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On characteristic \(p\) zeta functions. (English) Zbl 0841.11030

Let \(\mathbb{F}_q\) be the finite field with \(q= p^m\) elements. Let \(X\) be a smooth, projective, geometrically irreducible curve over \(\mathbb{F}_q\) with function field \({\mathbf k}\) and let \(\infty\in X\) be a fixed closed point. Let \({\mathbf A}\) be the affine ring of those functions in \({\mathbf k}\) which are holomorphic away from \(\infty\). It is well known that \({\mathbf A}\) is discrete and cocompact in the completion \({\mathbf K}= {\mathbf k}_\infty\) of \({\mathbf k}\) at \(\infty\) just as \(\mathbb{Z}\) is discrete and cocompact in \(\mathbb{R}\); let \(\mathbb{F}_\infty \subset {\mathbf K}\) be the field of constants. In this setup, Drinfeld constructed his elliptic modules (now called Drinfeld modules), and it is in this setup that the present author introduced his theory of \(L\)-functions.
One of the happy consequences of the paper being reviewed is that it led to communication between its author and the reviewer on the nature of these characteristic \(p\) functions, and, in turn, led the reviewer to a great simplification of their construction. Since these functions are the subject of the paper being reviewed, we now briefly present this construction.
One begins by choosing a sign function, \(\text{sgn}: {\mathbf K}^*\to \mathbb{F}^*_\infty\) (which is defined to be a homomorphism which restricts to the identity on \(\mathbb{F}^*_\infty \subset {\mathbf K}^*\)), and a positive uniformizer \(\pi\in {\mathbf K}\). (So sgn is the analog of the classical sign \(|x|/x\) of a non-zero real number.) An element of \(x\in {\mathbf K}^*\) is said to be positive if and only if \(\text{sgn} (x)=1\). Clearly the subgroup of positive elements is of index \(q^{\deg \infty}-1\) in \({\mathbf K}^*\). Let \({\mathbf C}_\infty\) be the completion of a fixed algebraic closure of \({\mathbf K}\) (equipped with its canonical topology) and set \(S_\infty:= {\mathbf C}^*_\infty \times \mathbb{Z}_p\). If \(a\in {\mathbf A}\) is monic, then we set \(\langle a\rangle:= \pi^{v_\infty (a)} a\), which is a 1-unit at \({\mathbf K}\). The mapping \(\langle\;\rangle\) is extended to the subgroup \({\mathcal P}^+\) of principal and positive ideals in the obvious fashion. The key simplification is to now note that the 1-units in \({\mathbf C}_\infty\) actually form a \(\mathbb{Q}_p\)-module since, it is obviously a \(\mathbb{Z}_p\)-module (use the binomial theorem) and one can take \(p\)-th roots. Thus, as \({\mathcal P}^+\) is of finite index in the group \({\mathcal I}\) of all fractional ideals, the mapping \(\langle\;\rangle\) has a unique extension to \({\mathcal I}\) (and is denoted the same way). So finally we define for any ideal \(I\) and \(s\in S_\infty\), \(I^s:= x^{\deg I}\cdot \langle I\rangle^y\), with the usual properties of exponentiation. It is a simple matter to use a fixed \(t\)-th root of \(\pi\) \((t:= \deg \infty)\) in \({\mathbf C}_\infty\) to embed the usual integer powers (on \({\mathcal P}^+)\) discretely into \(S_\infty\).
Let \(L\) be a finite extension of \({\mathbf k}\) and let \({\mathcal O}_L\) be the ring of \({\mathbf A}\)-integers. If \({\mathfrak B}\) is an ideal of \({\mathcal O}_L\) then we set \(n{\mathfrak B} \subseteq {\mathbf A}\) to be the norm of this ideal, and for \(s\in S_\infty\) \[ \zeta_{{\mathcal O}_L} (s):= \sum_{\mathfrak B} n{\mathfrak B}^{-s}= \prod_{{\mathfrak p} \text{ prime}} (1-n {\mathfrak p}^{-s})^{-1}. \] It can be established that these functions actually analytically continue to all of \(S_\infty\) such that for fixed \(y\in \mathbb{Z}_p\) we obtain an entire power series in \(x^{-1}\) and such that the zeros of these power series “flow continuously”. Through the use of a double congruence, one can show these analogs of classical Dedekind zeta functions have trivial zeros at negative integers. However, unlike classical theory we only have a lower bound on the order of the zeros. One of the important results of the paper being reviewed is to present examples where this lower bound is not exact. Moreover, the author also shows how the exact orders of vanishing sometimes follow interesting patterns involving \(q\)-adic digits, thus, in the words of the author “providing a challenge to understand them in a general framework as in the classical case”.
Once one has a sign function sgn, D. Hayes constructs certain special rank 1 Drinfeld modules which are sgn-normalized [see A brief introduction to Drinfeld modules, in: The arithmetic of function fields, 1-32 (1992; Zbl 0793.11015)]. Let \(\xi\) be a period for a sgn-normalized Drinfeld module and let \(L/{\mathbf k}\) now be an abelian totally-real (= totally split at \(\infty\)) extension of \({\mathbf k}\) which contains the totally-real Hilbert class field \({\mathbf H}\). In this situation one is able to prove an analog of Siegel’s formula as follows: For simplicity let us assume that \(\infty\) is rational and let \(j\) be a positive integer with \((q-1)|j\). Then one has \((\zeta_{{\mathcal O}_L}(j)/ \xi^{[L:k ]j})^2\in{\mathbf H}^*\). Classically one actually obtains that the square lies in \(\mathbb{Q}\), and so it was natural to wonder if one could replace \({\mathbf H}\) by \({\mathbf k}\) in the above result. The author also provides an elegant and important example showing that this cannot be done.

MSC:

11G09 Drinfel’d modules; higher-dimensional motives, etc.
11R42 Zeta functions and \(L\)-functions of number fields

Citations:

Zbl 0793.11015
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References:

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