Degrees of strongly special subvarieties and the André–Oort conjecture
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Christopher Daw
Abstract
In this paper we give a new proof of the André–Oort conjecture under the generalised Riemann hypothesis. In fact, we generalise the strategy pioneered by Edixhoven, and implemented by Klingler and Yafaev, to all special subvarieties. Thus, we remove ergodic theory from the proof of Klingler, Ullmo and Yafaev and replace it with tools from algebraic geometry. Our key ingredient is a lower bound for the degrees of strongly special subvarieties coming from Prasad’s volume formula for S-arithmetic quotients of semisimple groups.
Acknowledgements
The author is deeply indebted to Andrei Yafaev, who has been beyond reproach in his role as a supervisor. He is also grateful to the Department of Mathematics at University College London and to the organisers and sponsors of the ‘Around the Zilber–Pink conjectures’ summer school held in Paris during June/July 2012. He would like to thank the ANR Modig Programme for the opportunity to visit Emmanuel Ullmo at the IHES in October 2013, whose comments have been extremely helpful. Finally, the author would like to thank the referee, not only for his technical remarks, but also for his comments regarding the overall presentation and for his attention to detail.
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Articles in the same Issue
- Frontmatter
- Singularities with 𝔾m-action and\break the log minimal model program for ℳ¯g
- Normal functions, Picard--Fuchs equations, and elliptic fibrations on K3 surfaces
- Degrees of strongly special subvarieties and the André–Oort conjecture
- Ambitoric geometry I: Einstein metrics and extremal ambikähler structures
- The closure of spectral data for constant mean curvature tori in 𝕊3
- Modular p-adic L-functions attached to real quadratic fields and arithmetic applications
- Dimension invariants for groups admitting a cocompact model for proper actions
Articles in the same Issue
- Frontmatter
- Singularities with 𝔾m-action and\break the log minimal model program for ℳ¯g
- Normal functions, Picard--Fuchs equations, and elliptic fibrations on K3 surfaces
- Degrees of strongly special subvarieties and the André–Oort conjecture
- Ambitoric geometry I: Einstein metrics and extremal ambikähler structures
- The closure of spectral data for constant mean curvature tori in 𝕊3
- Modular p-adic L-functions attached to real quadratic fields and arithmetic applications
- Dimension invariants for groups admitting a cocompact model for proper actions
