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Looking in admiration at Deligne's re-definition of Kloosterman sums as traces of Frobenii acting on stalks of certain complexes of sheaves defined via pull-push from $\mathbb{G}_a$ to $\mathbb{G}_m$ via $\mathbb{G}_m^n$, I realized that this kind of geometrization/sheafification of explicit elementary formulas must be going on under the hood of highly abstract-seeming notions all the time. That's after all how simply stated longstanding classical problems are often solved as well as generalized by current mathematical machinery, but I found the Kloosterman reformulation particularly direct and neat.

I'd love to know other beautiful (in the eyes of the provider, of course) instances of such geometric reformulations of elementary formulas so that one can begin to discern the art of doing so. I'm primarily interested in number theory, but examples from other areas are welcome too.

P.S.1 Can the Kloosterman reformulation somehow be regarded a case of Serre's faisceaux-fonctions dictionary?

P.S.2 I don't think of coding of (non-existent) solutions in Fermat's last theorem as elliptic curves as a good example because while extremely clever, and obviously fruitful, it doesn't seem to me intrinsic or natural (hard-to-define characteristics I admit.) For the sake of delimiting the scope of the question too I'd like to keep it to geometric (or even just sheafy) re-interpretation of simple explicit formulas.

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    $\begingroup$ I must disagree (courteously) with your PS2. Both the encoding of Fermat's last theorem into an elliptic curve, and the rncoding of the Mordell conjecture into the Shafarevich conjecture (which is what Faltings used) are examples of encoding the solutions to a Diophantine equation as points on a moduli space, and then using the moduli structure as a fundamental tool in analyzing the solutions to the original problem. Of course, this is somewhat oversimplified, but helps to explain why the proof of FLT through elliptic curves is not simply an isolated trick. $\endgroup$ – Joe Silverman May 22 at 20:51

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