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I am currently discovering the algebraic geometry of Grothendieck. I have the impression that this theory, which leads to categories, schemas, topos etc. alone can encompass all modern mathematics (with the exception of probabilities). That is to say, to understand it, you really need to know everything. It also has extraordinary opportunities in the understanding of arithmetic (Pierre Deligne in the proofs of André Weil etc.).

However, I don't see any connection with the analytic number theory like the one undertaken by Dirichlet, Von Mangoldt, Chebyshev, Hardy, Littlewood, Ramanujan, and so on.

Does anyone have ideas of theorems, conjectures, or "approaches" that combine these two points of view?

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    $\begingroup$ I like the question at the end of your post, but I think that your claim "encompass all modern mathematics" is quite exaggerated. $\endgroup$ – EFinat-S Feb 24 at 14:28
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    $\begingroup$ Drinfeld's "Finitely additive measures on S2 and S3, invariant with respect to rotations" solves a measure-theoretic problem using automorphic forms and a consequence of Weil conjectures (although the latter is not crucial to the argument, one can do with a weaker version which can be proved without algebraic geometry) $\endgroup$ – user74900 Feb 24 at 16:24
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    $\begingroup$ If I remember rightly, one of Weil's original uses for the Riemann hypothesis for curves over finite fields was applications to bounding exponential sums. Grothendieck invented $\ell$-adic cohomology to prove the Weil conjectures (including the Riemann hypothesis for varieties that was eventually proved by Deligne), so it's not totally surprising that there is a connection to exponential sums. $\endgroup$ – Robert Furber Feb 24 at 23:20
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    $\begingroup$ The question would benefit from changing the speculative "I have the impression... everything." to something saying that it is connected with many subfields of maths, possibly with examples. $\endgroup$ – YCor Feb 25 at 0:36
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Do you consider $L$-functions of elliptic curves over $\mathbf Q$ (or other number fields) to be in the spirit of "analytic number theory undertaken by Dirichlet, Von Mangoldt, Chebyshev, Hardy, Littlewood, Ramanujan, and so on"? Those 19th and early 20th century folks did not have the definition, which only came much later in the 20th century, but the idea of defining such functions as an Euler product and then Dirichlet series, and seeking an analytic continuation and functional equation, is a task they would have understood. Deuring proved the analytic continuation and functional equation in a special case (CM elliptic curves) in the 1950s, but the case of all elliptic curves over $\mathbf Q$ was settled using ideas coming from the proof of Fermat's Last Theorem, hence using modern algebraic geometry.

The Sato-Tate conjecture is an analytic conjecture somewhat in the spirit of the prime number theorem. It was formulated in the 2nd half of the 20th century but could have been appreciated earlier. Like the prime number theorem, which is equivalent to nonvanishing of the zeta-function on the line ${\rm Re}(s) = 1$, the Sato-Tate conjecture was known to be a consequence of analyticity and nonvanishing of certain $L$-functions on vertical lines (boundary of right half-planes) and those $L$-function properties were proved about 10 years ago with algebro-geometric methods.

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    $\begingroup$ Could you tell where heavy estimates are needed ? One looks at the symmetric power L-functions $L(s,\text{sym}^mE)=\exp(\sum_{p^k} \frac{p^{-sk}}{k}\frac{\sin t_p k(m+1)}{\sin t_p k})$ where $p+1−\#E(\mathbf{F}_p)=2\cos(t_p)p^{1/2}$ having no pole at $s=1$ iff E has no CM, from the PNT-like asymptotic $\sum_{p \le x} \frac{\sin t_p m}{\sin t_p }=o(\pi(x))$ and the orthonormal basis for $(u,v)=\frac1\pi\int_0^\pi u(t)v(t)\sin^2(t)dt$ one finds $\sum_{p \le x}f(t_p) \sim (f,1)\pi(x)$ for any $f$ continuous, which implies Sato-Tate $\endgroup$ – reuns Feb 25 at 22:08
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    $\begingroup$ @reuns the deduction of Sato-Tate from analytic continuation and nonvanishing of symmetric-power $L$-functions on the line ${\rm Re}(s) = 1$ (which is the boundary of the half-plane of convergence of their Euler products) is discussed in the Murty-Murty book "Non-vanishing of $L$-functions and Applications." See Section 7 of Chapter IV, where the argument relies on an equidistribution theorem (Theorem 3.1 on p. 68), whose proof relies on Weyl's equidistribution theorem in a compact group (Cor. 2.2 on p. 67) and a Tauberian theorem (Thm. 1.1 on p. 7) $\endgroup$ – KConrad Feb 26 at 12:18
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    $\begingroup$ The Tauberian theorem used in the book is the Wiener-Ikehara one, but they could also have used Newman's Tauberian theorem, which gets the same conclusion (an asymptotic relation) by a less technical argument at the cost of an added hypothesis that is often easy to verify in practice. $\endgroup$ – KConrad Feb 26 at 12:20
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    $\begingroup$ Kumar Murty showed ("On the Sato-Tate conjecture", pp. 195-205 in Number Theory related to Fermat’s Last Theorem, 1982) that analytic continuation to ${\rm Re}(s) = 1$ of the $m$-th symmetric power $L$-functions for all $m \geq 1$ is enough, as their nonvanishing can be deduced from the analytic continuation. (The paper also has a result about Sato-Tate in the function field case, and unfortunately the MathSciNet review of this paper --- see MR0685296 -- mentions only the function field result.) $\endgroup$ – KConrad Feb 26 at 13:02
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There are lots of examples, so let me just tell one.

P. Deligne (1971) used Eichler–Shimura isomorphism to reduce the Ramanujan conjecture on the $\tau$ function to the Weil conjectures, that he later proved by using the full strength of Grothendieck's machinery.

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    $\begingroup$ I don't know if this is tacit in your answer, but Deligne's proof of the Weil conjectures used, in addition to the Grothendieck machinery, the method of de la Vallee-Poussin from analytic number theory. $\endgroup$ – meh Feb 24 at 18:07
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    $\begingroup$ @aginensky that I think is only true of Deligne's second paper ("Weil II"). $\endgroup$ – Piotr Achinger Feb 26 at 19:52
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You can look at Lectures on applied $\ell$-adic cohomology by Fouvry, Kowalski, Michel and Sawin : https://arxiv.org/abs/1712.03173

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From the point of view of analytic number theory the most important specific result which is proved using algebraic geometry is Burgess' bounds for character sums. The proof relies on Wiles bound for character sums, together with a rather complicated combinatorial argument. One could argue that as Stepanov, Schmidt, and Bombieri gave independent proofs of the required bounds, Weils bounds are not really required, but the "elementary" approach is certainly not easy either.

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    $\begingroup$ Wiles' bound...? $\endgroup$ – KConrad Feb 27 at 22:44

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