Uniform Mordell

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)

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  • Dimitrov, Gao, Habegger. Uniformity in Mordell-Lang for curves. Ann. of Math. (2) 194 (2021), no. 1, 237–298.
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Organizers: Fabrizio Barroero, Laura Capuano and Amos Turchet