A Shintani-type Formula for Gross–stark Units over Function Fields

Let F be a totally real number field of degree n, and let H be a finite abelian extension of F . Let p denote a prime ideal of F that splits completely in H. Following Brumer and Stark, Tate conjectured the existence of a p-unit u in H whose p-adic absolute values are related in a precise way to the partial zeta-functions of the extension H/F . Gross later refined this conjecture by proposing a formula for the p-adic norm of the element u. Recently, using methods of Shintani, the first author refined the conjecture further by proposing an exact formula for u in the p-adic completion of H. In this article we state and prove a function field analogue of this Shintani-type formula. The role of the totally real field F is played by the function field of a curve over a finite field in which n places have been removed. These places represent the “real places” of F . Our method of proof follows that of Hayes, who proved Gross’s conjecture for function fields using the theory of Drinfeld modules and their associated exponential functions.