A glimpse of F1 at ANTS

From the first day of ANTS, one highlight I'd like to share is the talk on Drinfeld modules by Antoine Leudière, reporting on joint work with Renate Scheidler.

In the context of Weil's Rosetta Stone, there is a specific analogy between elliptic curves (in the number field world) and something called Drinfeld modules (in the function field world). The talk was about certain computational problems related to submodules of a Drinfeld module, and you can check out the slides as well as the conference paper.

To give an idea of the flavour of the talk, here are two slides from the introduction:

At the very end, Leudière reflected a bit on the differences between the two worlds, and the fact that some algorithms in the function field world simply do not have a counterpart in the number field world, and cannot have so "until we get a field with one element".

Drinfeld modules is a fun topic to learn about because of the connections to the Langlands program and to $\mathbb{F}_1$, and I was surprised to hear in this talk that there are also many real-world applications, even to PQC (post-quantum cryptography).

One place to learn more is Leudière's thesis, which I found after the talk. Two other excellent references are these notes of Poonen (aimed more towards the Langlands connections) and these notes by Leudière and a bunch of other people (geared more towards computational applications).