##### About

The number of rational points $X(\mathbb{Q})$ on a hyperbolic curve $X$ defined over $\mathbb{Q}$ is finite. But how do you find those points?

Minhyong Kim’s non-abelian Chabauty method gives a $p$-adic approach to answer this question. The theoretical inspiration to use quotients of the fundamental group comes from anabelian geometry. The resulting algorithms are some of the best available to calculate rational points. And the methods along the way involve Jacobians, $p$-adic integration and $p$-adic heights on curves.

The aim of this workshop is to give an introduction into these beautiful ideas, mixed with exercise sessions and research talks to give a glimpse of what lies ahead.

##### Info

**Time:** 14th - 16th February 2024 (Wednesday morning - Friday noon).

**Place:** Heidelberg University, Germany

**Room:** Konferenzraum, 5th floor @ Mathematikon

**Organisers:** Marius Leonhardt, Tim Holzschuh

##### Speakers

##### Schedule

##### Wednesday

Time | |
---|---|

08:30 - 09:30 | Registration |

09:30 - 10:30 | Martin Lüdtke: Foundations of Chabauty–Kim I |

10:30 - 11:00 | Coffee break |

11:00 - 12:00 | Martin Lüdtke: Foundations of Chabauty–Kim II notes |

12:00 - 15:00 | Lunch + Discussions |

15:00 - 16:00 | Exercise Session sheet |

16:00 - 16:30 | Coffee break |

16:30 - 17:30 | Michael Stoll: Selmer Group Chabauty notes |

##### Thursday

Time | |
---|---|

09:30 - 10:30 | Steffen Müller: Algorithmic Aspects of Chabauty I |

10:30 - 11:00 | Coffee break |

11:00 - 12:00 | Steffen Müller: Algorithmic Aspects of Chabauty II |

12:00 - 14:30 | Lunch + Discussions |

14:30 - 15:30 | Stevan Gajovic: Computing p-adic heights on hyperelliptic curves and linear quadratic Chabauty notes |

15:30 - 16:00 | Coffee break |

16:00 - 17:00 | Netan Dogra: Nonabelian Chabauty V: The Jacobian strikes back |

17:10 - 17:50 | David Corwin: Beyond Quadratic Chabauty: Tannakian Selmer Varieties and $p$-adic Periods |

19:00 - | Workshop Dinner at Brauhaus Vetters |

##### Friday

Time | |
---|---|

09:30 - 10:30 | Enis Kaya: Computing Schneider $p$-adic heights on hyperelliptic Mumford curves notes |

10:30 - 11:00 | Coffee break |

11:00 - 12:00 | Oana Padurariu: Rational points on modular star quotients of genus two notes |

12:15 - 13:15 | Martin Lüdtke: Linear and quadratic Chabauty for affine hyperbolic curves notes |

13:15 - | Lunch |

##### Abstracts

**Steffen Müller**:*Algorithmic Aspects of Chabauty*I will discuss computational tools that make it possible to compute rational (or integral) points in practice using some instances of the Chabauty-Kim method. In particular, I will explain algorithms of Balakrishnan et al. to calculate Coleman integrals via reduction in $p$-adic cohomology. I will also discuss how to compute rational points using the quadratic Chabauty method due to Balakrishnan and Dogra, the simplest non-abelian instance of Chabauty-Kim.

**Michael Stoll**:*Selmer Group Chabauty*Frequently, the bottleneck in a (classical) Chabauty computation is the necessity to find generators of a subgroup of finite index of the Mordell-Weil group, as these points may be large and therefore hard to find. “Selmer Group Chabauty” is an approach to avoid this problem by using a suitable Selmer group as a proxy for the Mordell-Weil group, but is not guaranteed to work in all cases when the rank is less than the genus. I will explain the underlying ideas and give some examples.

**Stevan Gajovic**:*Computing $p$-adic heights on hyperelliptic curves and linear quadratic Chabauty*We present an algorithm to compute $p$-adic heights on hyperelliptic curves with good reduction. Our algorithm improves a previous algorithm of Balakrishnan and Besser by being simpler and faster and allowing even degree models. We discuss the linear quadratic Chabauty method, which relies on the $p$-adic heights, for computing integral points on certain hyperelliptic curves. This is joint work with Steffen Müller.

**Netan Dogra**:*Nonabelian Chabauty V: The Jacobian strikes back*The Chabauty–Coleman–Kim is sometimes characterised as an attempt to circumvent the use of the Jacobian in the calculation of the rational points of a curve $X$. In this talk I will try to explain some (perhaps under-appreciated) ways in which explicit calculations with algebraic cycles are necessary to use Chabauty–Coleman–Kim to determine rational points. I will focus on the example of the relevance of the Albanese kernel of $\operatorname{CH}^{2} (X \times X)$ in the calculation of the ‘full’ $X(\mathbb{Q}p)_{2}$ (not just the part involving $p$-adic heights). Finally, I will explain how these difficulties can be overcome in the case of hyperelliptic curves, and explain some work in progress with Alex Best which gives new instances of curves for which Chabauty–Coleman–Kim can be used to determine $X(\mathbb{Q})$.

**David Corwin**:*Beyond Quadratic Chabauty: Tannakian Selmer Varieties and $p$-adic Periods*Kim’s diagram provides a general framework for studying and bounding rational points on hyperbolic curves. However, it’s not clear how to compute all the maps in the diagram in general. Quadratic Chabauty relies on the bilinearity of $p$-adic height pairings to compute. Until recently, the only work beyond the quadratic setting was in the case of $\mathbb{P}^{1} - 0,1, \infty $, which relied on the Tannakian category of mixed Tate motives of Deligne–Goncharov and the $p$-adic period map of Chatzistamatiou–Unver. In general, there is no known abelian category of mixed motives, so a substitute must be used. In this talk, I explain how to extend this Tannakian approach to

*all*Selmer varieties using either Galois representations or motivic structures, as well as a $p$-adic period map of the author and Dan-Cohen that allows one to compute the localization map from the local to global Selmer varieties.**Enis Kaya**:*Computing Schneider $p$-adic heights on hyperelliptic Mumford curves*There are several definitions of $p$-adic height pairings on curves in the literature, and algorithms for computing them play a crucial role in, for example, carrying out the quadratic Chabauty method, which is a $p$-adic method that attempts to determine rational points on curves of genus at least two. The $p$-adic height pairing constructed by Peter Schneider in 1982 is particularly important since the corresponding $p$-adic regulator fits into $p$-adic versions of Birch and Swinnerton-Dyer conjecture. In this talk, we present an algorithm to compute the Schneider $p$-adic height pairing on hyperelliptic Mumford curves. We illustrate this algorithm with a numerical example computed in the computer algebra system SageMath. This talk is based on a joint work in progress with Marc Masdeu, J. Steffen Müller and Marius van der Put.

**Oana Padurariu**:*Rational points on modular star quotients of genus two*In this talk I will explain how one can use quadratic Chabauty together with the Mordell-Weil sieve to provably compute the set $X_{0}(N)^{*}(\mathbb{Q})$ when $X_{0}(N)^{*}$ has genus $2$ and has a rank $2$ Jacobian over $\mathbb{Q}$.

**Martin Lüdtke**:*Linear and quadratic Chabauty for affine hyperbolic curves*For a smooth projective hyperbolic curve the classical Chabauty method applies whenever the rank-genus inequality $g - r > 0$ holds; the quadratic Chabauty method applies whenever $g - r + \rho - 1 > 0$ holds, where $\rho$ is the Picard number of the Jacobian. I will present the analogous inequalities for affine hyperbolic curves and explain how they can be used to obtain effective bounds on the number of $S$-integral points. This is joint work with M. Leonhardt and J. S. Müller.

##### Registration & Financial Support

The deadline for registration was 10/12/2023.
If you would still like to participate, send us an email to *workshop-chabauty[at]mathi.uni-heidelberg.de*.

Here is the document necessary for reimbursement.

##### Acknowledgements

This event is funded by CRC326 - GAUS: “Geometry and Arithmetic of Uniformizing Structures”.