It seems to me that if I have a vertical antenna with the ATU at the
base of it with coax back to the shack, that all I need is a series
inductor to tune it for frequencies for which it is shorter than 1/4
wave and a series capacitor for frequencies for which it is longer
than 1/4 wave.

What do I do, however, for those frequencies for which it is close
to a half wave? PI network? Or L network?