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.
