Organisers: Harry Schmidt and Ju-Feng Wu

General information: In number theory, Malle's conjecture (roughly) predicts the asymptotic growth of the number of number fields with bounded discriminant; Batyrev–Manin conjecture (roughly) predicts the asymptotic growth of the number of rational points with bounded height. These two conjectures turn out to have a very similar form. To understand such a similarity, the key ingredient is a notion of heights on algebraic stacks. In this study group, we follow [ESZB23] to learn how such a notion is constructed. We also take a special focus on the examples. 

Time: Fridays 1 pm

Room: B3.03

Schedule:

Talk 1 (26. Apr.): Introduction. Speaker: Ju-Feng

Talk 2 (03. May): Sites. Speaker: Katerina

[Stacks] and [Wu23, Lecture 8]

Talk 3 (10. May.): Algebraic spaces and algebraic stacks. Speaker: Muhammad

[Stacks] and [Hei10]

Talk 4 (17. May): Tuning stacks. Speaker: Kenji

[ESZB23, Sect. 2.1]

Talk 5 (24. May.): Matrisable vector bundles. Speaker: Joseph

[ESZB23, Sect. A]

Talk 6 (31. May.): Heights on stacks. Speaker: Ju-Feng

[ESZB23, Sect. 2.2 & 2.3]

Talk 7 (07. Jun.): Example I: BG and Bμ_n. Speaker: Kenji

[ESZB23, Sect. 3.1 & 3.2]

Talk 8 (14. Jun.): Example II: Weighted projective spaces. Speaker: James

[ESZB23, Sect. 3.3]

Talk 9 (21. Jun.): Example III: Abelian varieties. Speaker: Harry

[ESZB23, Sect. 3.4]

Talk 10 (24. Jun.): A conjecture of Batyrev–Manin–Malle type. Speaker: Harry

[ESZB23, Sect. 4.1 - 4.4]

Reference

[ESZB23] Jordan S. Ellenberg, Matthew Satriano, and David Zureick-Brown, 'Heights on stacks and a generalized Batyrev–Manin–Malle conjecture'. In: Forum of Mathematics, Sigma (2023)

[Hei10] Jochen Heinloth, 'Lectures on the Moduli Stack of Vector Bundles on a Curve'. In: Affine Flag Manifolds and Principal Bundles. Trends in Mathematics. Springer, Basel (2010)

[Stacks] Stacks project authors, 'The Stacks project'.

[Wu23] Ju-Feng Wu, 'MA939 - Topics in Number Theory: Introduction to p-adic Geometry'. Lecture notes (2023)