Cfrgtkky

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?

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.

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.

You can look at Lectures on applied $ell$-adic cohomology by Fouvry, Kowalski, Michel and Sawin : https://arxiv.org/abs/1712.03173

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.