The grandest project in mathematics has received a rare gift, in the form of a mammoth 350-page paper posted in February that will change the way researchers around the world investigate some
of the field’s deepest questions. The work fashions a new geometric
object that fulfills a bold, once fanciful dream about the relationship
between geometry and numbers.
“This truly opens up a tremendous amount of possibilities. Their
methods and constructions are so new they’re just waiting to be
explored,” said Tasho Kaletha of the University of Michigan.
The work is a collaboration between Laurent Fargues of the Institute of Mathematics of Jussieu in Paris and Peter Scholze of the University of Bonn. It opens a new front in the long-running
“Langlands program,” which seeks to link disparate branches of
mathematics — like calculus and geometry — to answer some of the most
fundamental questions about numbers.
Their paper realizes that vision, giving mathematicians an entirely
new way of thinking about questions that have inspired and confounded
them for centuries.
At the center of Fargues and Scholze’s work is a revitalized
geometric object called the Fargues-Fontaine curve. It was first
developed around 2010 by Fargues and Jean-Marc Fontaine, who was a
professor at Paris-Sud University until he died of cancer in 2019. After
a decade, the curve is only now achieving its highest form.
“Back then they knew the Fargues-Fontaine curve was something
interesting and important, but they didn’t understand in which ways,”
said Eva Viehmann of the Technical University of Munich.
The curve might have remained confined to the technical corner of
mathematics where it was invented, but in 2014 events involving Fargues
and Scholze propelled it to the center of the field. Over the next seven
years they worked out the foundational details needed to adapt Fargues’
curve to Scholze’s theory. The final result doesn’t so much bridge
numbers and geometry as collapse the ground between them.
“It’s some kind of wormhole between two different worlds,” said
Scholze. “They really just become the same thing somehow through a
different lens.”
By the 1500s mathematicians had discovered tidy formulas for
calculating the roots of polynomials whose highest powers are 2, 3 or 4.
They then searched for ways to identify the roots of polynomials with
variables raised to the power of 5 and beyond. But in 1832 the young
mathematician Évariste Galois discovered the search was fruitless,
proving that there are no general methods for calculating the roots of
higher-power polynomials.
Galois didn’t stop there, though. In the months before his death in a
duel in 1832 at age 20, Galois laid out a new theory of polynomial
solutions. Rather than calculating roots exactly — which can’t be done
in most cases — he proposed studying the symmetries between roots, which
he encoded in a new mathematical object eventually called a Galois
group.
In the example x2 − 2, instead of making the
roots explicit, the Galois group emphasizes that the two roots (whatever
they are) are mirror images of each other as far as the laws of algebra
are concerned.
“Mathematicians had to step away from formulas because usually there were no formulas,” said Brian Conrad of Stanford University. “Computing a Galois group is some measure of computing the relations among the roots.”
Throughout the 20th century mathematicians devised new ways of
studying Galois groups. One main strategy involved creating a dictionary
translating between the groups and other objects — often functions
coming from calculus — and investigating those as a proxy for working
with Galois groups directly. This is the basic premise of the Langlands
program, which is a broad vision for investigating Galois groups — and
really polynomials — through these types of translations.
The Langlands program began in 1967, when its namesake, Robert Langlands, wrote a letter to a famed mathematician named André Weil. Langlands proposed that
there should be a way of matching every Galois group with an object
called an automorphic form. While Galois groups arise in algebra
(reflecting the way you use algebra to solve equations), automorphic
forms come from a very different branch of mathematics called analysis,
which is an enhanced form of calculus. Mathematical advances from the
first half of the 20th century had identified enough similarities
between the two to make Langlands suspect a more thorough link.
“It’s remarkable that these objects of a very different nature somehow communicate with each other,” said Ana Caraiani of Imperial College London.
If mathematicians could prove what came to be called the Langlands
correspondence, they could confidently investigate all polynomials using
the powerful tools of calculus. The conjectured relationship is so
fundamental that its solution may also touch on many of the biggest open
problems in number theory, including three of the million-dollar
Millennium Prize problems: the Riemann hypothesis, the BSD conjecture and the Hodge conjecture.
Given the stakes, generations of mathematicians have been motivated
to join the effort, developing Langlands’ initial conjectures into what
is almost certainly the largest, most expansive project in the field
today.
“The Langlands program is a network of conjectures that touch upon almost every area of pure mathematics,” said Caraiani.
Beginning in the early 1980s Vladimir Drinfeld and later Alexander Beilinson proposed that there should be a way to interpret Langlands’ conjectures
in geometric terms. The translation between numbers and geometry is
often difficult, but when it works it can crack problems wide open.
To take just one example, a basic question about a number is whether
it has a repeated prime factor. The number 12 does: It factors into 2 × 2
× 3, with the 2 occurring twice. The number 15 does not (it factors
into 3 × 5).
In general, there’s no quick way of knowing whether a number has a
repeated factor. But there is an analogous geometric problem which is
much easier.
Polynomials have many of the same properties as numbers: You can add,
subtract, multiply and divide them. There’s even a notion of what it
means for a polynomial to be “prime.” But unlike numbers, polynomials have a clear geometric guise. You can
graph their solutions and study the graphs to gain insights about them.
For instance, if the graph is tangent to the x-axis at any
point, you can deduce that the polynomial has a repeated factor
(indicated at exactly the point of tangency). It’s just one example of
how a murky arithmetic question acquires a visual meaning once converted
into its analogue for polynomials.
“You can graph polynomials. You can’t graph a number. And when you
graph a [polynomial] it gives you ideas,” said Conrad. “With a number
you just have the number.”
The “geometric” Langlands program, as it came to be called, aimed to
find geometric objects with properties that could stand in for the
Galois groups and automorphic forms in Langlands’ conjectures. Proving
an analogous correspondence in this new setting by using geometric tools
could give mathematicians more confidence in the original Langlands
conjectures and perhaps suggest useful ways of thinking about them. It
was a nice vision, but also a somewhat airy one — a bit like saying you
could cross the universe if you only had a time machine.
“Making geometric objects that serve a similar role in the setting of
numbers is a much more difficult thing to do,” said Conrad.
So for decades the geometric Langlands program remained at a distance
from the original one. The two were animated by the same goal, but they
involved such fundamentally different objects that there was no real
way to make them talk to each other.
“The arithmetic people sort of looked bemused by [the geometric
Langlands program]. They said it’s fine and good, but completely
unrelated to our concern,” said Kaletha.
The new work from Scholze and Fargues, however, finally fulfills the
hopes pinned on the geometric Langlands program — by finding the first
shape whose properties communicate directly with Langlands’ original
concerns.
In September 2014, Scholze was teaching a special course at the
University of California, Berkeley. Despite being only 26, he was
already a legend in the mathematics world. Two years earlier he had
completed his dissertation, in which he articulated a new geometric
theory based on objects he’d invented called perfectoid spaces. He then
used this framework to solve part of a problem in number theory called
the weight-monodromy conjecture.
But more important than the particular result was the sense of
possibility surrounding it — there was no telling how many other
questions in mathematics might yield to this incisive new perspective.
The topic of Scholze’s course was an even more expansive version of
his theory of perfectoid spaces. Mathematicians filled the seats in the
small seminar room, lined up along the walls and spilled out into the
hallway to hear him talk.
“Everyone wanted to be there because we knew this was revolutionary stuff,” said David Ben-Zvi of the University of Texas, Austin.
Scholze’s theory was based on special number systems called the p-adics. The “p” in p-adic stands for “prime,” as in prime numbers. For each prime, there is a unique p-adic number system: the 2-adics, the 3-adics, the 5-adics and so on. P-adic
numbers have been a central tool in mathematics for over a century.
They’re useful as more manageable number systems in which to investigate
questions that occur back in the rational numbers (numbers that can be
written as a ratio of positive or negative whole numbers), which are
unwieldy by comparison.
The virtue of p-adic numbers is that they’re each based on
just one single prime. This makes them more straightforward, with more
obvious structure, than the rationals, which have an infinitude of
primes with no obvious pattern among them. Mathematicians often try to
understand basic questions about numbers in the p-adics first, and then take those lessons back to their investigation of the rationals.
“The p-adic numbers are a small window into the rational numbers,” said Kaletha.
All number systems have a geometric form — the real numbers, for
instance, take the form of a line. Scholze’s perfectoid spaces gave a
new and more useful geometric form to the p-adic numbers. This enhanced geometry made the p-adics,
as seen through his perfectoid spaces, an even more effective way to
probe basic number-theoretic phenomena, like questions about the
solutions of polynomial equations.
“He reimagined the p-adic world and made it into geometry,”
said Ben-Zvi. “Because they’re so fundamental, this leads to lots and
lots of successes.”
In his Berkeley course, Scholze presented a more general version of
his theory of perfectoid spaces, built on even newer objects he’d
devised called diamonds. The theory promised to further enlarge the uses
of the p-adic numbers. Yet at the time Scholze began teaching, he had not even finished working it out.
“He was giving the course as he was developing the theory. He was
coming up with ideas in the evening and presenting them fresh out of his
mind in the morning,” said Kaletha.
It was a virtuosic display, and one of the people in the room to hear it was Laurent Fargues.
At the same time Scholze was giving his lectures, Fargues was attending a special semester at the Mathematical Sciences Research Institute just up the hill from the Berkeley campus. He had thought a lot about the p-adic numbers, too. For the past decade he’d worked with Jean-Marc Fontaine in an area of math called p-adic
Hodge theory, which focuses on basic arithmetic questions about these
numbers. During that time, he and Fontaine had come up with a new
geometric object of their own. It was a curve — the Fargues-Fontaine
curve — whose points each represented a version of an important object
called a p-adic ring.
As originally conceived, it was a narrowly useful tool in a technical
part of mathematics, not something likely to shake up the entire field.
“It’s an organizing principle in p-adic Hodge theory, that’s
how I think of it. It was impossible for me to keep track of all these
rings before this curve came up,” said Caraiani.
But as Fargues sat listening to Scholze, he envisioned an even
greater role for the curve in mathematics. The never-realized goal of
the geometric Langlands program was to find a geometric object that
encoded answers to questions in number theory. Fargues perceived how his
curve, merged with Scholze’s p-adic geometry, could serve
exactly that role. Around mid-semester he pulled Scholze aside and
shared his nascent plan. Scholze was skeptical.
“He mentioned this idea to me over a coffee break at MSRI,” said
Scholze. “It was not a very long conversation. At first I thought it
couldn’t be good.”
But they had more conversations, and Scholze soon realized the
approach might work after all. On December 5, as the semester wound
down, Fargues gave a lecture at MSRI in which he introduced a new vision
for the geometric Langlands program. He proposed that it should be
possible to redefine the Fargues-Fontaine curve in terms of Scholze’s p-adic
geometry, and then use that redefined object to prove a version of the
Langlands correspondence. Fargues’ proposal was a final, unexpected turn
in what had already been a thrilling season of mathematics.
“It was like this grand finale of this semester. I remember just being in shock,” said Ben-Zvi.
The original Langlands conjectures are about matching representations
of the Galois groups of the rational numbers with automorphic forms.
The p-adics are a different number system, and there is a
version of the Langlands conjectures there, too. (Both are still
separate from the geometric Langlands program.) It also involves a kind
of matching, though in this case it’s between representations of the
Galois group of the p-adic numbers and representations of p-adic groups.
While their objects are different, the spirit of the two conjectures
is the same: to study solutions to polynomials — in terms of rational
numbers in one case and p-adic numbers in the other — by
relating two seemingly unrelated kinds of objects. Mathematicians refer
to the Langlands conjecture for rational numbers as the “global”
Langlands correspondence, because the rationals contain all the primes,
and the version for p-adics as the “local” Langlands correspondence, since p-adic number systems deal with one prime at a time.
In his December lecture at MSRI, Fargues proposed proving the local
Langlands conjecture using the geometry of the Fargues-Fontaine curve.
But because he and Fontaine had developed the curve for a completely
different and more limited task, their definition required more powerful
geometry that could provide the structure and complexity the curve
would ultimately need to support these enlarged plans.
The situation was similar to how you could arrive at a three-sided
shape that’s independent of any particular geometric theory, but if you
combine that shape with the theory of Euclidean geometry, suddenly it
takes on a richer life: You get trigonometry, the Pythagorean theorem
and well-defined notions of symmetry. It becomes a fully fledged
triangle.
“[Fargues] was taking the idea of the curve and using the powerful
geometry that Scholze developed to flesh out that idea,” said Kaletha.
“That allows you to formally state the beautiful properties of the
curve.”
Fargues’ strategy came to be known as the “geometrization of the
local Langlands correspondence.” But at the time he made it, existing
mathematics didn’t have the tools he needed to carry it out, and new
geometric theories don’t come along every day. Luckily, history was on
his side.
“[Fargues’ conjecture] was a bold idea because Fargues needed
geometry that didn’t exist. But as it turned out Scholze at that very
moment was developing it,” said Kaletha.
Following their time together in Berkeley, Fargues and Scholze spent
the next seven years establishing a geometric theory that would allow
them to reconstruct the Fargues-Fontaine curve in a form suitable for
their plans.
“In 2014 it was basically already clear what the picture should be
and how everything should fit together. It was just that everything was
completely ill-defined. There were no foundations in place to talk about
any of this,” said Scholze.
The work took place in several stages. In 2017 Scholze completed a paper called “Étale Cohomology of Diamonds,”
which formalized many of the most important ideas he had introduced
during his Berkeley lectures. He combined that paper with another massive work that he and co-author Dustin Clausen of the University of Copenhagen released as a series of lectures in
2020. That material — all 352 pages of it — was needed to establish a
foundation for a few particular points that had come up in Scholze’s
work on diamonds.
“Scholze had to come up with a whole other theory which was just
there to take care of certain technical issues that came up on the last
three pages of his [2017] paper,” said Kaletha.
Altogether, these and other papers allowed Fargues and Scholze to
devise an entirely new way of defining a geometric object. Imagine that
you start with an unorganized collection of points — a “cloud of dust,”
in Scholze’s words — that you want to glue together in just the right
way to assemble the object you’re looking for. The theory Fargues and
Scholze developed provides exact mathematical directions for performing
that gluing and certifies that, in the end, you will get the
Fargues-Fontaine curve. And this time, it’s defined in just the right
way for the task at hand — addressing the local Langlands
correspondence.
“That’s technically the only way we can get our hands on it,” said
Scholze. “You have to rebuild a lot of foundations of geometry in this
kind of framework, and it was very surprising to me that it is
possible.”
After they’d defined the Fargues-Fontaine curve, Fargues and Scholze
embarked on the next stage of their journey: equipping it with the
features necessary to prove a correspondence between representations of
Galois groups and representations of p-adic groups.
To understand these features, let’s first consider a simpler
geometric object, like a circle. At every point on the circle it’s
possible to position a line that’s tangent to the shape at exactly that
point. Every point has a unique tangent line. You can collect all those
many lines together into an auxiliary geometric object, called the
tangent bundle, that’s associated to the underlying geometric object,
the circle.
In their new work, Fargues and Scholze do something similar for the
Fargues-Fontaine curve. But instead of tangent planes and bundles, they
define ways of constructing many more complicated geometric objects. One
example, called sheaves, can be associated naturally to points on the
Fargues-Fontaine curve the way tangent lines can be associated to points
on a circle.
Sheaves were first defined in the 1950s by Alexander Grothendieck,
and they keep track of how algebraic and geometric features of the
underlying geometric object interact with each other. For decades,
mathematicians have suspected they might be the best objects to focus on
in the geometric Langlands program.
“You reinterpret the theory of representations of Galois groups in terms of sheaves,” said Conrad.
There are local and global versions of the geometric Langlands
program, just as there are for the original one. Questions about sheaves
relate to the global geometric program, which Fargues suspected could
connect to the local Langlands correspondence. The issue was that
mathematicians didn’t have the right kinds of sheaves defined on the
right kind of geometric object to carry the day. Now Fargues and Scholze
have provided them, via the Fargues-Fontaine curve.
Specifically, they came up with two different kinds: Coherent sheaves correspond to representations of p-adic
groups, and étale sheaves to representations of Galois groups. In their
new paper, Fargues and Scholze prove that there’s always a way to match
a coherent sheaf with an étale sheaf, and as a result there’s always a
way to match a representation of a p-adic group with a representation of a Galois group.
In this way, they finally proved one direction of the local Langlands
correspondence. But the other direction remains an open question.
“It gives you one direction, how to go from a representation of a p-adic group to a representation of a Galois group, but doesn’t tell you how to go back,” said Scholze.
The work is one of the biggest advances so far on the Langlands program — often mentioned in the same breath as work by Vincent Lafforgue of the Fourier Institute in Grenoble, France, on a different aspect of
the Langlands correspondence in 2018. It’s also the most tangible
evidence yet that earlier mathematicians weren’t foolish to attempt the
Langlands program by geometric means.
“These things are a great vindication for the work people were doing in geometric Langlands for decades,” said Ben-Zvi.
For mathematics as a whole, there’s a sense of awe and possibility in
the reception of the new work: awe at the way the theory of p-adic
geometry Scholze has been building since graduate school manifests in
the Fargues-Fontaine curve, and possibility because that curve opens
entirely new and unexplored dimensions of the Langlands program.
“It’s really changed everything. These last five or eight years, they have really changed the whole field,” said Viehmann.
The clear next step is to nail down both sides of the local Langlands
correspondence — to prove that it’s a two-way street, rather than the
one-way road Fargues and Scholze have paved so far.
Beyond that, there’s the global Langlands correspondence itself.
There’s no obvious way to translate Fargues and Scholze’s geometry of
the p-adic numbers into corresponding constructions for the
rational numbers. But it’s also impossible to look at this new work and
not wonder if there might be a way.
“It’s a direction I’m really hoping to head into,” Scholze said.