For context, I'm a sophomore high school student. I'll be done with undergrad Abstract Algebra, Topology, and Analysis by the end of the school year, and I'll be burning through the necessary graduate Analysis, (Rudin) and Algebra (Lang) over the summer for Hartshorne, and doing some Differential Geometry from Tu along with Hartshorne.
I'm hoping that by the time I make it to Diophantine Geometry, I'll be able to use it to solve Olympiad number theory problems (Diophantine Equations in particular). However, that's something I can't really find out for myself until I actually learn basic Diophantine Geometry, which will take at least two years.
So, in order to get around that, I thought "Why not ask people who know Diophantine Geometry?"
Although this question is mainly about Arithmetic Geometry, answers about other high-level math subjects applicable to Olympiad problems are welcome.
And here we are. By the way, I also enjoy the Algebra by itself, it's not like I want to slog through more for the express purpose of solving competition math problems. I just wanted to know if it was an alternative or supplement to studying classical methods.
Best Answer
Algebra and analysis: you should definitely know about the derivative and limits of sequences, though I doubt that you will need integration; continuity can help with functional equations. It doesn't hurt to have an understanding of fields and rings if you want to really understand what is going on with polynomials, like Gauss's lemma and Bezout's lemma for polynomials and irreducibility criteria.
Geometry: projective and inversive geometry; Barycrentric coordinates can be made formal using higher mathematics. Euclidean geometry is not a part of higher math. Be familiar with determinants.
Combinatorics: generating functions, including the derivative. There is occasionally a bit of graph theory though nothing very advanced. On that note, I guess knowing the general version of Ramsey's theorem isn't a bad idea. The probabilistic method is good to know. Solving homogeneous linear recurrences in general requires some linear algebra. Combinatorial Nullstellensatz and Polya enumeration theorem are probably as sophisticated as olympiad combinatorics gets.
Number theory: There are some non-trivial number theory theorems that pop up in olympiads, such as Thue's lemma, Lifting the Exponent lemma, Hensel's lemma, Zsigmondy's theorem, the solution to Pell's equation. Gaussian integers is also a possibility, as are aspects of algebraic number theory. You might consider checking out some of Titu Andreescu's books on number theory as he likes to occasionally bring in higher level stuff into his curriculum.