# Special Values of the Goss L-function and Special Polynomials

- Publication Year:
- 2010

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Learn more- Repository URL:
- http://hdl.handle.net/1969.1/ETD-TAMU-2010-08-8251

- Author(s):

- Tags:
- Function Fields; Drinfeld Modules; L-functions

##### book description

Let K be the function field of an irreducible, smooth projective curve X defined over Fq. Let [lemniscate] be a fixed point on X and let A [a subset of or is equal to] K be the Dedekind domain of functions which are regular away from [lemniscate]. Following the work of Greg Anderson, we define special polynomials and explain how they are used to define an A-module (in the case where the class number of A and the degree of [lemniscate] are both one) known as the module of special points associated to the Drinfeld A-module [rho]. We show that this module is finitely generated and explicitly compute its rank. We also show that if K is a function field such that the degree of [lemniscate] is one, then the Goss L-function, evaluated at 1, is a finite linear combination of logarithms evaluated at algebraic points. We conclude with examples showing how to use special polynomials to compute special values of both the Goss L-function and the Goss zeta function.