Hello,
Thanks Bahner for your answer. I am sorry for the late reply, but I was in vacation.
The town mark which is present on the lid, on the rim and on the base is exactly the same, but different parts of it have been erased on each of them. But they are the same 'pyr'. I tried to rebuild the mark by taking each legible part of the 3 marks and I obtain the following drawing (which is not exact but approximated) :

When I search this hallmark I have only at disposal the 'book of old silver' by Seymour Wyler in which Augsburg marks are documented.
I find only two candidate marks which are close enough. Mark #144 (documented as 1625-1630) and mark #201 (documented as 1696-1705).
But I am not fully satisfied with these marks and I hesitate.
Here are their drawings :

If I look at the maker's mark, I find two candidates in the same book. Mark #499 (documented as Hans Andreas Antoni, 1630 ?) and mark #526 (documented as Hans Peters 1672 ? or Hans Christoff Petters, 1677). Here are the drawings of these marks:

Thus we could have a good fit for both makers, and either the cup is by Antoni, 1630 or it is by Peters, 1696.
But I am not fully satisfied with these matches, and I would need confirmations for these marks (town and maker) using more specialized references.
I was also wondering when the tremolierstrich has been used for the first time in Augsburg, and if it was usual to have it at that time.
Any suggestion will be highly appreciated.
Thanks,
Dinio