So, let me first start out with some geometry. I will introduce the notion of a “fundamental group”. First, take any surface. Could be a plane, could be a torus, could be a sphere, whatever. The fundamental group of that surface is the set of all closed loops that can’t be deformed into eachother. By “deformed”, I mean squeezed, rescaled, moved around, pulled, whatever. Like a rubber band.
Now, to take a primitive example - a plane. Any loop you make can be deformed into any other loop, so the fundamental group has one element (which is a function that describes the loop but we won’t get into that). If you take a torus, it’s more complicated - there are two loops, the red and the blue:
You can’t stretch and squeeze the red to make the blue, and vice-versa, so the fundamental group of a torus has two elements.
Here’s where it get’s tricky - inter universal geometry isn’t studying fundamental groups. It’s studying etale fundamental groups. It seems like a small change but it isn’t - it’s a lot trickier. You know how we defined our fundamental group by “deforming” loops? Well. it turns out there is a type of “loop transformation” that’s stronger than deformation - they’re called “deck transformations”. You don’t need to know what they are, but you just need to know that if we have a fundamental group, and we take any loop from it, a deck transformation allows us to transform that loop into any other loop in the fundamental group.
What this means is that the deck transformation defines the fundamental group - all you have to do is start out with one loop and just apply every deck transformation to it and you have the rest of the fundamental group. In this way, we can define the fundamental group of some surface as the set of deck transformations of one of it’s loops (this is just restating what I said in the previous sentence, but in a cleaner way). An etale fundamental group is what happens if you replace the “deck transformation” with something else:
Etale Fundamental Group of a surface = The set of etale transformations1 of one it’s loops2
Here, an etale transformation is simply a transformation that preserves the structure of the loop, which is a fancy way of saying that the loop can change, but some of it’s fundamental properties can’t (imagine a real rubber band - you can stretch it, sure, but not past a certain point, and you can’t really shape it any way you want - try making the shape of a giraffe with one :P)
Now, we have defined the etale fundamental group of a surface. What does this have to do with the abc conjecture? Well, the abc conjecture is equivalent to another problem called Szpiro’s conjecture. Szpiros conjecture deals with elliptic curves, which are simply equations of the form y^2=x^3+ax+b. Szpiro’s conjecture establishes an inequality between two numbers having to do with elliptic curves - the discriminant, and the conductor. You don’t need to know what these two numbers are (I’ll be happy to explain if you want), nor what Szpiro’s conjecture says. You just need to know that Szpiros conjecture deals with elliptic curves, and that elliptic curves are just that - curves. Just like a plot of x=y is a line, elliptic curves make a curve:
Szpiros Conjecture names a certain property, let’s call it “Property Szpiro” (remember this, you’ll need it later), and then says that all elliptic curves have Property Szpiro. Of course, there is no proof, because it’s a conjecture. And it turns out that Szpiro’s conjecture is equivalent to the abc conjecutre.
Now, we can think of the ‘curve’ that the graph of an elliptic curve draws out as an object, and actually take the etale fundamental group of it. Now, here, I must digress, my analogies break down. You can’t think of “loops” on a 1 dimensional curve. But, you can almost think of it as a “hack”. We define the etale fundamental group of a surface, and we formalize it mathematically, and then we realize that we can plug in an elliptic curve into it and actually get something out. Just like if you write a program to calculuate the nth fibionacci number, and suddenly you realize you can plug in negative numbers. Obviously, there is no such thing as the -2nd fibionacci number, but the “negative fibionaccis” have a pattern nonetheless. In our case, the etale fundamental group was made for surfaces, but because of the way that mathematicians formalized it, we can take the etale fundamental group of an elliptic curve and it will give us something that makes sense.
Here is where everything comes together. Mochizuki’s work is simply answering the following question:
Given an etale fundamental group of some unknown curve, how much can we figure out about the curve?
In fact, even more specifically:
Given an etale fundamental group of some unknown curve, can we re-construct the entire curve?
That’s what inter-universal geometry is. Constructing elliptic curves (or more generally, any curve) from their etale fundamental group only.
Mochizuki’s work gives an affirmative answer to that question (if his work is correct, of course). He then uses this to prove the abc conjecture by:
- Giving a way to construct an elliptic curve from it’s etale fundamental group.
- Reminding us that all etale fundamental groups have a certain property, let’s call it “Property X”. This wasn’t a new property, it was something that is pretty clear from the formal definition of etale groups.
- Morphing the set of all etale fundamental groups into the set of all elliptic curves using his method.
- Showing that “Property X”, when put through his process, becomes “Property Szpiro” (remember that?). Therefore, all elliptic curves have Property Szpiro.
- All elliptic curves having Property Szpiro means Szpiro’s conjecture is true, meaning the abc conjecture is true.
Of course, this is only the motivation behind inter universal geometry - how any of this is actually accomplished is a lot more difficult. If anyone is interested in learning, though, take a look a few short surveys that give you an idea of the math involved. Be warned, you need a strong background in class field theory and galois theory to understand any of it. As a stepping stone, try learning about the Neukirch-Uchida theorem and the Hodge-Arakelov theory of elliptic curves.
Actually, they’re called “etale morphisms”, but morphism basically means transformation, so I called it a transformation to keep the terminology consistent.
2This is kind of imprecise, just because the etale fundamental group is so hard to define. Technically, it’s the inverse limit of finite automorphism groups. But even if you know what that means, that gives you no intuition as to what the etale fundamental group is.