Uniform Mordell

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).

#	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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Organizers: Fabrizio Barroero, Laura Capuano and Amos Turchet