Sixteenth Algorithmic Number Theory Symposium (ANTS XVI)
Massachusetts Institute of Technology
July 15–19, 2024

I will be interested in some aspects of our understanding of the arithmetic of curves over local fields, focusing on algorithmic applications, the study of families of curves, and applications to global questions such as understanding rational points on curves and their Jacobians. Using the L-function of an elliptic curve as a starting point, I will derive several key local arithmetic invariants and discuss how we might want to address their computations to suit the three types of applications mentioned above. I will present examples for hyperelliptic curves over local fields of odd residue characteristic, and end with a short discussion on general curves and the case of even residue characteristic.

The talk is based on the following joint work:

• Arithmetic of hyperelliptic curves over local fields, with T. Dokchitser, V. Dokchitser, A. Morgan.
• Computing Euler Factors of genus 2 curves at odd primes of almost good reduction, with A. Sutherland.
• 2-parity conjecture for elliptic curves with isomorphic 2-torsion, with H. Green.
• Conductor computations via cluster pictures for the modular method, with M. Azon, M. Curco Iranzo, M. Khawaja, D. Mocanu (in progress).

I will briefly introduce Oscar our "new" computer algebra system under active development (and available (and even useful)) since roughly 2020. But mainly I will talk about norm equation in several settings:

• normal extensions of number fields
• non-normal extensions
• the Diophantine case - where one wants integral solutions

I'll talk about my experience working with scientists from DeepMind on a project to use large language models to generate large cap sets (subsets of \(\mathbb{F}_3^n\) in which no three element sum to 0). The key novelty in this project is that, rather than search the (very large) space of subsets of \(\mathbb{F}_3^n\), we search the (also large but very different) space of computer programs whose output is a subset of \(\mathbb{F}_3^n\). I, as a pure mathematician, thought this was a very weird idea when I first heard about it, which means maybe you will too, and I'll try to make the case that it's actually a pretty promising direction for applications of machine learning to pure math! At the end, I'll step back and talk more generally about what the experience of working with industrial scientists has taught me about the prospects for future progress at the math/ML interface.

Let E be an elliptic curve with complex multiplication by a ring R, where R is an order in an imaginary quadratic field or quaternion algebra. We define sesquilinear pairings (R-linear in one variable and R-conjugate linear in the other), taking values in an R-module, generalizing the Weil and Tate-Lichtenbaum pairings. We discuss some applications to the hard problems in isogeny-based cryptography (this portion is joint work with Joseph Macula).