×

Why is it difficult to compute the Mordell-Weil group. (English) Zbl 1219.11099

Zannier, Umberto (ed.), Diophantine geometry. Selected papers of a the workshop, Pisa, Italy, April 12–July 22, 2005. Pisa: Edizioni della Normale (ISBN 978-88-7642-206-5/pbk). Centro di Ricerca Matematica Ennio De Giorgi (CRM) Series (Nuova Serie) 4, 197-219 (2007).
The main contribution of the paper is a Brauer-Siegel type conjecture for abelian varieties over number fields. The conjecture states that the product of the order Tate-Shafarevich group and the Mordell-Weil regulator grows like the height when we run through the family of all abelian varieties over a fixed number field. This conjecture explains why it is so difficult to compute the group of rational points on abelian varieties over number fields: either the height of generators of the group is very large (comparable to the height of the variety itself), or the obstructions to the Hasse principle appear in great numbers. This conjecture is analogous to the well-known Brauer-Siegel theorem for number fields (where the order of the Tate-Shafarevich group plays the role of the class number, the Mordell-Weil regulator is just the ordinary regulator of a number field and the height of an abelian variety is replaced by the discriminant of a number field). Assuming the Birch and Swinnerton-Dyer conjecture, the generalized Riemann hypothesis for \(L\)-functions of abelian varieties and the generalized Szpiro conjecture the author reduces his conjecture to a plausible conjecture concerning the asymptotic behaviour of the special values of \(L\)-functions of abelian varieties at the central point of the critical strip.
For the entire collection see [Zbl 1113.11003].

MSC:

11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
11G05 Elliptic curves over global fields
11G10 Abelian varieties of dimension \(> 1\)
11G50 Heights
PDF BibTeX XML Cite
Full Text: Link