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Determination of the algebraic relations among special \(\Gamma\)-values in positive characteristic. (English) Zbl 1064.11055

A well-known conjecture of Rohrlich states that all multiplicative dependence relations among values of the Euler Gamma function at rational arguments follow from the standard functional equations. This conjecture has been strengthened by S. Lang into a conjecture of algebraic independence: all linear equations with algebraic coefficients among monomials in the numbers \(2i\pi\) and \(\Gamma(s)\) with \(s\in\mathbb Q\), \(-s\not\in\mathbb Z\) follow from the two-terms relations provided by the Deligne-Koblitz-Ogus criterion. These conjectures seem out of reach. However the authors succeed in the present paper in proving the precise analogs of these conjectures for Thakur’s geometric Gamma function over the field \(k=\mathbb F_{q}(T)\): they determine the exact transcendence degree of the field generated by elements \(\omega\) and \(\Gamma(s)\) where \(s\in k\), \(s\not\in A_{+}\); here \(A_{+}\) denotes the set of monic polynomials in the ring \(A=\mathbb F_{q}[T]\) while \(\omega\) is Carlitz’ \(\mathbb F_{q}(T)\)-analog of \(2i\pi\). The proof of this deep result involves dual motives, Coleman functions, geometric complex multiplication and Jing Yu’s transcendence method.
This paper represents an important step in the theory of transcendence in positive characteristic. The method has been further developed by the third author [Matthew A. Papanikolas, Tannakian duality for Anderson-Drinfeld motives and algebraic independence of Carlitz logarithms, math.NT/0506078 (http://arXiv.org/abs/math/0506078)].

MSC:

11J93 Transcendence theory of Drinfel’d and \(t\)-modules
11G09 Drinfel’d modules; higher-dimensional motives, etc.
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