?? duality .... reciprocity .... ??? .... ?? "symbols" of certain sorts as matrix coefficients for some sort of fourier transform ?? .... ???? ...

?? nth roots of unity ... multiplicative group of z/n ..... ???? ......

?? fourier duality between multiplicative group of quotient of number ring, and corresponding (?? ...) torsion subgroup of abelian variety .... ????? ..... ???? .... ???? galois ambiguity .... naturality ...... ?????? ...... ??? confusion here ???? ..... ???? attempted bit about jugendtraum as saying something like "galois correspondent of x as practically same thing as x" ... ??? except with duality twist ???? .... ??? well, but why not just "untwist" ???? ... =?= use different basis (???) ??? ........ confusion ... ???? ...... ?? hmmm, try to straigthen out .... "quadratic part of given cyclotomic extension" .... ???? ..... ????? .......

?? "artin symbol" ..... ?????? "kronecker symbol" ?????? ...... "jacobi symbol" .... ???? .....

?? subgroup / quotient group relationship ... with self vs with fourier dual .... ???

?? allegation that .... ???? ... ?? explicit fourier duality approach developed by tate reduces exceptionalness of archimedean places ?? .... ???? ....

??? toric geometry as relative of fourier duality .... ???? ....