# Sixteenth Algorithmic Number Theory Symposium (ANTS XVI)

Massachusetts Institute of Technology

July 15–19, 2024

## Schedule

Unless otherwise noted, all activities will take place in room 10-250.

Coffee/tea/snacks will be provided at the morning and afternoon breaks, but participants will need to forage for lunch (there are many options nearby).

## Monday, July 15

## Tuesday, July 16

09:40 | setup (10-250) | First chair: Renate Scheidler | |

09:45 | Céline Maistret | Local arithmetic of curves and applications (slides) | |

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).
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10:45 | morning break | Next chair: Jennifer Balakrishnan | |

11:15 | Andrew Sutherland | Computing Euler factors of genus 2 curves at odd primes of almost good reduction (slides) | |

with Céline Maistret | |||

11:45 | Xavier Caruso | Computation of classical and v-adic L-series of t-motives (slides) | |

with Quentin Gazda | |||

12:15 | Edgar Costa | Hypergeometric L-functions in average polynomial time II (slides) | |

with Kiran Kedlaya and David Roe | |||

12:45 | Group photo | ||

12:55 | lunch break | Next chair: Alyson Deines | |

14:30 | Maarten Derickx | Torsion subgroups of elliptic curves over quadratic fields and a conjecture of Granville (slides) | |

with Barinder Banwait | |||

15:00 | Freddy Saia | Bielliptic Shimura curves \(X_0^D(N)\) with nontrivial level (slides) | |

with Oana Padurariu | |||

15:30 | afternoon break | Next chair: John Voight | |

16:00 | lightning talks | Voting link on Zulip | |

17:45 | break | Next chair: Joshua Holden | |

18:00 | business meeting | Agenda on Zulip |

## Wednesday, July 17

09:40 | setup (10-250) | First chair: John Cremona |

09:45 | Claus Fieker | Norm equations and more in Oscar (slides) |

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
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10:45 | morning break | Next chair: Jonathan Love |

11:15 | Sumin Leem | Solving norm equations in global function fields (slides) |

with Michael Jacobson and Renate Scheidler | ||

11:45 | Felipe Voloch | Factoring polynomials over function fields (slides) |

12:15 | Erik Mulder | Fast square-free decomposition of integers using class groups (slides) |

## Thursday, July 18

09:40 | setup (10-250) | First chair: David Zureick-Brown |

09:45 | Jordan Ellenberg | Machine learning and pure mathematics: Experiments and speculations (slides) |

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. | ||

10:45 | morning break | Next chair: David Roe |

11:15 | Enis Kaya | Algorithms for p-adic heights on hyperelliptic curves of arbitrary reduction (slides) |

with Francesca Bianchi and Steffen Müller | ||

11:45 | Martin Lüdtke | Refined Chabauty-Kim computations for the thrice-punctured line over ℤ[1/6] (slides) |

12:15 | Aashraya Jha | Finding integral points of elliptic curves over imaginary quadratic fields (slides) |

12:45 | lunch break | Next chair: Everett Howe |

14:30 | Stefano Marseglia | Modules over orders, conjugacy classes of integral matrices, and abelian varieties over finite fields (slides) |

15:00 | Jonathan Webster | Advances in tabulating Carmichael numbers (slides) |

with Andrew Shallue | ||

15:30 | afternoon break | |

16:00 | Jon Sorenson | An algorithm and computation to verify Legendre's Conjecture up to 7·10 (slides)^{13} |

with Jonathan Webster | ||

16:30 | Aaron Pollack | Computation of Fourier coefficients of automorphic forms of type G (slides)_{2} |

17:00 | bubble tea (Lowell court) | Next chair: Benjamin Smith |

18:00 | rump session (2-190) | Application instructions on Zulip |

## Friday, July 19

09:40 | setup (10-250) | First chair: Travis Morrison |

09:45 | Katherine Stange | Sesquilinear pairings on elliptic curves and applications to isogenies (slides) |

Let | ||

10:45 | morning break | Next chair: Jonathan Webster |

11:15 | Seungki Kim | On counterexamples to the Mertens conjecture (slides) |

with Phong Nguyen | ||

11:45 | Colby Brown | An almost linear time algorithm testing whether the Markoff graph modulo p is connected (slides) |

12:15 | Joseph Silverman | A heuristic subexponential algorithm to find paths in Markoff graphs over finite fields (slides) |

12:45 | Selfridge prize | Chair: Andrew Sutherland |

12:50 | lunch break | |

15:45 | waffle truck (Lowell court) |