Aleksandar Bahat
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Registered User
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Mar 16 |
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Where was the arithmetic zeta function of a scheme first defined? Changed tags |
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Mar 16 |
asked | Where was the arithmetic zeta function of a scheme first defined? |
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Mar 11 |
awarded | ● Good Answer |
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Feb 22 |
awarded | ● Fanatic |
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Feb 14 |
awarded | ● Civic Duty |
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Feb 8 |
awarded | ● Autobiographer |
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Feb 7 |
comment |
What do theta functions have to do with quadratic reciprocity? Correction: that should read "generating functions" rather than "generated functions". |
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Feb 7 |
awarded | ● Critic |
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Feb 6 |
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Algebraic number theory: building and simplifying The problem is, however, that schemes and cohomology could just as well fall under the "building new things" category... |
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Feb 6 |
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Algebraic number theory: building and simplifying I'm not sure if this counts as a "simplification" per se (which is why I'm posting it as a comment), but certainly arithmetic geometry can make some number theoretic results more "natural", at least for someone like me who likes to think geometrically. Although schemes and cohomology may not fit your criterion of accessibility to undergraduates, they could rightfully be considered a means of simplifying parts of algebraic number theory. |
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Feb 6 |
awarded | ● Commentator |
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Feb 4 |
awarded | ● Scholar |
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Feb 2 |
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fourier analytic proofs I believe the first appearance of this proof is in the paper "Uber die Gausschen Summen" by Issai Schur. |
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Jan 31 |
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What do theta functions have to do with quadratic reciprocity? I suspected there was something Langlands-y about this. That's probably the best "big picture" explanation. Unfortunately, I don't know enough about Langlands to get very far with this. I like what you're getting at in the last paragraph; Jacobi's proof is "natural" (in the sense that generated functions are natural), and so I guess it's not a big leap to try to generalize that. |
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Jan 31 |
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What do theta functions have to do with quadratic reciprocity? I've heard of Hecke's generalization before, but I still feel my "Why should we expect..." question is unresolved. Although the usefulness of analytic functions in number theory is no longer surprising to me, I'd like to understand specifically why somebody would see quadratic reciprocity and think, "Hmm, $\theta(z)=z^{-1/2}\theta(1/z)$ is relevant." Why this piece of analysis in particular? |
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Jan 28 |
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What do theta functions have to do with quadratic reciprocity? Interesting! I didn't know about Dedekind sums. I'll have to read up on this some time soon. Yet another piece of the grand puzzle... |
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Jan 28 |
awarded | ● Nice Question |
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Jan 28 |
asked | What do theta functions have to do with quadratic reciprocity? |
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Jan 5 |
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Can we categorify the formula for the quadratic Gauss sum? Thanks for your comment. I've heard of that result before somewhere, but never managed to track down a reference. It's very interesting, but doesn't really answer my question. If I had some result about objects that decategorified to this Gauss sum identity, then this paper might give me a proof "for free"; the problem is, I don't know what to prove about the objects yet! |
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Jan 3 |
awarded | ● Student |
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Jan 3 |
asked | Can we categorify the formula for the quadratic Gauss sum? |
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Jan 3 |
awarded | ● Nice Answer |
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Jan 2 |
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New grand projects in contemporary math Also, thank you for pointing that out from the paper. There is definitely still a long way to go before we "really find" $\mathbb{F}_1$, but from what I know, this is one of the most substantial theoretical developments so far. |
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Jan 2 |
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New grand projects in contemporary math Well, if Durov's definition solved the Riemann hypothesis, I think we would have heard! |
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Jan 2 |
answered | New grand projects in contemporary math |
