By Gisbert Wüstholz

Alan Baker's sixtieth birthday in August 1999 provided a fantastic chance to prepare a convention at ETH Zurich with the objective of providing the cutting-edge in quantity conception and geometry. the various leaders within the topic have been introduced jointly to give an account of study within the final century in addition to speculations for attainable additional learn. The papers during this quantity disguise a extensive spectrum of quantity idea together with geometric, algebrao-geometric and analytic points. This quantity will entice quantity theorists, algebraic geometers, and geometers with a bunch theoretic history. despite the fact that, it is going to even be precious for mathematicians (in specific learn scholars) who're attracted to being knowledgeable within the kingdom of quantity thought firstly of the twenty first century and in attainable advancements for the longer term.

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**Additional info for A panorama of number theory, or, The view from Baker's garden**

**Example text**

The factor ( f ℘ log p)2n is, of course, not desirable in any upper bound for ord℘ . Further, Kummer descent requires ζq be in K , that is, a ﬁeld extension of large degree in general, which one wishes to avoid. Thus, in order to overcome the essential difﬁculty in Kummer descent, we are forced, in Yu (1990, 1994), to appeal to the following Corollary to the Vahlen–Capelli Theorem Let q be a prime, k a positive integer, and E a ﬁeld. When q = 2 and k ≥ 2 we suppose further that ζ4 ∈ E. k If a ∈ E and a ∈ E q , then the polynomial x q − a is irreducible in E[x].

V. Chudnovsky in the late 1970s) leading to this reﬁnement. The complete proof of this result will be published in our forthcoming article (David & Hirata-Kohno 2002). Let us ﬁrst start with a short account of the history of the theory of linear forms in elliptic logarithms. Let K be an algebraic number ﬁeld of degree D over the rational number ﬁeld Q. We denote by Q the algebraic closure of Q in C. Let k be a rational integer ≥ 1. Let E1 , . . , Ek be k elliptic curves deﬁned over K . We assume that these curves are deﬁned by Weierstraß’ equations, normalized as follows†: y 2 = 4x 3 − g2,i x − g3,i : g2,i , g3,i ∈ K , 1 ≤ i ≤ k.

O. (1940), Sur la divisibilit´e de la difference des puissances de deux nombres enti`eres par une puissance d’un id´eal premier, Mat. Sbornik 7, 7–26. O. (1952), Transcendental and Algebraic Numbers, Moscow; English translation, Dover (1960). Hasse, H. (1980), Number Theory, Springer-Verlag. R. (1992), Zero-free regions for Dirichlet L-functions, and the least prime in an arithmetic progression, Proc. London Math. Soc. 64 (3), 265–338. M. (1971), Bounds for linear forms in the logarithms of algebraic numbers with p-adic metric, Vestnik Moskov.