Reading Seminar on Dimitrov-Gao-Habegger Theorem

The series of seminars focuses on the seminal work of Dimitrov-Gao-Habegger on the “Uniform Mordell” conjecture. The aim is to go over the proof of the main theorem of the paper of V.Dimitrov, Z.Gao, and P.Habegger Uniformity in Mordell–Lang for curves, Ann. of Math. (2) 194 (2021), no. 1, 237–298.

**Theorem (DGH '21).** Let \(g\geq 2\) and \(d \geq 1\) be integers. Then, there exists a constant \(c = c(g,d) \geq 1\) with the following property: if \(\mathcal{C}\) is a smooth curve of genus \(g\) defined over a number field \(k\) with \([k: \mathbb{Q}] \geq d\) then \(\# \mathcal{C}(k) < c^{1 + \rho}\), where \(\rho\) is the rank of \(\mathrm{Jac}(\mathcal{C})(k)\).

We will mainly follow Gao’s Survey - Recent developments of the Uniform Mordell-Lang Conjecture.

The plan is to divide the presentation into seminars that will be shared among the participants (taking into account backgrounds and expertise).

Plan of the seminars

# | Content | Speaker | Date and Time |
---|---|---|---|

1 | Introduction: Faltings' Theorem, proofs, Uniformity results. | Amos Turchet | Nov 17 - 12:00 (M2) |

2 | Heights: Definitions, Weil Machinery, Néron-Tate Heights | Cangini, Ferrigno, Pagliaro | Dec 1 - 12:00 |

3 | Abelian Varieties: Definitions, isogenies, abelian schemes, moduli spaces and universal families | Brahimi, Pieroni, Sammarco | Dec 15 - 12:00 |

4 | The Betti map and the Betti form. | Filippo Viviani | Jan 17 - 14:00 (M1) |

5 | Non-degenerate subvarieties: definition and constructions. | Fabrizio Barroero | Feb 10 - 14:00 (M2) |

6 | Gao-Habegger Height inequality | Laura Capuano | Mar 1 - 14:15 (M1) |

7 | The new-gap principle and proof of main theorem | Amos Turchet | Mar 15 - 14:15 (M1) |

Bibliography

- Bombieri. The Mordell conjecture revisited. Annali della Scuola Normale Superiore di Pisa-Classe di Scienze, 17(4):615–640, 1990.
- Caporaso, Harris, Mazur. Uniformity of rational points. J. Amer. Math. Soc. 10 (1997), no. 1, 1–35.
- de Diego: Points rationnels sur les familles de courbes de genre au moins 2. J. Number Theory 67 (1997), no. 1, 85–114.
- Dimitrov, Gao, Habegger. Uniformity in Mordell-Lang for curves. Ann. of Math. (2) 194 (2021), no. 1, 237–298.
- Dimitrov, Gao, Habegger. Uniform bound for the number of rational points on a pencil of curves. Int. Math. Res. Not. IMRN 2021, no. 2, 1138–1159.
- Faltings. Endlichkeitsätze für abelsche varietäten über zahlkörpern. Inventiones mathematicae, 73(3):349–366, 1983.
- Gao, Habegger. Heights in families of abelian varieties and the geometric Bogomolov conjecture. Ann. of Math. (2) 189 (2019), no. 2, 527–604.
- Gao. Generic rank of Betti map and unlikely intersections. Compos. Math., 156(12):2469–2509, 2020.
- Katz, Rabinoff, and Zureick-Brown. Uniform bounds for the number of rational points on curves of small Mordell-Weil rank. Duke Math. J., 165(16):3189–3240, 2016.
- Mazur. Arithmetic on curves. Bulletin of the American Mathematical Society, 14(2):207–259, 1986.
- Rémond. Inégalité de Vojta en dimension supéieure. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 29(1):101–151, 2000.
- Stoll. Uniform bounds for the number of rational points on hyperelliptic curves of small mordell-weil rank. J. Eur. Math. Soc. (JEMS), 21:923–956, 2019.
- Vojta. Siegel’s theorem in the compact case. Ann. of Math. (2), 133(3):509–548, 1991.

Organizers: Fabrizio Barroero, Laura Capuano and Amos Turchet