An effective result of André-Oort type

Abstract

Using transcendence theory we prove the André-Oort conjecture in case of the Shimura variety $\mathbb{A}_{\mathbb{C}}^{2}$. It is well known that this result implies the André-Oort conjecture for a product of two arbitrary modular curves. In contrast to all previous proofs we obtain a result that is at once effective and unconditional.

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      SERIES = {Appunti. Scuola Normale Superiore di Pisa (Nuova Serie) [Lecture Notes. Scuola Normale Superiore di Pisa (New Series)]},
      VOLUME = {8},
      NOTE = {With an appendix by Francesco Amoroso},
      PUBLISHER = {Edizioni della Normale, Pisa},
      YEAR = {2009},
      PAGES = {xvi+237},
      ISBN = {978-88-7642-341-3; 88-7642-341-3},
      MRCLASS = {11-01 (11D04 11D09 11D45 11D59 11G50)},
      MRNUMBER = {2517762},
      MRREVIEWER = {Damien Roy},
      ZBLNUMBER = {1186.11001},
      }
  • [Zannier2010] U. Zannier, Some Problems of Unlikely Intersections in Arithmetic and Geometry, Princeton, NJ: Princeton Univ. Press, 2012, vol. 181.
    @book{Zannier2010,
      author={Zannier, Umberto},
      TITLE={Some Problems of Unlikely Intersections in Arithmetic and Geometry},
      NOTE={with appendixes by {D}avid {M}asser},
      SERIES = {Ann. of Math. Studies},
      VOLUME = {181},
      PUBLISHER = {Princeton Univ. Press},
      ADDRESS = {Princeton, NJ},
      YEAR = {2012},
      ZBLNUMBER = {06022187},
     }

Authors

Lars Kühne

Departement Mathematik
ETH Zürich
Rämistrasse 101
8092 Zürich
Switzerland