By Gisbert Wüstholz

ISBN-10: 0511063903

ISBN-13: 9780511063909

ISBN-10: 0521807999

ISBN-13: 9780521807999

Alan Baker's sixtieth birthday in August 1999 provided a great chance to arrange a convention at ETH Zurich with the objective of providing the cutting-edge in quantity thought and geometry. the various leaders within the topic have been introduced jointly to give an account of analysis within the final century in addition to speculations for attainable additional study. The papers during this quantity hide a extensive spectrum of quantity conception together with geometric, algebrao-geometric and analytic points. This quantity will entice quantity theorists, algebraic geometers, and geometers with a bunch theoretic historical past. even if, it's going to even be invaluable for mathematicians (in specific examine scholars) who're attracted to being educated within the country of quantity thought first and foremost of the twenty first century and in attainable advancements for the long run.

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**Example text**

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. Univ. Ser. I, 26, 3–10. Lang, S. (1990), Old and new conjectured Diophantine inequalities, Bull. ) Amer. Math. Soc. 23 (1), 37–75. Langevin, M. (1993), Cas d’´egalit´e pour le Th´eor´eme de Mason et application de la conjecture (abc), Comptes Rendus Acad.

Yu, Kunrui (1998), P-adic logarithmic forms and group varieties I, J. Reine Angew. Math. 502, 29–92. Yu, Kunrui (1999), p-adic logarithmic forms and group varieties II, Acta Arith. 89, 337–378. 3 Recent Progress on Linear Forms in Elliptic Logarithms Sinnou David and Noriko Hirata-Kohno 1 Introduction In this article, we describe recent progress on the theory of linear forms in logarithms associated with elliptic curves deﬁned over a number ﬁeld. g. complex multiplication), we obtain the ﬁrst best possible dependence in the height of the linear form.

Z k ) = 0. Put also G(t) = P (β1 z 1 (t1 ) + · · · + βk z k (tk ), (w1 (t1 ), t1 , −2) , . . , (wk (tk ), tk , −2)) , a meromorphic function, analytic at t = (t1 , . . , tk ) = 0. For τ1 , . . , τk ∈ Z, τ1 , . . , τk ≥ 0, put τ = (τ1 , . . , τk ), |τ | = τ1 + · · · + τk . We deﬁne τ z F(z) = 1 τ1 ! . τk ! ∂ ∂z 1 τ1 ◦ ··· ◦ 1 ∂z k τk F(z), Recent Progress on Linear Forms in Elliptic Logarithms 33 and similarly τ t G(t) = 1 τ1 ! . τk ! ∂ ∂t1 τ1 ◦ ··· ◦ ∂ ∂tk τk G(t). Then we have the following two properties.

### A Panorama of Number Theory or The View from Baker's Garden by Gisbert Wüstholz

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