Log-algebraicity of twisted \(A\)-harmonic series and special values of \(L\)-series in characteristic \(p\)
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 \textit{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 \textit{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 \textit{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)].
log-algebraicity of twisted \(A\)-harmonic series
special values of \(L\)-series in characteristic \(p\)
Carlitz module analog of cyclotomic units
scientific article; zbMATH DE number 934974