Galois orbits and equidistribution of special subvarieties: towards the André-Oort conjecture

By E. Ullmo, A. Yafaev. 2012

80 citations recorded by the source corpus.

In this paper we develop a strategy and some technical tools for proving the Andr e-Oort conjecture. We give lower bounds for the degrees of Galois orbits of geometric components of special subvarieties of Shimura varieties, assuming the Generalised Riemann Hypothesis. We proceed to show that sequences of special subvarieties whose Galois orbits have bounded degrees are equidistributed in a suitable sense.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/acd8386a48d52c39ad6e993712d90f745ac42608

How researchers see this paper

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In Discrete Mathematics and Combinatorics

1. Jacob Tsimerman
2. Ziyang Gao
3. Bruno Klingler
4. Jonathan Pila
5. A. Yafaev
6. Manjul Bhargava
7. Benjamin Bakker

In Geometry and Topology

1. Ziyang Gao
2. Jacob Tsimerman
3. Jonathan Pila
4. Bruno Klingler
5. A. Yafaev
6. E. Ullmo
7. P. Habegger
8. Benjamin Bakker

In Mathematical Physics

1. Ziyang Gao
2. Jacob Tsimerman
3. Jonathan Pila
4. Bruno Klingler
5. A. Yafaev
6. E. Ullmo
7. Benjamin Bakker
8. Frans Oort

Citations in the corpus

• The André-Oort conjecture (2012, methodology): “2 in [36] (and its proof) that φ|Z : Z −→ Z ′ := φ(Z) is generically injective.”
• Applications of the hyperbolic Ax–Schanuel conjecture (2017, unknown): “Furthermore, by [36], Lemma 3.7, the number of Shimura subdata ( G F , Y ) of ( G, X ) is finite and, by [22], Corollary 5.3, the number of connected components Y of Y is also finite.”
• Ax-Lindemann for $\mathcal{A}_g$ (2014, methodology): “This gives a new proof of AO for A g for all g assuming GRH, by different methods from the ones employed by Klingler-Ullmo-Yafaev [18], [42].”
• The hyperbolic Ax-Lindemann-Weierstraß conjecture (2015, unknown): “5 of [31] which applies verbatim in our case, we see that there exists an element σ in (Y) such that σ Y contains at least T c1 4n elements γ ∈ (Y) ∩ G(Z) such that H(γ ) ≤ T 1 2n .”
• Towards the Andr\'e-Oort conjecture for mixed Shimura varieties: the Ax-Lindemann theorem and lower bounds for Galois orbits of special points (2013, methodology): “nditionally or under GRH without using o-minimality) by Moonen [39], Andr´e [2], Edixhoven [23] [24], Edixhoven–Yafaev [26] and Yafaev [66] [67]. It was proved under GRH by Klingler, Ullmo and Yafaev [61], [35] where equidistribution results of Clozel-Ullmo [16] were used. Our generalized version of”