On April 10, James Arthur collected his notes, put them in his bag and walked out of the Fields Institute. He headed to his office on the sixth floor of the University of Toronto’s Bahen Centre for Information Technology, conveniently located across the courtyard at the back of the building. Moments earlier, the renowned mathematician wrapped up the final lecture in his graduate course on Automorphic Forms and Representation Theory. The course was billed as an introduction to the Langlands Program, a series of conjectures that supply the internal structure connecting phenomena in geometry and number theory with representation theory. The Program, first proposed by Robert Langlands in 1967, is considered one of the most significant mathematical discoveries of the last century, even as its difficulty has sometimes led to misunderstanding among mathematicians.

James Arthur’s retirement from teaching is also significant. More than anyone, Arthur has worked to take the Langlands Program from conjecture to realization. Automorphic forms are considered the key to unifying huge areas of mathematics and Arthur is credited with placing the trace formula at the centre of his approach, thereby moving the Langlands program from conjecture towards its future resolution. Key to this is the so-called trace formula, on which Arthur is the world’s expert. Langlands always believed that the trace formula would be the most powerful tool that could be brought to bear on his conjectures. In Arthur’s hands, it has attained this status, having in the process been named the Arthur-Selberg trace formula after his contributions. Robert Langlands himself described Arthur, his former graduate student, as “the leading mathematician in Canada.” The next generation agrees. According to Arthur’s own graduate student, Clifton Cunningham, “he is the Langlands Program in Canada.” 

Arthur’s retirement has not received a great deal of attention so far. There are two main reasons why. The first is that the Langlands Program is still sometimes regarded as exotic. It requires a new way of thinking that can present a barrier to other mathematicians, especially since it can be difficult to explain. The second reason is that Arthur himself is sometimes low-key and gets focused on his own work. “As a new grad student, you get excited about courses and conferences,” Cunningham says. “Jim’s advice was always stay home, write the papers, then go when you have something to say. That’s his approach to everything. He quietly produces amazing work on his own terms decade after decade and people notice. There’s volume to that quiet voice. It’s very effective.”

But people are, in fact, listening. When Fields Director, Kumar Murty, heard Arthur needed a classroom for his last University of Toronto course, he jumped at the chance to bring him to the Fields Institute. Murty considered it a rare opportunity for the Shared Graduate Courses Program, part of Fields Academy that opens up advanced graduate-level courses taught at any of our Principal Sponsoring Universities to qualified students across the province and the world. “We have a responsibility to preserve – and even grow – what Arthur has built up. We at Fields, sitting right beside the University of Toronto, have to see how we can continue the study and develop this expertise,” Murty explains, before diving further into the Fields ethos. “Canada has invested a lot in research and we have the world’s leading expert on this really important subject. Fields wants to make sure that knowledge becomes freely available and stimulates more research.”

Watch: James Arthur's graduate course on Automorphic Forms and Representation Theory is available on our YouTube channel.

Lightning in a bottle 

Part of what makes the Langlands Program so important is that it supplies the internal structure of some of the greatest phenomena discovered in algebraic geometry, number theory and representation theory. These fields were largely regarded as separate, but Langlands’ intuitions linked them.

Langlands caused a stir when he first presented these ideas in 1967, and it garnered him the kind of notice reserved for giants such as Alexander Grothendieck. To understand the magnitude of his work, it’s important to begin with a phenomenon in math research where two things that at first seem like they shouldn’t be connected are, in fact, very much connected. From this realization, amazing things can happen. One early example of this comes from Gauss. Today, the law of quadratic reciprocity is taught in a first course in elementary number theory, but it wasn’t always the case. Gauss was so mystified by this law, he gave eight  proofs trying to figure out why it was happening (and today there are more than a hundred of them). But what remained mysterious was why it was true and not just that it was true.

For the next 100 years, the subject of number theory continued to build on Gauss’s proofs in more and more generality. Then, lightning in a bottle: Emil Artin published a series of papers between 1924-1930 proving that an equality of certain L-functions allows formulation of a generalization of quadratic reciprocity to n-dimensional representations. Quadratic reciprocity became the Artin reciprocity law, setting the stage for a related breakthrough several decades later.

Modern family of conjectures

When quadratic reciprocity became Artin reciprocity, mathematicians already thought it was a big deal. Lightning in a bottle would strike again at the 1955 International Symposium on Algebraic Number Theory in Japan. There, Yutaka Taniyama presented what would become the Taniyama conjecture, which he refined and corrected in 1957. It was further refined by Goro Shimura, while initial progress toward proving it was later made by Andre Weil in 1967.

The Taniyama conjecture takes an elliptic curve from geometry. It has an L-function associated with it L(E, s). Taniyama conjectured that there is an automorphic representation whose L-function is the same as the L-function of the elliptic curve. The automorphic representation is analysis, whereas the elliptic curve lies in the realm of geometry. Here, Taniyama and Weil found a link between these two disparate fields: they are connected through the intermediary of the L-function. And this is a completely new reciprocity law.

After Weil’s work, Langlands changed everything at a famous conference in 1977. At the American Mathematical Society’s month-long Summer Symposium in Corvallis, Oregon, he asserted that this sort of thing must happen all the time – not just for the Taniyama conjecture. There must be something more general. The elliptic curve is one of the first concrete examples of what Grothendieck would call a motive. Langlands said, why just this one motive? They should all be connected to automorphic forms. Any motive has an L-function and that L-function should be the same as that of an automorphic representation. 

With this intuition, and various other fundamental papers, Langlands linked his ideas to those of Grothendieck. These days, Langlands and Grothendieck are often compared for the historical influence their work will have. They are sometimes seen as the two greatest mathematicians of the second half of the twentieth century.

Beyond Endoscopy

In 1979, Langlands gave a series of lectures at the École normale supérieure de jeunes filles (later published as a book) on a new theory, now known as Endoscopy, which supplemented his 1967 conjectures. It postulated a refinement of the Arthur-Selberg trace formula, which he called the stable trace formula, and which would govern the finer properties of automorphic representations. In 2002, Arthur established the stable trace formula. This then served as the foundation for a project Arthur completed in 2013 that provides an endoscopic classification of the automorphic representations for many groups. His monograph on the subject remains the best progress to date of the Theory of Endoscopy.

Finally, around the year 2000, Langlands proposed a further theory, which he called Beyond Endoscopy. It outlines a strategy for using the stable trace formula to attack his original 1967 conjectures, and perhaps also his later conjecture on motives. As might be expected, this will be very difficult, requiring fundamental analytic and arithmetic questions that have never before been broached. Nevertheless, Arthur believes that Beyond Endoscopy represents the natural way to approach the Langlands conjectures and he is confident that it will eventually be solved. 

The message is in the motive

This brings us back to Arthur’s Fields Institute lectures. The first two thirds of the course were a basic introduction to the Langlands program that largely followed Langlands’ 1969 Bochner paper in which he first outlined his conjectures. The remaining third delved into motives. Grothendieck’s idea of a motive is often regarded as his greatest mathematical insight, and one of the great achievements in pure abstract thought of any kind. He is said to have arrived at the name from the notion of a motif in music, art or literature – a fundamental hidden principle that governs what we hear, see or read. Arthur’s lectures on motives are largely speculative. They explore new ideas in the fundamental relationships between Grothendieck’s motives and Langlands’ automorphic representations.

Fields was honoured to host James Arthur’s final graduate class, and we offer the full course in video form on our YouTube channel. The hope is that he will continue to make use of our building, setting up an office where he can continue his important work and continue to inspire future generations of students and researchers. And one thing we can get better at in the meantime is promoting the major contributions Canadian mathematicians continue to make. In this case, there is already plenty of proof available.