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Log-algebraicity of twisted \(A\)-harmonic series and special values of \(L\)-series in characteristic \(p\). (English) Zbl 0868.11031

Let \(R\) be an arbitrary commutative ring with unit \(1_R\). There is an obvious map from \(\mathbb{Z}\to R\) which has \(1\mapsto 1_R\); thus a dual morphism of affine schemes \(\text{Spec}(R)\to \text{Spec}(\mathbb{Z})\). Therefore every scheme may thought of as lying over \(\text{Spec}(\mathbb{Z})\). In practice this means that “\(\mathbb{Z}\)-concepts” such as abelian group, class group, unit groups, etc., can be translated to arbitrary schemes. In other words, classical arithmetic geometry is based on concepts derived from \(\mathbb{Z}\), which is not terribly surprising.
Now let \(\mathcal X\) be a projective, smooth, geometrically connected curve over the finite field \(\mathbb{F}_q\) \((q=p^n)\). Let \(\infty\in{\mathcal X}\) be a closed point and \({\mathbf A}\) the affine ring of those functions on \({\mathcal X}\) regular outside \(\infty\); let \({\mathbf k}\) be the function field of \({\mathcal X}\) (= fraction field of \({\mathbf A}\)). In this situation one has the theory of Drinfeld \({\mathbf A}\)-modules where \({\mathbf A}\) plays the role of \(\mathbb{Z}\), \({\mathbf k}\) of \(\mathbb{Q}\), and so on. By fiat, \({\mathbf A}\) is the “bottom” of the theory; that is, the theory is defined only for schemes lying over \(\text{Spec} ({\mathbf A})\). The importance of the paper being reviewed is that, for the first time, objects related to “\({\mathbf A}\)-concepts” are created for Drinfeld modules over global \({\mathbf A}\)-fields. More precisely, the paper presents the technology needed to find, in certain basic circumstances, the \({\mathbf A}\)-analog of “units” and “cyclotomic units” for rank one Drinfeld modules. It therefore opens up many fascinating areas of inquiry and some of them, such as an amazing analog of the classical Kummer-Vandiver conjecture, will be mentioned below.
As is known to beginning calculus students, one has \(\exp(- \sum^\infty_{n=1} z^n/n)=1-z\). In number theory this, and its \(p\)-adic analogue, are used to establish relationships between the values of \(L\)-series at \(s=1\) and cyclotomic units. The main thrust of the paper being reviewed is to establish a version of this basic identity in the theory of Drinfeld modules (at least under the assumption that \(\infty\) is a rational point). More precisely, let \(\rho\) be a sgn-normalized rank one Drinfeld module for a fixed choice of sign function sgn. (Such a Drinfeld module plays the role of \(\mathbb{G}_m\) for the theory and is the natural generalization of the Carlitz module). Such Drinfeld modules were constructed by D. Hayes [see, for example, A brief introduction to Drinfeld modules, in: The arithmetic of function fields, de Gruyter, 1-32 (1992; Zbl 0793.11015)] and are therefore called Drinfeld-Hayes modules or just Hayes modules. Let \({\mathbf H}\) be the Hilbert class field of \({\mathbf k}\) in which \(\infty\) splits completely and let \(L\) be any extension of \({\mathbf H}\) such that \(L/{\mathbf k}\) is abelian and unramified over all finite places of \({\mathbf A}\). Let \(B\subset L\) be the ring of \({\mathbf A}\)-integers and let \(I\) be an \({\mathbf A}\)-ideal. Associated to \(I\) and \(\rho\) is a monic polynomial \(\rho_I(x):= \rho_{I,0}x+ \rho_{I,1}x^q+\dots\), where the roots of \(\rho_I(x)\) are precisely the group of \(I\)-division elements (inside some fixed algebraic closure of \({\mathbf k}\)). Let \(b(t)= \sum b_it^i\in L[t]\); then one sets \(I* b(t):=\sum b_i^{(I,B/{\mathbf A})}(\rho_I(t))^i\), where \((I,B/{\mathbf A})\) is the Artin symbol of \(I\). One checks that \(I*(J*b)= (IJ)*b\) and that the action \(I*b\) encapsulates the explicit reciprocity law associated to all division values of \(\rho\). To \(b\) we also assign the formal power series \[ \ell(b;z):= \sum_i\frac {(I*b)(t)}{\rho_{I,0}} z^{q^{\deg I}}\in L[t] [[z]]. \] The author then establishes the following result.
Theorem: Let \(\exp_\rho(u)\) be the exponential of \(\rho\) and assume that \(b(t)\in B[t]\). Then \(\exp_\rho (\ell(b;z))\) is actually a polynomial in \(z\) (i.e., \(\in L[t][z]\)).
The theorem is established in the following fashion: one equips \(L[t]\) with a norm for which \(B[t]\) is discrete. Next one shows that the coefficients of \(\exp_\rho (\ell(b;z))\) are actually elements of \(B[t]\). Finally one also shows that such coefficients must go to zero – therefore by discreteness they vanish sufficiently far along. (In [Duke Math. J. 73, 491-542 (1994; Zbl 0807.11032)], the author had previously established a weaker version of the above result using “soliton” methods. The present work uses only techniques from Drinfeld modules.)
Now let \({\mathbf A}= \mathbb{F}_q[T]\). The author uses the theorem in the following remarkable fashion. Let \({\mathbf p}\subset{\mathbf A}\) be a monic irreducible prime of degree \(d\) and let \(\rho=C\) be the Carlitz module. Let \({\mathbf e}(x):= \exp_C(\xi x)\), where \(\xi\) is a generator of the lattice associated to \(C\). Let \(m\) be a nonnegative integer and let \(S_m(t,z):= \exp_C(\ell(t^m;z))\) (with the notation just above). Let \(\pi_{\mathbf p}:={\mathbf e}(1/{\mathbf p})\), \(\mathbb{F}_{\mathbf p}:={\mathbf A}/{\mathbf p}\) and \({\mathcal O}:={\mathbf A}[\pi_{\mathbf p}]\). One knows that \({\mathcal O}\) is the ring of \({\mathbf A}\)-integers in the abelian extension \({\mathbf k}({\mathbf p})/{\mathbf k}\) obtained by adjoining the \({\mathbf p}\)-division points of the Carlitz module. One also knows that \(\text{Gal} ({\mathbf k}({\mathbf p})/{\mathbf k})\simeq \mathbb{F}^*_{\mathbf p}\). Let \({\mathcal S}({\mathbf p})\) be the \({\mathbf A}\)-submodule of \(C[{\mathcal O}]\) (\(={\mathcal O}\) viewed as an \({\mathbf A}\)-module via \(C\)) generated by \(S_m({\mathbf e}(b/{\mathbf p}),1)\), where \(b\in\mathbb{F}^*_{\mathbf p}\) and where \({\mathbf e}(b/{\mathbf p})\) has the obvious meaning – this is the module of “special points”. Through the use of the value of characteristic \(p\) \(L\)-series at \(s=1\), the author shows that \({\mathcal S}({\mathbf p})\) is finitely generated over \({\mathbf A}\) and calculates the rank. All of this is so analogous to classical theory that one views \({\mathcal S}({\mathbf p})\) as being the Carlitz module analog of cyclotomic units. It is well known in classical cyclotomic theory that the cyclotomic units are of finite index in the group of all units; thus the group of all \(u\) such that \(u^n\) is a cyclotomic unit for some \(n>0\) is precisely the unit group of the cyclotomic field and is thus finitely generated. Let \(\sqrt{\mathcal S}({\mathbf p}):= \{u\in{\mathcal O}\mid C_a(u)\in{\mathcal S}({\mathbf p})\}\) for some \(0\neq a\in{\mathbf A}\). By a basic result of B. Poonen [Compos. Math. 97, 349-368 (1995; Zbl 0839.11024)], \(\sqrt{\mathcal S}({\mathbf p})\) is also finitely generated over \({\mathbf A}\). Thus the finite \({\mathbf A}\)-module \(\sqrt{\mathcal S}/\mathcal S\) forms a good analog of {units/cyclotomic units}. The analog of the Kummer-Vandiver conjecture is that \({\mathbf p}\) should not divide the Fitting ideal of this module (or, what is the same, \({\mathbf p}\) should act invertibly on \(\sqrt{\mathcal S}/{\mathcal S}\)). This conjecture can actually be proved if the \(p\) class group of \({\mathbf k}({\mathbf p})\) is trivial, and so on.
Once one knows what the “units” are for the Carlitz module, it makes sense to inquire about an analog of the class group (again as some finite \({\mathbf A}\)-module). One such construction has been proposed by Poonen. It is a well-known piece of folk-lore in algebraic number theory that the Tate-Shafarevich group of the units is the ideal class group. Thus Poonen suggests trying to do an analogous Tate-Shafarevich construction for the author’s special points. However, this seems to require a functorial definition of “special points” for Drinfeld modules over both global and local fields which is not yet known and is still another interesting problem.
Finally the author constructs certain “root numbers” in his paper and asks whether these are connected to the Gauss sums of D. Thakur. In fact, there is a deep connection as pointed out by J. Zhao [see: On root numbers connected with special values of \(L\)-functions over \(\mathbb{F}_q(T)\), J. Number Theory 62, No. 2, 307-321 (1997)]. Buoyed by this, the author proceeds to use these Gauss sums to give another, and quite mysterious, description of the special points [see subsection 10.6 of the reviewer’s monograph “Basic structures of function field arithmetic”, Springer (1996)].

MSC:

11G09 Drinfel’d modules; higher-dimensional motives, etc.
11R58 Arithmetic theory of algebraic function fields
11R27 Units and factorization
11R70 \(K\)-theory of global fields
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