#### Euler Systems

SLMath - April 2024

In the mid-19th century, while attempting to prove Fermat’s last theorem, German mathematician Ernst Kummer started investigating arithmetic on novel number systems. Fermat’s last theorem states that if \(n\) is a whole number greater than \(2\), the equation \(x^n+y^n=z^n\) has no solutions where \(x\), \(y\), and \(z\) are all nonzero whole numbers. Kummer’s approach was a huge advancement over previous work, proving the theorem for an infinite set of exponents \(n\), but he was never able to prove the theorem in full generality; that would not happen until the work of Andrew Wiles in the 1990s. The new number fields Kummer introduced, however, played a foundational role in several fields of number theory.

In the realm of the familiar whole numbers, prime factorization is unique. Any whole number greater than \(1\) can be broken down into prime factors: for example, \(12\) is \(2 \times 2 \times 3\). Other than shuffling the order of the factors, there is no way to change the prime factorization. However, in the new number fields that Kummer developed, prime factorization may not be unique. For example, if you combine the imaginary number \(i\) (the square root of \(−1\)) with the familiar whole numbers, you can form the number \(6\) either as \(2 \times 3\) or \((1+ i\sqrt 5) \times (1− i\sqrt 5)\). Kummer combined the rational numbers (those that can be written as ratios of whole numbers) with imaginary numbers to create new number systems that failed to have unique factorization. Kummer began studying how much unique factorization failed in different number fields constructed similarly, developing a notion called the ideal class number that applied not only to number fields but also to other abstract algebraic objects.

Some aspects of Kummer’s work connecting numbers called cyclotomic units to ideal class numbers “were somewhat forgotten for over a century, until the work of Francisco Thaine resurrected them and exploited them to give a surprisingly simple proof of a result that was one of the mainstays of the subject,” says Henri Darmon, a mathematician at McGill University in Montreal and one of the organizers of the spring 2023 program Algebraic Cycles, \(L\)-Values, and Euler Systems at the Simons Laufer Mathematical Sciences Institute. “Something that is remarkable about mathematics, compared to other scientific disciplines, is that even the mathematics that was done 200 years ago — or 1000 years ago — is still very relevant today,” Darmon says.

The utility of Kummer’s work was rediscovered in the 1980s, in part at a 1986–1987 program at SLMath, then known as the Mathematical Sciences Research Institute. There, mathematicians used Kummer’s work to create the first examples of what are now known as Euler systems and used them to study the connections between abstract algebraic objects called motives and \(L\)-functions. The first decade of research on Euler systems was exciting, but the field seemed to stagnate in the mid-1990s. “There was a sense that we had a small, quite limited collection of examples,” says Darmon, rather than a diverse menagerie of different Euler systems and a robust theory about how to construct them.

Mathematicians conjecture a broad, deep connection between motives and \(L\)-functions, but so far the connection is only poorly understood. Euler systems form a bridge between these two objects that allows mathematicians to study relationships between them. Hence, mathematicians have been seeking new ways to produce Euler systems in the hopes that it will allow them to fully understand and verify the connection for more motives and \(L\)-functions.

In the 2010s, mathematicians started to find new ways to create Euler systems using ideas from topology, a seemingly distant area of mathematics. One of the highlights of the SLMath semester program was a series of lectures by Princeton mathematicians Chris Skinner and Marco Sangiovanni-Vincentelli on their work developing a more systematic approach to producing new Euler systems. “Approaches for producing new Euler systems had been in the air for at least the previous decade,” Darmon says, “but there was a sense that some of the new perspectives crystallized and became more firmly understood during the SLMath semester, which marked something of a turning point in this subject.”

Evelyn Lamb

In the realm of the familiar whole numbers, prime factorization is unique. Any whole number greater than \(1\) can be broken down into prime factors: for example, \(12\) is \(2 \times 2 \times 3\). Other than shuffling the order of the factors, there is no way to change the prime factorization. However, in the new number fields that Kummer developed, prime factorization may not be unique. For example, if you combine the imaginary number \(i\) (the square root of \(−1\)) with the familiar whole numbers, you can form the number \(6\) either as \(2 \times 3\) or \((1+ i\sqrt 5) \times (1− i\sqrt 5)\). Kummer combined the rational numbers (those that can be written as ratios of whole numbers) with imaginary numbers to create new number systems that failed to have unique factorization. Kummer began studying how much unique factorization failed in different number fields constructed similarly, developing a notion called the ideal class number that applied not only to number fields but also to other abstract algebraic objects.

Some aspects of Kummer’s work connecting numbers called cyclotomic units to ideal class numbers “were somewhat forgotten for over a century, until the work of Francisco Thaine resurrected them and exploited them to give a surprisingly simple proof of a result that was one of the mainstays of the subject,” says Henri Darmon, a mathematician at McGill University in Montreal and one of the organizers of the spring 2023 program Algebraic Cycles, \(L\)-Values, and Euler Systems at the Simons Laufer Mathematical Sciences Institute. “Something that is remarkable about mathematics, compared to other scientific disciplines, is that even the mathematics that was done 200 years ago — or 1000 years ago — is still very relevant today,” Darmon says.

The utility of Kummer’s work was rediscovered in the 1980s, in part at a 1986–1987 program at SLMath, then known as the Mathematical Sciences Research Institute. There, mathematicians used Kummer’s work to create the first examples of what are now known as Euler systems and used them to study the connections between abstract algebraic objects called motives and \(L\)-functions. The first decade of research on Euler systems was exciting, but the field seemed to stagnate in the mid-1990s. “There was a sense that we had a small, quite limited collection of examples,” says Darmon, rather than a diverse menagerie of different Euler systems and a robust theory about how to construct them.

Mathematicians conjecture a broad, deep connection between motives and \(L\)-functions, but so far the connection is only poorly understood. Euler systems form a bridge between these two objects that allows mathematicians to study relationships between them. Hence, mathematicians have been seeking new ways to produce Euler systems in the hopes that it will allow them to fully understand and verify the connection for more motives and \(L\)-functions.

In the 2010s, mathematicians started to find new ways to create Euler systems using ideas from topology, a seemingly distant area of mathematics. One of the highlights of the SLMath semester program was a series of lectures by Princeton mathematicians Chris Skinner and Marco Sangiovanni-Vincentelli on their work developing a more systematic approach to producing new Euler systems. “Approaches for producing new Euler systems had been in the air for at least the previous decade,” Darmon says, “but there was a sense that some of the new perspectives crystallized and became more firmly understood during the SLMath semester, which marked something of a turning point in this subject.”

Evelyn Lamb