?? "polynomial with coefficients in sections of anti-tautological line bundle over p^1 ...... ????? .....

?? by degree ....

??? 0 ....

constant ....

degenerate ... no roots ... "inconsistent" ... 0=1 .... all fibers empty .... ??? ....

1

?? c + [ax+by]z = 0 ... ... ?? don't seem to have "homogeneous" aspect straight here yet .... ??? .....

?? variable / unknown / new generator z in grade 1 ... with x and y ..... ????? .....

?? equation to live in grade n ... ?? say n=3 for now .... ??? ....

?? coefficient of z^0 should live in grade n ??

?? coefficient of z should live in grade n-1 ??

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

az + bx+cy = 0

az^2 +[bx+cy]z + [dx^2+exy+fy^2] = 0 .... ???? .....

??level slip here ... ?? ... concerning whether we're thinking of p^1 as analogous to moduli stack of ec's (?? and contemplating adding extra structure to the ec's ... ?? probably more or less what we thought we had in mind when we started here ?? ...), or as analogous to particular ec .... (????? case where that particular one is the generic one ??? .... ????? ......) .... (and contemplating it being double cover of (??? the / a ????? .... ???? ....) p^1 ... (?? identity analogy of p^1 hsowing up here ??? ....) ... ??? or also, now that i think about it, contemplating "hecke-flavored" covering maps to / from other ec's .... ???? ..... ?? connected of course with extra structure mentioned in other parenthetical remark above ... ?? lots of level-slipping here ... ???? ....) .... ??? .....

?? ag theory of line object l equipped with g2 : l^2 -> 1 and g3 : l^3 -> 1 ....

?? l as "generic cqa" ... more or less ...

?? other objects and so forth in here ... connected with fleshing out ec corresponding to cqa .....

?? ag theory of line object l equipped with g2 : l^2 -> 1, g3 : l^3 -> 1, x,y : 1 -> 1, y^2 = x^3 + g2*x + g3

?? ....

?? this theory as "ag propositional" relative to the first one ??? ... ?? and thus ??? ... ?? corresponding to commutative monoid in the first one ?? ... ?? hopefully in evident way .... ???? ....

??? intermediate ag theory here ... ??? .... ag propositional relative to first, second ag propositional relative to it in turn ... ??? ....

line object l, g2 : l^2 -> 1, g3 : l^3 -> 1, x : 1 -> 1

?? corresponding to hopefully evident comm monoid object in first theory .... ??? hmm, evidently really external comm monoid, manifestly .... ???? ..... (?? conflict / confusion here ??? ... ??? .... ?? maybe not too bad ?? ... ??? how much did we used to know about for example taking x as covering projection instead of y ?? .... ???? usual special "2" confusion about "taking y ..." as equivalent to "forgetting y ..." and thus to "focusing on x ..." .... ????? ....)

?? ag theory of "commutative algebra of polynomials in x" .... ??? ..... ?? trying to express property that "spectrum is discrete 2 except at 4 special points ..." .... ???? ..... ???? .... ?? i was going to say that that "externality" conflict and confusion that i mentioned was happening here, but ... ??? maybe it's really ok ??? ... ?? via ... ?? y ??? .... ??? ... well, nevertheless i'm confused ... but ... ?? somewhat hopeful of getting unconfused .... ??? ...

?? plain old (pointed ...) ec as .... lying naturally over ... not _the_ p^1, but _a_ p^1, right ??? ..... here's some confusion .... ???????????????? ....... ?? hmm, so maybe we forgot about the "intermediateness" here ... ?? we have _x_ in the theory, right ??? ...... ?? but still confusion about "x as variable vs as constsnt" .... ???? ...... ?? seems like sort of both ??? ..... ?? maybe bit about "some variables as more variable than others (aka parameters ...)" ... ?? ... ?? relationship to "intermediateness" here ??? .... ??? ..... ???? .... still confusion .... ??? ..... ????? ......

?? if straighten this out should try to work "h" and "theta function" into this ...

?? idea of program to relate structure on ec to structure on cqa as maybe becoming "tautological" (... ?? or at least ... make some progress ... ??? ...) if things get straightened out here .... ???

??? idea of actually thinking of complex p^1 as real s^2, and trying to organize theory (?? ...) this way .... ???? always use same 4 branch points and same covering torus, but vary conformal structure on 2-sphere ... and hence on torus, by pulling back ... ??? hmmm, "how much" do you have to vary the conformal structure on the 2-sphere here to get the variety (?? ....) you need ???? .... ???? ..... ?? "ellipsoidal" structures seem not to give enough variety because ... hmm, actually not sure about that ... ?? confusing / confusion here ??? ....

?? "logic programming" ... ???? ....

?? "ag theory programming" ... ??? ....

?? endo-span of p^1 given by ec .... ????? ..... ?? possibility of intuitive geometric description ... ??? "for most points, particular pair of other points ... except in 4 special cases the pair degenerates ..." ... ??? .... ?? not much symmetry to exploit though .... ??? .....

?? "branched" (??? ....) double cover of real p^1 with branch points at 0, 1, -1, infinity .... ???? .... ?? hmmm .... ??? "branching" manifesting as "zig-zagging" here ??? ..... ?? or maybe not ??? .... equator ?? 2 completely separate "perfectly formed" equators .... ??? ... perhaps right the first time ... trying to remember the picture i sketched the other day ... ok, branch point seems to manifest as "4-valent vertex" .... ???? .... hmmmm ..... ???? .... "rectangle boundary with 2 side edges for each one original" ... ?? eisenstein case ??? .... ?? ??? real coefficients vs real roots here ????? ...... ???? .....

?? hmmm, confusion between .... ?? real elliptic curve, and part of complex elliptic curve lying over real projective line under projection to complex projective line ... ?? "meridianal equators vs longitudinal equators" ... ??? real vs imaginary square roots of real quantities .... ?? definitely still some confusion here, but seems like not too far from straightening it out .... ??? .... ?? x^3-x as positive from 1 to infinity, and from -1 to 0 ..... ??????? extent to which this resolves some old confusions we had about ... ?? knowing that real ec has to be ab gp but thinking that it looked like it had singularities ... ??? ... ??? ....

?? gauss square ... ??? eisenstein rhombus ??? .... ???? ??? other rhombuses though .... other rectangles besides square ........ ?????.... ?? sides of modular triangle ... ??? ..... ?? what's the third side ??? ... ??? eisenstein and "tate" ... ??? ... ?? particular visual commonality between the parallelograms (??? ...) doesn't jump out at me .... ????? ..... ?? not sure how much i might have things screwed up here .... ???? .....

??? modularity and buckyballs .... ????? .... ??? simplex-based .... ???? ....