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Mathematicians Find Long-Sought Building Blocks for Special Polynomials | Quanta Magazine
Hilbert’s 12th problem asked for novel analogues of the roots of unity, the building blocks for certain number systems. Now, over 100 years later, two mathematicians have produced them.
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So the roots of unity are the exact building blocks you need to perfectly describe the roots of abelian polynomials with rational coefficients. On the flip side, any combination of the roots of unity will result in a number that is the root of some abelian polynomial with rational coefficients. The two are inextricably linked.
What Hilbert wanted when he posed his 12th problem was for mathematicians to find the building blocks of the roots of abelian polynomials with coefficients from number systems beyond rational numbers. In other words, what is the analogue of the roots of unity for other number systems?
Go Beyond It’s an ambitious question, but that’s why it landed on Hilbert’s list in the first place. He suspected it could be answered because, as he was writing it, he had an idea about how to describe the building blocks for one other type of number system, known as imaginary quadratic fields. (Roughly, this system includes only the rational numbers and the square root of a negative number.) His guess was proved correct several decades later.
“There are two cases — the case of [the rationals] and the case of the imaginary quadratic fields — that guided Hilbert in formulating his question,” said Alice Pozzi of Imperial College London.
Hilbert expected the building blocks of other number systems to be described in similar terms to the two cases he already knew about. This meant using complex analysis, the branch of mathematics that studies functions using complex numbers.
But in the 1970s — decades after Hilbert laid the foundations for his 12th problem — the mathematician Harold Stark conjectured that L-functions might help crack it instead. These are a type of function that adds together infinitely many numbers. The Riemann zeta function, which is the subject of another problem on Hilbert’s list, is a famous example:
ζ(s) = 1 + 1/2s + 1/3s + 1/4s + 1/5s+ 1/6s + 1/7s + ….
For centuries, mathematicians have known that L-functions produce mysterious and intriguing results. They show how infinite sequences of simple fractions can be used to build numbers related to pi and other important constants.
Building on this intuition, Stark was able to come up with an analogue of the roots of unity for other number systems using L-functions. But while mathematicians believe that Stark’s conjecture is true, having tested it extensively using computer analysis, they have not had any success proving it.
“The Stark conjecture is really difficult as far as we know,” said Darmon. “There’s been virtually no progress at all, and it’s been 50 years.”
Ultimately, what Stark did was provide a recipe that claims to find, using L-functions, the building blocks for the roots of abelian polynomials with coefficients from other number systems. It’s just that no one knows how to show that the recipe works.
Worse yet, Stark’s recipe only gives half of the information you need to actually describe the building blocks. This is equivalent to having only the longitude of a location — you also need the latitude to find a specific spot.
In the 1980s, Gross continued the work by publishing a modified version of Stark’s recipe, this time using newer ingredients. Stark, like Hilbert, had thought in terms of complex numbers, but Gross instead used the p-adic numbers. These are alternatives to the standard numbers that use different methods to decide when two numbers are close together.
Benedict Gross was the first to use p-adic numbers to look for the numerical building blocks that Hilbert’s 12th problem asks for. The approach proved successful, decades later.
“You can build up the whole theory of calculus from scratch again, where you use this new notion of what it means for things to be close,” said Dasgupta.
Many concepts in mathematics can be rewritten using p-adic numbers, and that includes L-functions. In fact, within modern number theory, p-adic L-functions are viewed as a natural companion to complex L-functions.
“They comprise a very coherent family,” said Barry Mazur of Harvard. “They work together.”
Even so, at first Gross’ translation from complex numbers to p-adic numbers didn’t seem to bring mathematicians any closer to proving Stark’s conjecture.
“I was just totally intimidated by it, because I thought, ‘Well, this is as hard as the original conjecture, but it’s just in a different form,’” said Gross. But in the following decades, Gross’ p-adic conjecture began to look more amenable to solving than its complex counterpart, as number theorists developed the theory of p-adic numbers.
“There’s a whole world now of p-adic analysis that’s very rich and has led to a lot of interesting results,” said Darmon. Many important problems in mathematics turned out to be easier to solve using p-adic numbers rather than complex numbers — Hilbert’s 12th problem included.
Chipping Away In March of this year, Dasgupta and Kakde posted a paper that used p-adic L-functions to answer, for the first time, Hilbert’s question for a separate large collection of number systems. Known as totally real fields, these systems are an extension of the rational numbers that also incorporates one root of a given polynomial. (For example, incorporating the square root of 2 expands the rational numbers to include numbers like 2/√3, 1 + √2, and 2 – 3√2)
Dasgupta first came up with the final formula they’d need — a refinement of Gross’ conjecture — in his 2004 doctoral thesis. “Half my life I’ve been working on this,” said Dasgupta. “So it’s very satisfying to have finally completed the proof.” The first step in the process, which took place over the past decade, was to finally prove Gross’ conjecture in a series of two papers, using the latest developments in p-adic number theory. But this gave only half the information, because Gross’ conjecture — like Stark’s —gives only one of the two numbers needed to precisely describe the building blocks. For the last three years, Dasgupta and Kakde have worked to prove a version of Gross’ conjecture that provides both numbers, even when it still seemed impossible. “Maybe it’s just that both of us are very optimistic,” said Kakde. “At times it did seem that these obstacles were quite serious, and at times they were quite serious, but fortunately, progress kept happening.”
Last year they had a breakthrough. They were able to prove that the precise building blocks associated to totally real fields exist. In other words, they knew that the desired objects were somewhere out there, and this insight led them in the right direction. It gave them the key equations for proving the exact formula that fully describes the building blocks.
To confirm that it’s correct, two students working with Dasgupta wrote a computer program that can actually generate the building blocks for a given number system — finally baking the now-completed recipe and demonstrating it works. Alongside the theoretical proof, the computer program helps to demonstrate the accuracy of Dasgupta and Kakde’s formulas — a particularly important element of a solution to such an abstract problem, prone to subtly wrong answers. (Hilbert himself incorrectly stated a partial solution to his 12th problem.)
“I view it as a collaborative effort,” said Dasgupta. Indeed, the recent work is the result of three generations of mathematicians: He was a student of Darmon’s, who was in turn a student of Gross’. “[It] has taken a long time and [was] finally brought to fruition by these recent papers.”
Hilbert’s 12th problem asks for a precise description of the building blocks of roots of abelian polynomials, analogous to the roots of unity, and Dasgupta and Kakde’s research gives the building blocks for a family of number systems — albeit with a decidedly modern take, in the form of p-adic L-functions.
But there’s one final wrinkle: Since Hilbert explicitly wrote that the building blocks should be formed from complex numbers, the way the solution departs from Hilbert’s original instructions showcases the versatility of mathematics. It gives an answer to Hilbert’s question using p-adic analysis, while still leaving the original question — using complex analysis — open for future generations of mathematicians to explore. There may be many ways to describe the building blocks, and someday someone might be able to describe them using complex numbers, satisfying Hilbert’s initial request.
“It’s all a relay race,” said Gross. “You’re just passing the baton as you get exhausted to the next gen