A higher-dimensional generalization of the Goss zeta function
This paper is both very original and very thought provoking. It contains the seeds of a quite remarkable generalization of the arithmetic of function fields of transcendency 1 over a finite field to function fields of arbitrary transcendency degree. It also poses very interesting challenges in non-archimedean analysis and arithmetic geometry as will be mentioned further on. We begin with a review of the transcendency 1 case: Let \(X\) be a smooth geometrically connected curve over the finite field \(\mathbb{F}_ q\) \((q=p^ t)\) with a fixed closed point \(\infty\). As is usual, we let \({\mathbf A}\) be the affine ring of \(X-\infty\), \({\mathbf k}\) the quotient field of \({\mathbf A}\) (= the function field of \(X\)) and \({\mathbf K}= {\mathbf k}_ \infty\) the completion of \({\mathbf k}\) at \(\infty\). Choose a fixed algebraic closure \(\overline {\mathbf K}\) of \({\mathbf K}\) with the canonical extension of \(| \cdot |_ \infty\) to it. Thus \({\mathbf K}\) is a local field and \({\mathbf A}\) is a discrete, co-compact subring of \({\mathbf K}\). The analogy -- in the 1-dimensional case -- is always with \(\mathbb{Z}\subset \mathbb{R}\). We let \(\pi\) be a fixed uniformizer in \({\mathbf K}\) (for instance, when \({\mathbf A}= \mathbb{F}_ q [T ]\) we may take \(\pi= 1/T\)). Let \(a\in {\mathbf k}^*\); one can then write \(a\) uniquely as \(a= \pi^ m u\) for some integer \(m\) and unit \(u\in {\mathbf K}^*\). We say \(a\) is ``positive'' (or ``monic'') if \(u\) is actually a 1-unit, in which case we set \(\langle a\rangle:= u\). Clearly \(\langle ab\rangle =\langle a\rangle \langle b\rangle\). Now set \(S_ \infty= \overline {\mathbf K}^* \times \mathbb{Z}_ p\); \(S_ \infty\) is a space of ``exponents'' in the same sense that \(\mathbb{C}\) is in classical number theory. Indeed, let \(a\) be positive with \(\deg a\) the degree of \(a\) \((= \deg_{\mathbb{F}_ q} {\mathbf A}/ (a))\) and let \(s= (x,y)\in S_ \infty\). Then we set \(a^ s:= x^{\deg a}\cdot \langle a\rangle^ y\), where \(\langle a\rangle^ y\) is defined via the binomial theorem. So we view \(x\) as the ``real'' part and \(y\) as the ``imaginary'' part of \(s\) in analogy with \(e^{x+ iy}= e^ x\cdot e^{iy}\). It is important to note that the usual integral powers \(a^ i\), for \(i\in \mathbb{Z}\), are recaptured as the elements \(s_ i:= (\pi_ i^{-i}, i)\in S_ \infty\), where \(\pi_ 1\in \overline {\mathbf K}\) is a fixed root of \(y^ t- \pi\) and \(t\) is the degree of \(\infty\). Thus we have a good definition of ``\(a^ s\)'' for the multiplicative group of positive elements and so for the group \({\mathcal P}^ +\) of \({\mathbf A}\)- principal ideals generated by them; this group is of finite index in the full group \({\mathcal P}\) of all \({\mathbf A}\)-fractional ideals. It is very important that the definition of \(I^ s\) if \(I\in {\mathcal P}^ +\) may be canonically extended to all of \({\mathcal P}\) as follows: Let \(U_ 1\in \overline {\mathbf K}\) be the group of 1-units. Since we are now over \(\overline {\mathbf K}\), as opposed to \({\mathbf K}\), one can make \(U_ 1\) a \(\mathbb{Q}_ p\)-vector space in the obvious fashion; in particular, \(U_ 1\) is a uniquely divisible group. Thus, the map \(\langle\;\rangle: {\mathcal P}^ +\to U_ 1\) automatically extends uniquely to \({\mathcal P}^ +\) and will be denoted by the same symbol; so finally we put, with \(s\) as above, \(I^ s:= x^{\deg I}\cdot \langle I\rangle^ y\). With this definition, and some estimates based on the Riemann Roch theorem and multilinear algebra, one can now define and analytically continue the zeta function \(\zeta_{\mathbf A} (s):= \sum I^{-s}\) of \({\mathbf A}\) [see the reviewer in `The arithmetic of function fields', 313-402, De Gruyter (1992; Zbl 0806.11028)]. The function \(\zeta_{\mathbf A} (s)\) becomes a continuous family of entire power series in \(x^{-1}\). We can now present Kapranov's ingenious extension of ``\(a^ s\)'' to higher dimensional varieties. Let \(X\) be a smooth, geometrically connected, projective variety of dimension \(n\) over \(\mathbb{F}_ q\). We assume that \(X\) comes equipped with a maximal flag ``\(\infty\)'' of smooth subvarieties \(\{X= X_ n\supset X_{n-1} \supset\cdots \supset X_ 0\}\), where \(\dim X_ i=i\) and \(X_{i-1} \subset X_ i\) is an ample divisor; thus \(X_ i- X_{i-i}\) is affine. An obvious example occurs when \(X\) is a curve and \(\infty\) a closed point as above. Another may be easily constructed out of projective \(n\)-space. Let \({\mathbf k}\) be the field of rational functions on \(X\). Due to Parshin one has the notion of ``completing \({\mathbf k}\) along the flag \(\infty\)''; one obtains an ``\(n\)- dimensional local field'' \({\mathbf K}:= {\mathbf k}_ \infty:= \mathbb{F}_{{\mathbf q}^ m} ((t_ 1))\dots ((t_ n))\) for appropriate parameters \(t_ 1,\dots, t_ n\); thus one has a complete field whose residue field is also complete and so on. Let \({\mathbf A}\) be the affine ring of \(X- X_{n-1}\). Using the expansion of \(a\) in terms of \(t_ 1,\dots, t_ n\) one has a good notion of when \(a\) is ``monic'' (or ``positive''). For instance, one can now talk about ``monic polynomials'' of several variables, etc. Moreover, there is an appropriate generalization of \(\langle\;\rangle\) and so on. Finally one sets \(S= (\overline {\mathbf K}^*)^ n \times \mathbb{Z}_ p\); so one now has ``\(n\)- real variables''. Using this space, one is able to define \(a^ s\) for \(a\) positive and thus a zeta function associated to \({\mathbf A}\) as \(\zeta_{\mathbf A} (s):= \sum_{a \text{ monic}} a^{-s}\) and show that it is a continuous family of entire \(n\)-dimensional power series. Moreover, this function produces good special-values at negative integers (as elements of \({\mathbf A}\)) which satisfy congruences and so on. We finish by mentioning some of the questions and problems which now arise. 1) Find the appropriate generalization of Drinfeld module that goes along with \(\zeta_{\mathbf A} (s)\). We note that there are many subtleties once \(n>1\): The ring \({\mathbf A}\) no longer uniquely defines the flag \(\infty\). Moreover, although the topology on \({\mathbf k}_ \infty\) is the one derived from \(X_{n-1}\), the use of the Parshin completion is necessary in order to define \(a^ s\). Thus, presumably, the correct generalization of Drinfeld module should also take the flag \(\infty\) into account. In the one dimensional case, Drinfeld modules give an elegant construction of class fields. Since class field theory is known to exist for higher dimensional \(X\) [see e.g., \textit{W. Raskind}: ``Abelian class field theory of arithmetic schemes'', in \(K\)-theory and algebraic geometry: Connections with quadratic forms and division algebras, Proc. Sympos. Pure Math. 58, Part I, 85-187 (1995)], can one use these generalized Drinfeld modules to also construct class fields of \(X\). 2) Does there exist an appropriate version of Euler's result \(({{\zeta (2i)} \over {\pi^{2i}}}\in \mathbb{Q})\) for \(\zeta_{\mathbf A} (s)\) at positive powers? We note that this has been done very successfully in the transcendency 1 case where there is only one ``period'' necessary. 3) In the 1-dimensional case we mentioned that there is a canonical extension of \(a^ s\) to the group of \({\mathbf A}\)-fractional ideals. This extension is vital in order to obtain the ``correct'' definition of \(\zeta_{\mathbf A} (s)\) as a sum over ideals. Is there some appropriate extension of Kapranov's \(a^ s\) to a larger group? (We note that this should not be needed in the factorial case of \({\mathbf A}= \mathbb{F}_ q [T_ 1, \dots, T_ n])\). 4) What object do the special values of \(\zeta_{\mathbf A} (s)\) at negative integers give information about? In the transcendency 1 case, this turns out to be the \(p\)-part of the class group of the function field. Perhaps in general it corresponds to the \(p\)-part of some Chow group.
scientific article; zbMATH DE number 729087