, and set \(\widetilde z(x,-N) = z(x/ \pi^N,-N)\). The key now is to notice that the coefficient of \(x^{-k}\) in \(\widetilde z(x,-N)\) is a sum over elements of the form \((\pi^ka)^N\), where \(\pi^ka\) is a 1-unit in \(\mathbb{F}_r((1/T))\) and so may be raised to any \(p\)-adic power \(y\). In fact, the polynomials \(\widetilde z(x,-N)\) continuously interpolate to a function \(\zeta (x,y): \overline {\mathbb{F}_r((1/T))}^* \times \mathbb{Z}_p \to \overline {\mathbb{F}_r((1/t))}\) (where \(\overline ?\) stands for a fixed algebraic closure). It is actually now quite easy to see that \(\zeta (x,y)\) can be given via an Euler product, etc. One can study special values of this function with results very familiar from classical theory and so on. Carlitz's statements given above allow one to compute exactly the Newton polygons of \(\zeta (x,y)\) for fixed \(y\). In turn, these Newton polygons turn out to be so elementary that all the zeros of \(\zeta (x,y)\) lie in \(\mathbb{F}_r ((1/T))\) and are simple. It is well known that, upon making a certain linear substitution of variables, the Riemann zeta function can be represented as an entire real power series whose zeros are also expected to be real; thus the comparison between the two theories. The Newton polygons were actually first computed by \textit{D. Wan} for \(\mathbb{F}_p [T]\) (with some results for general \(r)\) by much different methods [J. Number Theory 58, 196-212 (1996; Zbl 0858.11030)]. In the paper being reviewed, the author, following the excellent suggestion of Thakur, attacks the computation of the Newton polygons via (a slight variant of) Carlitz's ideas given as above; however, the author is again only successful for \(\mathbb{F}_p [T]\). Finally, in a recent preprint (``The Riemann hypothesis for the Goss zeta function'') \textit{J. Sheats} claims to have established Carlitz's statement for all \(\mathbb{F}_r [T]\) and so is then able to establish the ``RH'' for all \(r\). As a corollary the polynomials \(z(x,-N)\) are separable; it thus may be an interesting problem to compute their associated Galois groups. Carlitz's formula for the degrees is purely formal in that it only depends on properties of natural numbers. Thus using it to compute the Newton polygons is similar to the formal use of the geometric series to compute the classical \(L\)-series of the projective line. Furthermore, it is very reasonable to ask that this new Riemann hypothesis be true for more general functions such as the \(L\)-series of Drinfeld modules defined over \(\mathbb{F}_r (T)\) (if one uses extensions of \(\mathbb{F}_r (T)\) then one needs to alter what extensions of \(\mathbb{F}_r ((1/T))\) that the roots can be expected to lie in). Thus the really crucial issue for the theory is to develop connections with the theory of Drinfeld modules (and not just simple polynomial sums). At this point, one really does not know too much what to expect.
scientific article; zbMATH DE number 935008