we discuss a refinement of rubin's integral version of stark's conjecture for abelian $l$-functions of arbitrary order of vanishing at $s=0$. this generalizes the $v$-adic refinement of the abelian, order of vanishing $1$ integral stark conjecture formulated by gross in the 1980s, and predicts a link between special values of derivatives of $p$-adic and global $l$--functions. time permitting, we will also show how our refinement relates to a recent strengthening of gross's conjecture due to tate.