TITLE: Manageable project to learn some arithmetic geometry
QUESTION [18 upvotes]: OK, this may be an unusual question, since I'm asking you for advice, the way I would ask a supervisor were I a grad student.
I am a thirty-something maths teacher with an unresolved attraction to arithmetic geometry, and would like to know more about themes I find fascinating when I read texts about arithmetic geometry, such as:

the local-global principle
the way geometry influences the arithmetic behaviour of Diophantine equations, like in Faltings's theorem
the way scheme theory helps number theory by making a lot of geometric constructions possible, starting from base change

One of the reasons this attraction is unresolved is that I never managed to seriously learn enough algebraic number theory, algebraic geometry, etc. in a "bottom-to-top" way: I have often started to read for instance R. Vakil's The rising sea, D. Eisenbud & J. Harris The geometry of schemes or Q. Liu's Algebraic geometry and arithmetic curves  and find them to be very good texts, but I don't have the stamina to ingest hundreds of pages without a more specific goal.
Because of this there are a number of subjects I'm vaguely familiar with, but don't know in any depth. For instance, I know what a scheme is, I even think I understand in part their raison d'être, but am probably unable to pass an scheme theory exam (and the same goes for elliptic curves, derived categories, class field theory...).
To change this, I would like to try and understand a specific result and working my way from the top down to acquire the needed skills. The result has to be interesting enough to preserve my motivation in the long run, but "small" enough so that I could realistically understand it in a few months. I would love it to really mix number theory and geometry.
So the question is:

what theorem would constitute a reasonable-size project?

Personal facts that might be relevant: my PhD is in low-dimensional topology, so I'm quite comfortable with things like cohomology and Riemann surfaces, and I like it when geometric intuition is useful. Large doses of homological algebra or category theory don't frighten me.

REPLY [5 votes]: This is complementary to hm2020's answer, in that learning about the Hurwitz theorem and Riemann Roch are fantastic places to see some beautiful geometry. I think one more arithmetic direction this can go is to prove the (easier parts of) the Weil conjectures for smooth curves over finite fields. These conjectures motivated massive amounts of algebraic geometry, and are very down to earth, its all just counting solutions to equations over finite fields!
This is all done very well in Dino Lorenzini's book An Invitation to Arithmetic Geometry, and he goes out of his way to emphasise the arithmetic=geometry dictionary.
In the last chapter he even proves the hard part of the Weil conjectures, the Riemann Hypothesis for curves, which gives a nice perspective of why the integer version might be true (that the primes are as well distributed as possible).