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**Gamma functions for function fields and Drinfeld modules.**
*(English)*
Zbl 0734.11036

From the introduction: “The purpose of this paper is to study gamma functions in the context of the theory of function fields of one variable over a finite field. In particular, we explore, among other things, their connection with the theory of Drinfel’d modules of rank one (which is function field cyclotomic theory), their interpolations, functional equations, the nature of their special values (transcendence, algebraicity, connection with “periods”) and various analogies.”

Let K be a function field in one variable over a finite field, A the ring of integers outside a fixed place \(\infty\) of K, \(K_{\infty}\) the completion, \(\Omega\) the completion of an algebraic closure of \(K_{\infty}\). The gamma functions to be considered are of two types:

1) A continuous function \(\Gamma\) from \({\mathbb{Z}}_ p\) \((p=char(K))\) to \(\Omega\), obtained by D. Goss by interpolating \(\infty\)-adically the Carlitz factorial (the latter being a function from \({\mathbb{N}}\) to K).

2) A meromorphic function (also named \(\Gamma\)) on \(\Omega\), defined by \[ \Gamma (x)=\frac{1}{x}\prod_{n\in A\quad monic}(1+\frac{x}{n})^{- 1}. \] Following the well known analogy \(K\leftrightarrow {\mathbb{Q}}\), \(A\leftrightarrow {\mathbb{Z}}\), \(K\leftrightarrow {\mathbb{R}}\), \(\Omega\leftrightarrow {\mathbb{C}}\), etc., both of these correspond to some extent to the classical gamma function. In both cases, there are analogues of the classical functional equations, the distribution formula... Moreover, in case 1, one may interpolate v-adically \((v=finite\) \(place=prime\) of A) instead of \(\infty\)-adically, which leads to v-adic gamma functions \(\Gamma_ v\). These latter have properties similar to those of the Morita p-adic gamma functions, including their relation to Gauss sums, the Gross-Koblitz formula etc.

These properties are developed in detail, following earlier work of the author. In the last section, the author proposes and discusses several conjectured (and partially proved) formulas that parallel known results of Chowla-Selberg, Gross-Koblitz, and Deligne in the number field context.

Let K be a function field in one variable over a finite field, A the ring of integers outside a fixed place \(\infty\) of K, \(K_{\infty}\) the completion, \(\Omega\) the completion of an algebraic closure of \(K_{\infty}\). The gamma functions to be considered are of two types:

1) A continuous function \(\Gamma\) from \({\mathbb{Z}}_ p\) \((p=char(K))\) to \(\Omega\), obtained by D. Goss by interpolating \(\infty\)-adically the Carlitz factorial (the latter being a function from \({\mathbb{N}}\) to K).

2) A meromorphic function (also named \(\Gamma\)) on \(\Omega\), defined by \[ \Gamma (x)=\frac{1}{x}\prod_{n\in A\quad monic}(1+\frac{x}{n})^{- 1}. \] Following the well known analogy \(K\leftrightarrow {\mathbb{Q}}\), \(A\leftrightarrow {\mathbb{Z}}\), \(K\leftrightarrow {\mathbb{R}}\), \(\Omega\leftrightarrow {\mathbb{C}}\), etc., both of these correspond to some extent to the classical gamma function. In both cases, there are analogues of the classical functional equations, the distribution formula... Moreover, in case 1, one may interpolate v-adically \((v=finite\) \(place=prime\) of A) instead of \(\infty\)-adically, which leads to v-adic gamma functions \(\Gamma_ v\). These latter have properties similar to those of the Morita p-adic gamma functions, including their relation to Gauss sums, the Gross-Koblitz formula etc.

These properties are developed in detail, following earlier work of the author. In the last section, the author proposes and discusses several conjectured (and partially proved) formulas that parallel known results of Chowla-Selberg, Gross-Koblitz, and Deligne in the number field context.

Reviewer: E.-U.Gekeler (Bonn)

### MSC:

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

11R58 | Arithmetic theory of algebraic function fields |