tailieunhanh - Đề tài " Determination of the algebraic relations among special Γ-values in positive characteristic "

We devise a new criterion for linear independence over function fields. Using this tool in the setting of dual t-motives, we find that all algebraic relations among special values of the geometric Γ-function over Fq [T ] are explained by the standard functional equations. Contents 1. Introduction 2. Notation and terminology 3. A linear independence criterion 4. Tools from (non)commutative algebra | Annals of Mathematics Determination of the algebraic relations among special r-values in positive characteristic By Greg W. Anderson W. Dale Brownawelln and Matthew A. Papanikolas Annals of Mathematics 160 2004 237 313 Determination of the algebraic relations among special r-values in positive characteristic By Greg W. Anderson W. Dale Brownawell and Matthew a. Papanikolas Abstract We devise a new criterion for linear independence over function fields. Using this tool in the setting of dual t-motives we find that all algebraic relations among special values of the geometric r-function over F T are explained by the standard functional equations. Contents 1. Introduction 2. Notation and terminology 3. A linear independence criterion 4. Tools from non commutative algebra 5. Special functions 6. Analysis of the algebraic relations among special n-values References 1. Introduction . Background on special r-values. . Notation. Let F be a field of q elements where q is a power of a prime p. Let A F T and k F T where T is a variable. Let A c A be the subset of monic polynomials. Let PF be the unique valuation of k for which ITF q. Let k F 1 T be the I F-completion of k let k be an algebraic closure of k x let C be the I I -completion of k x and let k be the algebraic closure of k in C . The second author was partially supported by NSF grant DMS-0100500. 238 G. W. ANDERSON W. D. BROWNAWELL AND M. A. PAPANIKOLAS . The geometric r-function. In Th Thakur studied the geometric T-function over Fg T r z 1 n 1 ny1 z G C n A which is a meromorphic function on C . Notably it satisfies several natural functional equations which are the analogues of the translation reflection and Gauss multiplication identities satisfied by the classical Euler T-function and to which we refer as the standard functional equations see . . Special r-values and the fundamental period of the Carlitz module. We define the set of special r-values to be r z I z G k -A U 0 c k. Up to .