I'm really not a π day hater, but I do roll my eyes when I read articles exclaiming π "goes on forever!" as if that is a rare and notable property that makes it unique among numbers. I mean, 1/3 "goes on forever" too.
Yeah. And the key here is that Eisenstein unites all these threads by stressing periodicity, which he enforces via what is now called "averaging over the group" (in this case, the additive group of the integers.) Hard to exaggerate how far these ideas reach.
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So what does this look like for other groups?
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Exactly the right question! The next group Eisenstein studied was the translation group in two dimensions, and its discrete subgroup of "integral translations." That gave him his theory of elliptic functions. Then you look at the modular group, and get automorphic functions.
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