Perhaps the problem is more with the ash rather than the annuals. Some ash species have allelopathic chemicals which sensitive plants will react to. In ash, allelopathy has been observed in the root systems.

And maybe the annuals have a difficult time competing for water, as well. Ash are well known for killing out other tree saplings, let alone annuals.

Anyway, something to think about?

I have heard that, but have 2 gardens under ash trees in different parts of the yard, and the shade perennials are doing very well. The large one has tulips, daffodils, hosta, astilbe, geranium sanguineum, and many others: the other garden is just a ring around 1 tree with heuchera and brunnera. In both the perennials are flourishing. Are annuals just more sensitive to these tree competition-killing chemicals?

As far as watering is concerned, I do water, but not heavily. The grade of one is terraced to catch the rain run off from a neighbors yard. I figure if hosta and ferns in one and heuchera in the other are doing well there must be enough moisture for most things. I can always spot water annuals if I put them in.

How about one of the short nicotianas? I've had them do well in pots on my porch in the shade, so they might do well under a tree.
Remy

Tuberous begonias as opposed to the wax begonias - nice foliage and splashy blooms. And if you're of the mind to, the tubers can be dug up and wintered over like dahlias or caladiums.