# Explaining genus to students

I need to do a presentation on my thesis, which is in arithmetic geometry. This presentation is meant for all students of mathematics, but I will assume some knowledge of abstract algebra (i.e. groups, rings, fields).

In the first part of my presentation, I want to explain what Faltings' Theorem is and why it is important (and nice). Defining a curve $C$ is not too hard to do (although I skip over some details) and defining $C(k)$ (or I might just stick with $C(\mathbb{Q})$) for $k$-rational points on the curve. I then introduce some examples of interesting curves, such as the circle, an elliptic curve, and hyperelliptic curves.

I'm having difficulty however, to explain in simple terms the concept of the genus $g(C)$ of the curve $C$. Most intuitive might be too think of it topologically "as the number of holes" and use the Weierstrass $\wp$ function to explain how we can think of the elliptic curve $\mathcal{E}$ as some torus $\mathbb{C}/\Lambda$. This takes up quite a lot of time, though.

Is there an easier, shorter, way to explain the genus $g(C)$ or might I be able to skip the whole genus and explain Faltings' in some other way?

• If it isn't too tangential, mentioning elliptic curve cryptography might help keep the interest of some students. If this is a talk aimed at undergraduates, you shouldn't worry too much about precision (since it would be impossible to achieve). The best you could hope for would be to convey 1 or 2 solid ideas together with the general flavor of the branch of mathematics. What is wrong with the number of holes approach? That would be one of the few things that students can visualize. – John Coleman Sep 27 '17 at 15:41
• Using the number of holes approach would take quite a lot of time, as you cannot properly visualise the number of holes when you write $\mathcal{E}$ as say $y^2 = x^3 + Ax + B$. So you'd have to explain or at least try to convince the students that we can count the holes of the corresponding torus $\mathbb{C}/\Lambda$. As I only have 45 minutes (and I would like to get to Faltings' in say, 20 minutes, I fear this might take a lot of time. – Krijn Sep 27 '17 at 16:05
• Don't worry about properly visualizing it. You don't need to worry about convincing the student that it is possible to count the holes in this context. They will take it on faith. Really, they have no alternative. It is somewhat inevitable that you will to a greater or lesser extent lose all of the students within a few minutes. Your goal should be lessen the extent to which they are lost, and to communicate a few ideas which shine through the unavoidable fog which is induced by too many definitions with not enough time to assimilate. – John Coleman Sep 27 '17 at 16:17

One thing I never do unless I'm talking to number theorists or algebraic geometers is get into the details of complex tori, the Weierstrass $\wp$-function, or anything like that. What is usually do for explaining the genus is something like this: