ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

3. Number Theory and Arithmetic Algebraic Geometry

1991 MS Classification: 11, 14 Leprévost Franck Université Paris 7, Technische Universität Berlin On some elliptic surfaces and elliptic curves related to discriminants of cubic or quartic polynomials We associate to each discriminant t ≠ 0 of a polynomial of degree n ≥ 1 an elliptic curve E_{t}. For each polynomial of degree n = 3, 4, we construct a rational point on E_{t}(ℚ) which is generically non torsion. For n = 4, we prove that E_{t} ≃ C, where C is the cubic associated by Gaàl, Pethö and Pohst to each quartic field. They also constructed a surface S connected to the cubic C. We compute the Kodaira dimension and the dimension of the Albanese variety of S and specify the class to which S belongs in the classification of algebraic surfaces. We also answer a question of Pethö, and give statistical results.