##
**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
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\textit{G. W. Anderson}, Duke Math. J. 73, No. 3, 491--542 (1994; Zbl 0807.11032)

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### References:

[1] | G. W. Anderson, A two-dimensional analogue of Stickelberger’s theorem , The Arithmetic of Function Fields (Columbus, OH, 1991) eds. D. Gross, D. R. Hayes, and M. I. Rosen, Ohio State Univ. Math. Res. Inst. Publ., vol. 2, W. de Gruyter, Berlin, 1992, pp. 51-73. · Zbl 0797.11056 |

[2] | G. W. Anderson and D. Thakur, Tensor powers of the Carlitz module and zeta values , Ann. of Math. (2) 132 (1990), no. 1, 159-191. JSTOR: · Zbl 0713.11082 |

[3] | L. Carlitz, On certain functions connected with polynomials in a Galois field , Duke Math. J. 1 (1935), 137-168. · Zbl 0012.04904 |

[4] | V. Drinfeld, Elliptic modules , Mat. Sb. (N.S.) 94(136) (1974), 594-627, 656. · Zbl 0321.14014 |

[5] | V. Drinfeld, Commutative subrings of certain noncommutative rings , Funkcional. Anal. i Priložen. 11 (1977), no. 1, 11-14, 96. · Zbl 0368.14011 |

[6] | F. Ducrot, Fibré déterminant et courbes relatives , Bull. Soc. Math. France 118 (1990), no. 3, 311-361. · Zbl 0742.14038 |

[7] | J. Fay, Theta Functions on Riemann Surfaces , LNM, vol. 352, Springer-Verlag, Berlin, 1973. · Zbl 0281.30013 |

[8] | J. Fay, On the even-order vanishing of Jacobian theta functions , Duke Math. J. 51 (1984), no. 1, 109-132. · Zbl 0583.14017 |

[9] | P. Griffiths and J. Harris, Principles of Algebraic Geometry , Wiley, New York, 1978. · Zbl 0408.14001 |

[10] | A. Grothendieck, Technique de descente et théorème en géométrie algébrique. V. Les schémas de Picard: Théorèmes d’existence , Séminaire Bourbaki, vol. 13, 1961/2. |

[11] | D. Hayes, Explicit class field theory in global function fields , Studies in Algebra and Number Theory ed. G. C. Rota, Adv. in Math. Suppl. Stud., vol. 6, Academic Press, New York, 1979, pp. 173-217. · Zbl 0476.12010 |

[12] | D. Hayes, The refined \(\mathfrak p\)-adic abelian Stark conjecture in function fields , Invent. Math. 94 (1988), no. 3, 505-527. · Zbl 0666.12009 |

[13] | D. Hayes, A brief introduction to Drinfeld modules , The Arithmetic of Function Fields (Columbus, OH, 1991) eds. D. Goss, D. R. Hayes, and M. I. Rosen, Ohio State Univ. Math. Res. Inst. Publ., vol. 2, W. de Gruyter, Berlin, 1992, pp. 1-32. · Zbl 0793.11015 |

[14] | F. Knudsen and D. Mumford, The projectivity of the moduli space of stable curves. I. Preliminaries on “det” and “Div” , Math. Scand. 39 (1976), no. 1, 19-55. · Zbl 0343.14008 |

[15] | I. G. Macdonald, Symmetric Functions and Hall Polynomials , Clarendon Press, Oxford, 1979. · Zbl 0487.20007 |

[16] | J. S. Milne, Jacobian varieties , Arithmetic Geometry (Storrs, Conn., 1984) eds. G. Cornell and J. Silverman, Springer-Verlag, New York, 1986, pp. 167-212. · Zbl 0604.14018 |

[17] | L. Moret-Bailly, Métriques permises , Astérisque (1985), no. 127, 29-87. · Zbl 1182.11028 |

[18] | D. Mumford, An algebro-geometric construction of commuting operators and of solutions to the Toda lattice equation, Korteweg- deVries equation, and related nonlinear equation , Proceedings of the International Symposium on Algebraic Geometry (Kyoto Univ., Kyoto, 1977) ed. M. Nagata, Kinokuniya, Tokyo, 1978, pp. 115-153. · Zbl 0423.14007 |

[19] | D. Mumford, Abelian Varieties , TIFR Studies in Mathematics, vol. 4, Oxford University Press, Oxford, 1970. · Zbl 0223.14022 |

[20] | G. Segal and G. Wilson, Loop groups and equations of KdV type , Inst. Hautes Études Sci. Publ. Math. (1985), no. 61, 5-65. · Zbl 0592.35112 |

[21] | J. Tate, Les Conjectures de Stark sur les Fonctions \(L\) d’Artin en \(s=0\) , Progress in Mathematics, vol. 47, Birkhäuser, Boston, 1984. · Zbl 0545.12009 |

[22] | D. Thakur, 1990, Letter to author dated April 24. |

[23] | D. Thakur, 1992, Letter to author dated December 14. |

[24] | D. Thakur, Drinfeld modules and arithmetic in the function fields , Internat. Math. Res. Notices (1992), no. 9, 185-197. · Zbl 0756.11015 |

[25] | D. Thakur, Shtukas and Jacobi sums , Invent. Math. 111 (1993), no. 3, 557-570. · Zbl 0770.11032 |

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