  We present a deterministic distributed algorithm, in the LOCAL model, that
computes a $(1+o(1))\Delta$-edge-coloring in polylogarithmic-time, so long as
the maximum degree $\Delta=\tilde{\Omega}(\log n)$. For smaller $\Delta$, we
give a polylogarithmic-time $3\Delta/2$-edge-coloring. These are the first
deterministic algorithms to go below the natural barrier of $2\Delta-1$ colors,
and they improve significantly on the recent polylogarithmic-time
$(2\Delta-1)(1+o(1))$-edge-coloring of Ghaffari and Su [SODA'17] and the
$(2\Delta-1)$-edge-coloring of Fischer, Ghaffari, and Kuhn [FOCS'17],
positively answering the main open question of the latter. The key technical
ingredient of our algorithm is a simple and novel gradual packing of
judiciously chosen near-maximum matchings, each of which becomes one of the
color classes.
