“We especially think that all the conjectures are true, but it is so leaving to see it really realized,” said Ana Caraianito the mathematician at the Imperial College in London. “And in a case that you really thought was out of reach.”

This is only the beginning of a hunt that will take years – the Mathematicians Ultimotelly want to show the modularity for the surface of each. But the result can help Almedady answer many open questions, as is the modularity of the elliptical curves has opened all the sorts of new research directions.

Through the glass in search

The elliptical curve is a type of equation founded by particles which uses only two variables –X and and Y. If you graphize its solutions, you will see what style to be simple curves. But these solutions are interdependent in a rich and complicated way, and they appear in many most important questions of the theory of numbers. The conjecture Birch and Swinneron -Dyer, for example – one of the most difficult open problems in mathematics, with a reward of $ 1 million for Whoaver proves it first – is the nature of the solutions to the elliptical curves.

Elliptical curves can be difficult to study directly. I know that sometimes mathematicians prefer to land them from a different angle.

This is where modular forms as in. A modular shape is a highly symmetrical function which appears in an ostensibly separate zone of the mathematical studio called analysis. Because they have so many beautiful symmetries, modular forms can be easier to work.

At the beginning, these objects seem that although the shound is not linked. But the proof of Taylor and Wiles revealed that each Elyptic curve stands at a specific modular shape. They have certain properties in common – for example, a set of numbers which describe the solutions to an elliptical curve will also appear in its associated modular form. Mathematicians can therefore use modular forms to obtain new perspectives on elliptical curves.

Buthematicians believe that the theorem of the modularity of Taylor and Wiles is only a case of a universal fact. There is a much more general class of objects beyond elliptical curves. And all these objects should also have a partner in the broader world of symmetrical functions such as modular forms. This is, in essence, what the Langlands program is.

An elliptical curve has only two variables –X and and Y– Thus, it can be represented on a flat sheet of paper. But if you add another variable, ZYou get a winding surface that lives in a three -dimensional space. This more complicated object is called an abelian surface, and as with elliptical curves, its solutions have an annate structure that mathematicians want to emphasize.

It seemed natural that Belian is surfaced with bodies were to more complex modular shapes. But the additional variable makes them much more difficult to build and their solutions much more difficult to find. Prove that they too, ensure a modularity theorem, seemed completely out of reach. “It was a known problem not to think because people thought about it and remained stuck,” said Gee.

But Boxer, Calegari, Gee and Pilloni wanted to try.

Deck

The four mathematicians were an involution in research on the Langlands program, and they wanted to test one of these conjectures for “an object that is in fact in real life, rather than something strange,” said Calegari.

Not only do abelian surfaces appear in real life, the real life of a mathematian, that is to say-but proving a modularity theorem to them would open new mathematical doors. “There are a lot of things you can do if you have this statement that you have no change to do otherwise,” said Calegari.

The mathematicians began working together in 2016, hoping to follow the same steps as Taylor and Wiles had elliptical curves in their proof. But each of these stages was much more complicated for Abélien surfaces.

They therefore focused on a type of abelian surface particles, called ordinary abelian surface, which was easy to work. For any surface, there is a set of numbers that describe the structure of its solutions. If they are to show that the same set of numbers could also be derived from a modular shape, they would have done. The figures would serve as a single tag, leaning them to associate each of their abelian surfaces with a modular shape.