Algebraic Relations among Goss's Zeta Values on Elliptic Curves

Abstract

In 2007 Chang and Yu determined all the algebraic relations among Goss's zeta values for the rational function field - these are also known as the Carlitz zeta values. Goss raised the problem about algebraic relations among Goss's zeta values for a general base ring A, but very little is known. In this paper we develop a general method and determine all algebraic relations among Goss's zeta values for the base ring A attached to an elliptic curve over a finite field. To our knowledge, these are the first non-trivial solutions of Goss's problem for a base ring whose class number is strictly greater than 1.