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I love asarum and grow a number of the chinese/japanese species and cultivars. I'm no expert, but I'd look at shuttleworthii and/or asaroides. The color of the leaf is not likely to be determinative.
And takoi is an unlikely but possible third option for the i.d. I grow the Japanese and Chinese species in pots--outdoors in a morning sun/afternoon shade situation during the season, and on a cool north facing windowsill for the winter. My thinking is that heat is problematic for them, especially in pots, so your summer should be just fine. They seem to respond well to being watered like orchids...almost dry before you give in and soak them again. When they show slight wilt, I give them a good drink and then leave them be.
My experience here in USDA zone 7 Maryland runs a bit counter to that, Carlo. I've had trouble with most asarums in heavy shade: they simply don't do that well, although they do survive. Last year I was moving things around in the garden and had some potted plants of Asarum nobilissimum and A. maximum which were placed - with the intention that they would be there only temporarily - in a frame which got full sun for much of the day. I got busy, and the roots grew from the pots into the ground. Seeing that the plant were now firmly rooted in the new spot, I left them there in the full sun throughout the growing season. These same plants spent the winter in a sunny cold frame: I'm convinced that they can handle sun very well - and benefit from it. If I forget to water them, they droop; but I've never seen any permanent foliage damage, and the leaves are large and healthy. So I think your advice about watering them like orchids agrees with my experience. But there is no doubt in my mind about the heat tolerance of the two species mentioned: they can take it!
Good to know. I'm sure that additional sun will improve flowering...What do you think about a winter rest in a shadier spot? I've got A. minimitanianum blooming there almost all winter long (and it's blooming again now!).