This is a very important and remarkable paper that deserves a wide audience. In it classical methods of algebraic number theory are combined with ideas from the theory of solitons (which arose in theory of water waves!). Let \(A:= \mathbb{F}_ q [T]\) \((q=p^ m)\), \(k:=\mathbb{F}_ q (T)\) and \(A_ + := \{a \in A \mid a\) monic}. We let \(C\) be the Carlitz module defined by \(C_ Tz = Tz+z^ q\); this is a rank one Drinfeld module with exponential function \[ \exp_ C (z): =z + \sum^ \infty_{n=1} {z^{q^ n} \over (T^{q^ n} - T^{q^{n-1}}) \cdots (T^{q_ n} - T)}. \] One knows that \(\exp_ C (z)\) is periodic with lattice \(\overline \pi A\) and \(\overline \pi: = \overline TT \prod^ \infty_{n=1} (1-T^{1 - q^ n})^{-1}\), where \(\overline T\) is a fixed \(q-1\)-st root of \(-T\). Fix \(f \in A_ +\); one then sets \(\zeta_ f: = \exp_ C (\overline \pi/f)\). The element \(\zeta_ f\) is a generator of the \(A\)-module of \(f\)-division points. Set \[ B: = B_ f: = A [\zeta_ f] = \mathbb{F}_ q [T, \zeta_ f] \subset \mathbb{F}_ q \bigl( (1/ \overline T) \bigr). \] Standard arguments establish that \(B\) is the ring of \(A\)-integers in the fraction field \(k_ f\) of \(B\). The Galois group of the geometric extension \(k_ f/k\) is precisely \(A/(f)^*\); \(a+(f) \mapsto \sigma_ a\).
Next we fix a smooth projective geometrically irreducible curve \(X\) over \(\mathbb{F}_ q\) with function field isomorphic to \(k_ f\) via a fixed isomorphism \(\xi\). It is important to keep \(\mathbb{F}_ q(X)\) separate in one’s mind from \(k_ f\). Let \(G\) be the group of automorphisms of \(X\) corresponding to the automorphisms of \(k_ f/ \mathbb{F}_ q(T)\). The scheme \(X/ \mathbb{F}_ q\) has the usual \(q\)-th Frobenius which we denote “Frob”; the point \(\text{Frob}^ N \xi\) is thus obtained by raising the coordinates of \(\xi\) to the \(q\)-th power. Via our fixed isomorphism \(\xi\) there is an open affine of \(X\) corresponding to \(\text{Spec} (B)\); we let \(U'\) be the open subscheme of \(U\) corresponding via \(\xi\) to \(\text{Spec} (B [f^{-1}])\). We let \(\infty_ X\) denote the unique reduced closed subscheme of \(X\) complimentary to \(U\) (and so lies over the place \(\infty\) of \(k)\). Finally, let \(V \subset X\) be the open subscheme containing \(\infty_ X\) and such that \(V \cap U = U'\). Then the author establishes the following:
Theorem: There exists a unique meromorphic function \(\varphi\) with the following properties: (1) \(\varphi\) is regular on \(V \times U\); (2) Let \(a \in A\) be prime to \(f\) with \(\deg (a)<\deg (f)\) and let \(N\) be a positive integer. Then \[ 1-\varphi (\text{Frob}^ N \xi, \sigma_ a \xi) = \prod_{{n \in A_ + \atop \deg (n) = N-1}} \left( 1+ {a \over fn} \right); \] (3) Each irreducible component of the divisor of \(\varphi\) and \(1-\varphi\) that is neither horizontal nor vertical is the graph of a power of Frobenius composed with an element of \(G\).
[(4) The multiplicities of all irreducible components of the divisors of \(\varphi\) and \(1-\varphi\) are computed with the exception of some horizontal and vertical components.]
One calls the function \(\varphi\) a “soliton” (for reasons given a bit later). The theorem is significant for two reasons:
1. The products appearing in property 2 are the basic building blocks of the \(\mathbb{F}_ q [T]\)-analogue of the \(\Gamma\)-function [see, eg. D. Goss, Duke Math. J. 56, No. 1, 163-191 (1988; Zbl 0661.12006), and D. Thakur, in The Arithmetic of Function Fields, de Gruyter 75-86 (1992)].
2. The divisor of the function \(1-\varphi\) induces, via the theory of correspondences, an endomorphism \(\Phi\) of the Jacobian of \(X\) – on the one hand, \(\Phi\) is an explicit polynomial in Frobenius with coefficients in the integral group ring of \(G\) (and is closely related to the Artin \(L\)-functions attached to the extension \(k_ f/ \mathbb{F}_ q(T))\); on the other hand, \(\Phi\) is the zero endomorphism since it is induced by the divisor of a function. Since the construction of the function \(1-\varphi\) is quite explicit, one gets an interesting annihilator of the Jacobian of \(X\) without appealing to the theory of \(\ell\)-adic representations! Seen in this light, the theorem generalizes a remarkable result of Coleman, [see R. Coleman, Proc. Am. Math. Soc. 102, 463-466 (1988; Zbl 0666.14014)].
Why the name “soliton”? The author is obscure on this point and so we attempt a brief explanation here. We begin with some background. The theory of Drinfeld modules is based on the injection of \(A\) into the ring of additive endomorphisms of \(\mathbb{G}_ a\) with a Lie condition similar to that involved in complex multiplication of elliptic curves. Let \(\tau\) be the \(p\)-th power mapping; so all endomorphisms are of the form \(p(\tau) = \sum^ t_{i=0} c_ j \tau^ j\). In particular the solutions to \(p(\tau)=0\) clearly form a finite dimensional space over the “\(\tau\)- constants” \(\mathbb{F}_ q\). Similarly, let \(D=d/dx\); then a complex differential operator \(P(D)\) has (under good conditions) only a finite dimensional space of solutions. We thus see (the beginnings of) a striking analogy between \(D\) and \(\tau\); in fact pseudo-differential operators are analogous to formal series \(\sum_{i \gg - \infty}c_ i \tau^{-i}\). A high point of this analogy was achieved by Krichever and Drinfeld. Krichever found an equivalence between commutative subrings of \(\mathbb{C}[[x]] [D]\) and complete \(\mathbb{C}\)-curves together with certain sheaf data. Inspired by this dictionary Drinfeld found a similar equivalence in finite characteristic upon replacing \(D\) by \(\tau\). Now consider the famous nonlinear partial differential equation known as the Korteweg-de Vries equation (KdV): \(\partial u/ \partial t = \partial^ 3 u/ \partial x^ 3 + 6u \partial u/ \partial x\). Using his dictionary, Krichever is able to produce soliton (= periodic or quasi-periodic) solutions to KdV. [see, D. Mumford, Proc. Int. Sympos. Algebraic Geometry, Kyoto, 115-153 (1977; Zbl 0423.14007)]. Return now to the function \(\varphi\) appearing in the theorem. The calculation of the divisors of \(\varphi\) and \(1-\varphi\) comes down to a two-dimensional power series calculation. In turn, these calculations (given in an ad-hoc form by the author) are motivated by calculations involving solitons over \(\mathbb{C}\) due to Krichever (above) and the Japanese school [see G. Segal and G. Wilson, Publ. Math. Inst. Hautes Étud. Sci. 61, 5-65 (1985; Zbl 0592.35112), especially Chap. 8]; thus the genesis of the use of “soliton” for \(\varphi\). The author has subsequently developed the soliton methodology much farther and applied it to the study of \(\zeta\)- values as well as \(\Gamma\)-values [see G. Anderson, Rank one Elliptic \(A\)-modules and \(A\)-harmonic Series, Duke Math. J. 73, No. 3, 491-542 (1994)]. Heuristically (in the author’s words): ‘Methods for solving “soliton equations”, transposed to characteristic \(p\), become methods to study function field arithmetic’.
For the entire collection see [Zbl 0771.00031].