?? in discussion with todd this morning, we found that we had slightly different approaches ... which however we could pretty straightforwardly relate to each other ... todd taking a more "legendre symbol" approach, with me taking a more "galois correspondence" approach ... ?? but so then there are various ideas which seem particularly obvious over on todd's side of the bridge, which it seems like it would be good to understand how they relate to my side of the bridge ... legendre symbols as forming character of (?? galois ir)rep ... tensor product of those irreps ... "artin reciprocity" .... ????.... representation-theoretic approach to galois group ... ???? lambda ring structure of character ring, and galois snake eating its own tail ?? ... ??? .... ?? zeta function and l-functions ... ?? factors (?? ...) corresponding to ideal classes ... ????? .... ??? .... ??? character of regular rep factoring into factors for irreps ... ???? .... ???? ....

?? also, seems like it would be a really good idea if i could get a working copy of mathematica real soon, for use with todd and summer course students ... ??? ....

x^2 + 5*y^2 ....

0 1 4 9 16 25 36 49 64 81 100

5 6 9 14 21 30 41 54 69 86 105

20 21 24 29 36 45 56 69 84 101 120

45 46 49 54 61 70 81 94 109 126 145

80 81 84 89 96 105 116 129 144 161 180

125 126 129 134 141 150 161 174 189 206 325

180 181 184 189 196 205 216 229 244 261 280

245 246 249 254 241 270 281 294 309 326 345

320 321 324 329 336 345 356 369 384 401 420

405 406 409 414 421 430 441 454 469 486 505

500 501 504 509 516 525 536 549 564 581 600

?? "congruence subgroup" ... ??? gl(1) vs sl(2) ... ??? ..... ???? ... ??elements of finite order in affine group gl(1) vs in "elliptic curve" .... ???? ..... ????....

fearless symmetry ... ??? .... ash & gross ... ?? ....

??? free AG theory on dimensional theory of fractional ideals ... vs AG theory with syntactic category given by modules ... ??? ....

"zeta vs theta" .... ???? ..... ????? ......

???any chance of .... ???derived category interpretation of l-function to fit with (joyalesque ...) "combinatorial" interpretation of zeta function ??? ... ?? ... "modulis stack ... " ... ???? .... ??? .....

?? trying to understand "analytic class number formula" (?? saying something about zeta function rather than l-function, at least naively perhaps ?? .... ??though presumably interesting to relate to l-functions ... ??? ....) in terms of ... ??? joyalesque stuff ... ??? ..... ???? .....