?? green structure type .... "filtration with semi-simple (?? ...) k-alg structure on associated grades" for k given number ring ... ???? ... ?? analyzing filtration / extension in terms of torsors ??? .... ??? bit about zeta function of number ring and evaluation at integers (?? ...) and torsors (?? of ... ??? additive group ?? ....) and "tate motive" (?? ...) .... ???? ..... ???? .....

?? "original" jugendtraum .... ("absolutely absolute" case ... ?? ...) ... ?? nth roots of unity as forming sort of "galois representation" .... (?? for example n = p ... ??? ....) ???? ..... ?? thinking of extension field as covering space ... ?? f_p-vsp structure on fibers of covering space .... ?? cyclotomic field vs group alg of group of nth roots of unity .... ???? ..... ?? this group / hopf algebra as corresponding to "galois representation" .... ???? philosophy of "any ag object defined over comm ring k as giving galois rep of abs galois gp of k" ?? .....

?? explicitly working out this example in bunch of ways ..... ??? .... ??? also analogous example over imaginary quadratic base .... say gaussian integers .... 2-torsion of the lemniscate .... ???? .... ???? .... explicit hopf alg ..... ??? .... ??? ....

??? kan (?? ...) extension of "galois representation" ... ?? from complete fields to incomplete ones ... ??? ....

?? galois rep as structured "galois set" vs vice versa ..... ???? .... ?? working over finite field to get the vice versa to work better, vs working over c (?? ...) to take advantage of certain nice properties ..... ??? "getting fourier transform to work better" ??? .... ???? ..... ??? then decategorification here .... l-function as special (??) zeta function vs vice versa ..... ????? "niceness" of (certain ??) l-functions vs typical zeta functions ????? ..... ???? "character of 1d irrep" .... ???? ..... ??? periodicity .... ???? ..... ???? simple reciprocity patterns ..... ???? hmmm, bit gunnarsen tried to explain to me .... ??? check for notes on that ???? ... ?? "... coefficients of modular form ..." as stretching-it case of "reciprocity pattern" .... ????

?? galois rep .... l-function ..... ??? "how dimension varies as finite field varies" .... ????? ...... ???? .... action ... invariant subspace .... ???? ... equalizer .... kernel ....

???? "galois stack" ?????? ....... ???? vague (?? ...) idea of "thing that has l-fn as coherent sheaf of some sort ..." ???? ..... ??? relationship between "galois stack" and "ag object defined over comm ring k" (?? as manifestly abs.galois(k)-equivariant ... ???? ....) idea ...

??? 4th roots and gaussian integers, 3rd and eisenstein .... ?? jugendtraum approach here ... imaginary quadratic base .... ?? certain 1/3 or 1/4 size subgp of mult gp of .... ???? ..... ???? "alternation between randomness and order" ..... ???? .... ?? "correlation ..." ?? ....

???? "primitive" ..... ???? level slip ???? ..... ???? ...... ?? "imprimitive" ??? .... ????? ...



?? problematicness of modding out real quadratic field as additive subgp of r or c because of its non-discreteness ??? ..... ????? .....

?? understanding spectrum of given number ring .... ?? just "in itself" (?? ...), or as "galois set" ...... ???? ...... ????? .....