The Langlands program in the words of Langlands himself

In the next chapter of the RH Saga, I want to discuss operations on L-functions, and say something meaningful about the relationship between these operations and a possible future theory of $\mathbb{F}_1$.

One of the most important perspectives on L-function operations comes from the Principle of Functoriality, which forms (as you may recall from Chapter 2), one of the two deep threads of the global Langlands program (the other being Reciprocity).

In this post I want to collect some of the talks and letters of Robert Langlands himself, in which he describes in his own words the origins and the meaning of Functoriality.

When Langlands was awarded the Abel prize in 2018, three Abel lectures were given (with links here to YouTube), by Langlands himself, by James Arthur, and by Edward Frenkel. The talks of Arthur and Frenkel constitute an honest effort to explain what the Langlands program is about, but they are aimed at a general audience and therefore avoid technical details. The talk of Langlands was short and a bit difficult to follow. If you have time, and you want something deeper, I would instead recommend a series of three talks filmed four years earlier in Langlands' own office at the Institute for Advanced Study (which was Albert Einstein's office in an earlier era!). All three talks are included as videos at the end of this post. They appear to be a kind of farewell message from Langlands to future generations of mathematicians, in which he talks about what the main problems are, giving many insights, remarks, and reading recommendations, like the observation that there seems to be no analogue of Reciprocity in the geometric versions of the Langlands program (over $\mathbb{C}$ or a finite field), but perhaps such a notion might exist, for one-parameter families of motives. If you go to the IAS archive page for the complete works of Langlands and scroll down to the very last item, you will find a pdf file written up by Langlands as a complement to the video lectures, as well as some additional remarks including the one on "geometric" reciprocity.

On that same page, you can of course sample many other treasures from a life of creative output by one of the greatest mathematical thinkers ever to live. There is also another site organising some of the same material, hosted by University of British Columbia. They have a subpage specifically for Functoriality, listing among other things these very interesting texts:

• The famous 1967 letter to André Weil, in some sense the beginning of the entire story.
• Problems the the Theory of Automorphic Forms, the 1969 lecture where these ideas were explained more in detail.
• A 1989 lecture on Representation theory in number theory. This is (by far) the most readable and accessible of all these Langlands writings, so check this one out if you're looking to read just one coherent lecture by the legend himself!).
• Where stands functoriality today? (1997)

Below are those lectures from 2014. Feel the wings of the history of science, and enjoy :-)