Characteristic \(p\) analogues of the Mumford--Tate and André--Oort conjectures for products of ordinary GSpin Shimura varieties

• Published in: arXiv.org 2024-02
• Main Author: Jiang, Ruofan
• Format: Article
• Language: English
• Subjects: Linearity; Toruses
• ISSN: 2331-8422

Let \(p\) be an odd prime. We state characteristic \(p\) analogues of the Mumford--Tate conjecture and the André--Oort conjecture for ordinary strata of mod \(p\) Shimura varieties. We prove the conjectures for arbitrary products of GSpin Shimura varieties (and their subvarieties). Important subvarieties of GSpin Shimura varieties include modular and Shimura curves, Hilbert modular surfaces, \(\mathrm{U}(1,n)\) unitary Shimura varieties, and moduli spaces of principally polarized Abelian and K3 surfaces. The two conjectures are both related to a notion of linearity for mod \(p\) Shimura varieties, about which Chai has formulated the Tate-linear conjecture. Though seemingly different, the three conjectures are intricately entangled. We will first solve the Tate-linear conjecture for single GSpin Shimura varieties, above which we build the proof of the Tate-linear conjecture and the characteristic \(p\) analogue of the Mumford--Tate conjecture for products of GSpin Shimura varieties. We then use the Tate-linear and the characteristic \(p\) analogue of the Mumford--Tate conjectures to prove the characteristic \(p\) analogue of the André--Oort conjecture. Our proof uses Chai's results on monodromy of \(p\)-divisible groups and rigidity theorems for formal tori, as well as Crew's parabolicity conjecture which is recently proven by D'Addezio.