##
**Rank one elliptic \(A\)-modules and \(A\)-harmonic series.**
*(English)*
Zbl 0807.11032

This paper is a tour de force on the part of the author. It represents a high point of a trend started by Drinfeld: Apply ideas from the theory of partial differential equations to the arithmetic of function fields over finite fields; in particular, basic to the author’s work is the concept of a “soliton” – an idea from the theory of water waves and the famous nonlinear Korteweg-de Vries equation (KdV):
\[
{{\partial\varphi} \over {\partial t}} = {{\partial^ 3\varphi} \over {\partial x^ 3}} +6\varphi{{\partial\varphi} \over {\partial x}}.
\]
(A “soliton” is – roughly – a periodic or quasi-periodic solution to KdV.)

Let \(X/\mathbb{F}_ q\) be a smooth, projective, geometrically irreducible curve and let \(\infty\in X\) be a rational point; one then sets \(A\) to be the affine ring of \(X-\infty\) and \(k\) to be the quotient field of \(A\). By fiat the field \(k\) contains exactly one place not in \(\text{Spec} (A)\) thus mimicking the classical case of \(\mathbb{Z}\subset \mathbb{Q}\). The field \(k\) forms a “bottom” for the theory, exactly as \(\mathbb{Q}\) is a bottom in algebraic number theory. In this situation we can define characteristic \(p\) analogs of classical \(L\)-functions and \(\Gamma\)-functions [see the reviewer in ‘The arithmetic of function fields’, de Gruyter, 313-402 (1992)]. In order to make these definitions “good” analytic functions (in a straightforward sense) one always “groups according to degree”. For instance, an \(L\)-series \(\sum c_ a a^{-s}\) is analytically continued by grouping the terms of the same degree; \[ \sum c_ a a^{- s}= \sum_ j \Bigl( \sum_{\deg a=j} c_ a a^{-s} \Bigr), \] etc. Thus each such function \(f\) is made up of a sequence \(f_ k\) of “degree parts”.

The author seeks to study such \(f\) via an “interpolation formula”: One seeks an algebraic function \(\varphi\) of two variables \(\{x,t\}\) such that \(f_ k\) (or, perhaps, \(f_{k+1}/ f_ k\) or \(f_{k+2} f_ k/ f^ 2_{k+1}\)) is recovered by specializing \(x\) and \(t\) (for instance, if \(A= \mathbb{F}_ q [T]\) then \(x=T\) and \(t= T^{q^ k}\)). The point being that such a \(\varphi\) is found by using soliton methods in characteristic \(p\) obtained by essentially replacing differentiation with the \(q\)-th power mapping, an idea that may seem far-fetched from afar but becomes more and more natural the closer it is examined. For instance, set \(\delta y:= y^ q- y\). One then finds trivially that \(\delta\) satisfies the derivation-identity: \[ \delta (xy)= x\delta y+ y^ q\delta x. \] In the case \(A= \mathbb{F}_ q[T]\) an ad-hoc version of this procedure was used to find \(\varphi\) [see G. W. Anderson in ‘The arithmetic of function fields’, 51-73 (1992; Zbl 0797.11056)].

The main tool in the author’s derivation of \(\varphi\) is his Theorem 4.1.1: one maps \(Y\times Y\) to \(J\) and looks at the locus \(Z\) of the pullback of the theta-divisor. One expresses \(Z\) as a sum over certain correspondences coming from graphs of (composites) of the Frobenius and elements of \(\operatorname{Aut}(Y/X)\) – so one is interested in the multiplicities that appear. The author establishes the deep fact that these multiplicities may be read off of an \(L\)-function evaluator (i.e., a formal power series with coefficients in \(\mathbb{Z}[ \operatorname{Aut} (Y/X)]\) used to define classical – characteristic 0 – \(L\)-series). The main result of the author is then established: A version for general sign-normalized rank one Drinfeld \(A\)-modules of the basic classical identity: \(\exp(- \sum_{n=1}^ \infty {{z^ n} \over n})= 1-z\). This result was inspired by some remarkable class number one calculations of D. Thakur [Int. Math. Res. Not. 1992, No. 9, 185-197 (1992; Zbl 0756.11015)]. Motivated by the present paper, in a recent preprint the author is able to construct regulators for zeta-functions (in the above sense).

Let \(X/\mathbb{F}_ q\) be a smooth, projective, geometrically irreducible curve and let \(\infty\in X\) be a rational point; one then sets \(A\) to be the affine ring of \(X-\infty\) and \(k\) to be the quotient field of \(A\). By fiat the field \(k\) contains exactly one place not in \(\text{Spec} (A)\) thus mimicking the classical case of \(\mathbb{Z}\subset \mathbb{Q}\). The field \(k\) forms a “bottom” for the theory, exactly as \(\mathbb{Q}\) is a bottom in algebraic number theory. In this situation we can define characteristic \(p\) analogs of classical \(L\)-functions and \(\Gamma\)-functions [see the reviewer in ‘The arithmetic of function fields’, de Gruyter, 313-402 (1992)]. In order to make these definitions “good” analytic functions (in a straightforward sense) one always “groups according to degree”. For instance, an \(L\)-series \(\sum c_ a a^{-s}\) is analytically continued by grouping the terms of the same degree; \[ \sum c_ a a^{- s}= \sum_ j \Bigl( \sum_{\deg a=j} c_ a a^{-s} \Bigr), \] etc. Thus each such function \(f\) is made up of a sequence \(f_ k\) of “degree parts”.

The author seeks to study such \(f\) via an “interpolation formula”: One seeks an algebraic function \(\varphi\) of two variables \(\{x,t\}\) such that \(f_ k\) (or, perhaps, \(f_{k+1}/ f_ k\) or \(f_{k+2} f_ k/ f^ 2_{k+1}\)) is recovered by specializing \(x\) and \(t\) (for instance, if \(A= \mathbb{F}_ q [T]\) then \(x=T\) and \(t= T^{q^ k}\)). The point being that such a \(\varphi\) is found by using soliton methods in characteristic \(p\) obtained by essentially replacing differentiation with the \(q\)-th power mapping, an idea that may seem far-fetched from afar but becomes more and more natural the closer it is examined. For instance, set \(\delta y:= y^ q- y\). One then finds trivially that \(\delta\) satisfies the derivation-identity: \[ \delta (xy)= x\delta y+ y^ q\delta x. \] In the case \(A= \mathbb{F}_ q[T]\) an ad-hoc version of this procedure was used to find \(\varphi\) [see G. W. Anderson in ‘The arithmetic of function fields’, 51-73 (1992; Zbl 0797.11056)].

The main tool in the author’s derivation of \(\varphi\) is his Theorem 4.1.1: one maps \(Y\times Y\) to \(J\) and looks at the locus \(Z\) of the pullback of the theta-divisor. One expresses \(Z\) as a sum over certain correspondences coming from graphs of (composites) of the Frobenius and elements of \(\operatorname{Aut}(Y/X)\) – so one is interested in the multiplicities that appear. The author establishes the deep fact that these multiplicities may be read off of an \(L\)-function evaluator (i.e., a formal power series with coefficients in \(\mathbb{Z}[ \operatorname{Aut} (Y/X)]\) used to define classical – characteristic 0 – \(L\)-series). The main result of the author is then established: A version for general sign-normalized rank one Drinfeld \(A\)-modules of the basic classical identity: \(\exp(- \sum_{n=1}^ \infty {{z^ n} \over n})= 1-z\). This result was inspired by some remarkable class number one calculations of D. Thakur [Int. Math. Res. Not. 1992, No. 9, 185-197 (1992; Zbl 0756.11015)]. Motivated by the present paper, in a recent preprint the author is able to construct regulators for zeta-functions (in the above sense).

Reviewer: D.Goss (Columbus / Ohio)

### MSC:

11G09 | Drinfel’d modules; higher-dimensional motives, etc. |

35Q51 | Soliton equations |

35Q53 | KdV equations (Korteweg-de Vries equations) |

### Keywords:

Drinfeld modules; soliton; arithmetic of function fields; Korteweg-de Vries equation; regulators for zeta-functions### References:

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