Approximate computations with modular curves

Jean-Marc Couveignes (1, 2, 3, 4, 5, 6, 7) , Bas Edixhoven (8)

1 IMB - Institut de Mathématiques de Bordeaux
2 LIRIMA - Laboratoire International de Recherche en Informatique et Mathématiques Appliquées
3 LFANT - Lithe and fast algorithmic number theory
4 UB - Université de Bordeaux
5 CNRS - Centre National de la Recherche Scientifique
6 Inria - Institut National de Recherche en Informatique et en Automatique
7 Bordeaux INP - Institut Polytechnique de Bordeaux
8 Mathematical institute

This article gives an introduction for mathematicians interested in numerical computations in algebraic geometry and number theory to some recent progress in algorithmic number theory, emphasising the key role of approximate computations with modular curves and their Jacobians. These approximations are done in polynomial time in the dimension and the required number of significant digits. We explain the main ideas of how the approximations are done, illustrating them with examples, and we sketch some applications in number theory.

Domaines: Théorie des nombres [math.NT]; Géométrie algébrique [math.AG]

Jean-Marc Couveignes, Bas Edixhoven. Approximate computations with modular curves. Carel Faber, Gavril Farkas, Robin de Jong. Geometry and Arithmetic, EMS, pp.91-112, 2012, EMS Series of Congress Reports, 978-3-03719-119-4. ⟨10.4171/119⟩. ⟨hal-00772114⟩