By Gisbert Wüstholz

Alan Baker's sixtieth birthday in August 1999 provided an awesome chance to arrange a convention at ETH Zurich with the aim of featuring the cutting-edge in quantity idea and geometry. a number of the leaders within the topic have been introduced jointly to offer an account of analysis within the final century in addition to speculations for attainable extra examine. The papers during this quantity conceal a extensive spectrum of quantity concept together with geometric, algebrao-geometric and analytic points. This quantity will attract quantity theorists, algebraic geometers, and geometers with a bunch theoretic historical past. even though, it's going to even be worthy for mathematicians (in specific learn scholars) who're drawn to being educated within the kingdom of quantity concept initially 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**

**Sample text**

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. We denote by ℘i , for 1 ≤ i ≤ k, (resp.

SSSR. Ser. Mat. 15, 153–176; English translation in Amer. Math. Trans. Ser. 2, 59, (1966), 246–270). I. (1958), Simultaneous approximation of the periods of an elliptic function by algebraic numbers, Izv. Akad. Nauk. SSSR, Ser. Mat. 22, 563–576; English translation in Amer. Math. Trans. Ser. 2, 59, (1966), 271– 284). I. O. Gel’fond, Trudy. Moskov 18, 65–76; English translation in Trans. Moscow Math. Soc. 18, (1968), 71–84). Hirata-Kohno, N. (1990), Formes lin´eaires d’int´egrales elliptiques, in S´em.

Now we explain (i)–(iii) as a whole. Subject to some cost – see Yu (1990), pp. 97–98 – we may assume that α1 , . . , αn are ℘-adic units. It is well-known that the multiplicative group of the residue class ﬁeld of K ℘ is a cyclic group of order G= (℘) = p f ℘ − 1, and that it is generated by the residue class represented by ζ , where ζ is a primitive Gth root of unity in K ℘ . So it is a natural choice to use this root of unity. Thus we can ﬁnd r1 , . . , rn ∈ Z with 0 ≤ r j < G such that ord℘ α j ζ r j − 1 ≥ 1 (1 ≤ j ≤ n).