?? was thinking about "moduli stack of elliptic curves" and various mental pictures of it .... ??? .... ?? got to thinking that it wasn't purely just matter of "a's classified up to b-equivalence vs b's classified up to a-equivalence" .... ????.... ?? rather ... ???? situation where certain actual space of a's gets re-intepreted as certain actual space of b's .... ???? .... ?? "gauge-fixing" and "in the presence of a c, a's and b's are equivalent" .... ????for example in the presence of a frame, just about anything is equivalent to just about anything else .... ?????? ......

??? specific example in mind here .... upper half plane as corresponding to .... ????what ???? .... well, points in the upper half plane, i guess, but how did they get involved here, exactly??? ..... they generate lattices extending the real integers .... ?????.... ??? but meanwhile i think that i have this other sort-of equivalent picture of the unit disk representing nice quadratic forms on something or other .... ????? ...... .... ?????...... ??so what _is_ going on here?? ... ???what is the space of lattices in C extending the real integers like ??? .... vs certain hopefully obvious usual quotient space .... ???? ......

presumably i've known the answers to some of theze questions before ... ???

well, actually there's enough similar questions that it's hard to remember which ones i actually knew the answers to ...

?? for example space of all lattices in the plane ....

?? trefoil complement .....

??? ..... ????? ......

??? various kinds of "normalization" / "gauge-fixing" ... ??? ..... projectivization .... real vs complex .... ????? .....




??? consider .... ??? most (?? ...) general sort of equivalence of stacks between pair of orbit stacks ... ??? whether in that (?? ...) generality it can be given some sort of conceptual interpretation .... ????? ..... ??? ...



??decoration on torsor ... ??? abelian case .... ?? "stripes" .... ????? ..... ??? "normal form for decoration" .... ???? .... ??? .....