# Quantitative Reduction Theory and Unlikely Intersections

@article{Daw2021QuantitativeRT, title={Quantitative Reduction Theory and Unlikely Intersections}, author={Christopher Daw and Martin Orr}, journal={International Mathematics Research Notices}, year={2021} }

We prove quantitative versions of Borel and Harish-Chandra’s theorems on reduction theory for arithmetic groups. Firstly, we obtain polynomial bounds on the lengths of reduced integral vectors in any rational representation of a reductive group. Secondly, we obtain polynomial bounds in the construction of fundamental sets for arithmetic subgroups of reductive groups, as the latter vary in a real conjugacy class of subgroups of a fixed reductive group. Our results allow us to apply the Pila…

## 5 Citations

Effective André–Oort for non-compact curves in Hilbert modular varieties

- Mathematics
- 2021

In the proofs of most cases of the André-Oort conjecture, there are two different steps whose effectivity is unclear: the use of generalizations of Brauer-Siegel and the use of Pila-Wilkie. Only the…

Height bounds for certain exceptional points in some variations of Hodge structures

- Mathematics
- 2022

We consider smooth projective morphisms f : X → S of Kvarieties with S an open curve and K a number field. We establish upper bounds of the Weil height h(s) by [K(s) : K] at certain points s ∈ S(K̄)…

Distinguished categories and the Zilber-Pink conjecture

- Mathematics
- 2021

We propose an axiomatic approach towards studying unlikely intersections by introducing the framework of distinguished categories. This includes commutative algebraic groups and mixed Shimura…

Lattices with skew-Hermitian forms over division algebras and unlikely intersections

- Mathematics
- 2021

This paper has two objectives. First, we study lattices with skewHermitian forms over division algebras with positive involutions. For division algebras of Albert types I and II, we show that such a…

Unlikely Intersections with ExCM curves in A2

- MathematicsANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE
- 2021

The Zilber--Pink conjecture predicts that an algebraic curve in A2 has only finitely many intersections with the special curves, unless it is contained in a proper special subvariety. Under a large…

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