By Gisbert Wüstholz

Alan Baker's sixtieth birthday in August 1999 provided an excellent chance to prepare a convention at ETH Zurich with the target of offering the state-of-the-art 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 idea together with geometric, algebrao-geometric and analytic facets. This quantity will entice quantity theorists, algebraic geometers, and geometers with a host theoretic historical past. in spite of the fact that, it is going to even be necessary for mathematicians (in specific learn scholars) who're drawn to being expert within the kingdom of quantity conception at the beginning of the twenty first century and in attainable advancements for the long run.

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1 ℘k (z k ) ℘k (z k ) which is a meromorphic function, analytic at z = (z 1 , . . , 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 !

K If a ∈ E and a ∈ E q , then the polynomial x q − a is irreducible in E[x]. See Capelli (1901) and R´edei (1967). 16 Kunrui Yu Now we choose the ‘Kummer prime’ q as 3 if p = 2 2 if p > 2. q= (3) Then Kummer descent requires that ζ3 ∈ K when p = 2; and ζ2 ∈ K (when p > 2) holds trivially. In order to be able to apply the above Corollary with the prime q given by (3), we may simply assume, in addition, that ζ4 ∈ K when p > 2 (see Yu 1990). , the dependence is a factor p 2 in the p-adic estimates when α1 , .

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