Special points on products of modular curves
By Bas Edixhoven. 2003
61 citations recorded by the source corpus.
We prove the Andre-Oort conjecture on special points of Shimura varieties for arbitrary products of modular curves, assuming the Generalized Riemann Hypothesis. More explicitly, this means the following. Let n be a positive integer, and let S be a subset of C^n (with C the complex numbers) consisting of points all of whose coordinates are j-invariants of elliptic curves with complex multiplications. Then we prove (under GRH) that the irreducible components of the Zariski closure of S are ``special subvarieties'', i.e., determined by isogeny conditions on coordinates and pairs of coordinates. A weaker variant is proved unconditionally.
Subfields: Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/138c73e0a59ef29b46bd2cd5e2be71568e862ffa
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In Geometry and Topology
- C. Cornut
- V. Vatsal
- Bruno Klingler
- A. Yafaev
- Benjamin Howard
- Jacob Tsimerman
- Jonathan Pila
- Robert H. Pollack
In Mathematical Physics
- V. Vatsal
- C. Cornut
- Bruno Klingler
- A. Yafaev
- Benjamin Howard
- Jacob Tsimerman
- Jonathan Pila
- Ziyang Gao
Citations in the corpus
- CM points and quaternion algebras (2005, background): “K (Hint: use section 7.3 of [6] and a variant of Proposition 2.1 of [ 5 ]).”
- The André-Oort conjecture (2012, background): “In the work [17], Edixhoven proves, assuming the GRH, the André-Oort conjecture for products of modular curves.”