Hard number theory problems that can be easily solved using algebraic geometry techniques I'm very interested in number theory, but it happens that now I'm taking an Algebraic Geometry course.

I was wondering what kind of number-theoretic problems could be easily solved using some skills and machinery from the algebro geometer theory.
I am not completely sure how to read the "hard" and the "easily" in your question, but as a starter you may consider the following (well-known) proposition.

There are infinitely many non-equivalent pythagorean triples.

This can be proven really easily (a brilliant high school student will understand the argument) observing that the set of inequivalent pythagorean triples is basically in one-to-one correspondence with the set of points with rational coordinates in the circle $x^2+y^2=1$ and the latter set is in one-to-one correspondence with the set of lines with rational slopes through the point $(1,0)$.

This is a basic application of the (geometric!) fact that a non-singular conic is isomorphic to the projective line, with a bit of arithmetic thrown in.
The most crazy example: Fermat's crazy Theorem.