“We mainly believe that all conjectures are real, but it’s exciting to see that it actually comes true,” said Ana Caraiani, a mathematician at Imperial College London. “In situations where you really think you won’t be able to reach.”
This is just the beginning of hunting, which will take years – mathematicians ultimately want to show modularity for each Abelian surface. However, the results can already help answer many open questions, just as the modularity of elliptic curves proves various new research directions.
By looking at the glass
Elliptic curves are a particularly basic type of equations, using only two variables –x and y. If you draw its solution, you will see curves that look simple. But these solutions are interrelated in rich and complex ways and appear in many of the most important problems of theories. For example, the conjecture of birch and Swinnerton-Dyer is one of the most difficult open problems in mathematics, and the reward to prove this first is a $1 million reward, which is about the nature of the solution to the elliptic curve.
Elliptic curves may be difficult to study directly. Therefore, sometimes mathematicians prefer to approach them from different angles.
This is where the modular form is. The modular form is a highly symmetrical function, and areas called analysis appear in surface-separated mathematical research areas. Because they show many nice symmetry, modular forms can be easier to use.
At first, these objects shouldn’t seem to be related to it. But Taylor and Wiles’ proof shows that each elliptic curve corresponds to a specific modular form. They share certain properties in common – for example, a set of numbers describing an elliptic curve solution will also appear in their associated modular form. Therefore, mathematicians can use modular forms to gain new insights into elliptic curves.
But mathematicians believe that Taylor and Wells’ modular theorem is just an example of a universal fact. There are more general object categories beyond elliptic curves. All these objects should also have companions in the wider world of symmetric functions, such as modular forms. Essentially, that’s all about the Lanlanz plan.
There are only two variables in the elliptic curve –x and y– So the graphics can be drawn on flat paper. However, if you add another variable, zyou will get a curved surface that lives in three-dimensional space. This more complex object, called the Abelian surface, has a solution like an elliptical curve with gorgeous structures that mathematicians want to understand.
It seems natural that the Abelian surface should correspond to a more complex modular form. However, the extra variables make them harder to build and it’s hard to find their solution. Prove that they also satisfy the modular theorem seems completely out of reach. “It’s a known issue, don’t think about it because people have already thought about it and are in trouble,” Gee said.
But boxers, Calegary, Ji and Piloni want to try it out.
Finding a Bridge
Calegari said all four mathematicians were involved in the Langlands project research and wanted to prove one of the conjectures that “the objects that actually appear in real life, not something strange.”
In real life, not only does the real life of mathematicians appear on the surface of Abel, that is, proof that modular theorems of them will open up new doors to mathematics. “If you have a statement like this, there are a lot of things you can do, no other chances,” Calegary said.
Mathematicians have collaborated since 2016, hoping to follow the same steps Taylor and Wells have in proof of elliptic curves. However, for the Abel Surface, each of these steps is more complicated.
Therefore, they focus on a specific type of Abelian surface, called the regular Abelian surface, which is easier to use. For any such surface, there is a set of numbers that describe the structure of its solution. If they can prove that the same set of numbers can also be derived from modular forms, they can be done. These numbers will be used as unique labels, allowing them to pair each Abelian surface with a modular form.
