Geometry and Arithmetic

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pp: 91–112

DOI: 10.4171/119-1/6

Approximate computations with modular curves

Jean-Marc Couveignes[1] and Bas Edixhoven[2]

(1) Université Bordeaux 1, Talence, France
(2) Universiteit Leiden, Netherlands

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.

Keywords: Galois representations, modular curves, Ramanujan tau-function, inverse Jacobi problem