## A Shintani-type formula for Gross--Stark units over function fields

J. Math. Sci. Univ. Tokyo
Vol. 16 (2009), No. 3, Page 415--440.

Dasgupta, Samit ; Miller, Alison
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.