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Modular forms, elliptic curves and the abc-conjecture. (English) Zbl 1046.11035
Wüstholz, Gisbert (ed.), A panorama in number theory or The view from Baker’s garden. Based on a conference in honor of Alan Baker’s 60th birthday, Zürich, Switzerland, 1999. Cambridge: Cambridge University Press (ISBN 0-521-80799-9/hbk). 128-147 (2002).
This article surveys the Masser-Oesterlé “abc” conjecture (and some of its variants) and its relations with known conjectures for elliptic curves and their moduli spaces. These latter conjectures include the Szpiro conjecture bounding the discriminant of an elliptic curve in terms of the conductor; Frey’s “degree conjecture” bounding the degree of a modular elliptic curve in terms of the conductor; the author’s “period conjecture” giving a lower bound for the periods of Frey-Hellegouarch elliptic curves in terms of the conductor; a conjectured upper bound for the size of the Tate-Shafarevich group Ш in terms of the conductor; and the author’s “modular symbol conjecture”. For the latter two conjectures, the equivalence with abc-like conjectures is conditional on other conjectures for elliptic curves.
MSC:
11G05Elliptic curves over global fields
11F67Special values of automorphic L-series, etc