ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

11. Mathematical Physics

1991 MS Classification: 11 Cohen, Paula B., CNRS, Université des Sciences et Technologies de Lille, France Siegel’s Lemma, Padé Approximations and Jacobians (Joint work with Enrico Bombieri.) Let C be a non-singular positive genus projective curve over a number field K. Let K(C) = K(x, y) for rational functions x, y with x non-constant of degree n. Assume x^{−1}(0) has n distinct points Q_{1}, …, Q_{n} and fix Q ∈ x^{−1}(0). Then y can be chosen as a local uniformiser at Q with a unique branch u(x) at (0, 0). We show that the growth of the height of classical Padé approximation polynomials for (1, u(x), …, u(x)^{n−1}) as a function of their degree is large if the rational equivalence class of the divisor nQ − (Q_{1} + Q_{2} + … + Q_{n}) is not a torsion point of the Jacobian of C. This reproves and generalises results of D. V. and G. V. Chudnovsky for elliptic curves. Reference: E. Bombieri, and P. B. Cohen, with Appendix by U. Zannier, Siegel’s Lemma, Padé Approximations and Jacobians, to appear in the De Giorgi volume, Annali Scuola Normale Superiore, Pisa.