By J. V. Armitage

For a few years, French mathematicians have run usual quantity concept meetings to which they've got invited quantity theorists from many nations. To pay off their hospitality, the London Mathematical Society prepared for the 1980 'Journees Arithmeiques' to be held in Exeter. The papers released listed here are both in keeping with the most invited lectures or on chosen study talks given on the convention. They conceal all branches of the topic: combinatorial and easy tools; analytic quantity concept; transcendence concept; Galois module thought and algebraic quantity idea commonly; elliptic curves and modular capabilities; neighborhood fields; additive quantity thought; Diophantine geometry, and uniform distribution. will probably be beneficial interpreting for all these venture study in quantity idea.

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**Example text**

P 7 , respectively. 19 We are interested in functions ential equations. satisfying algebraic differ- The most general definition of an algebraic dif- ferential equation over lowing [29],[38]. 1J satisfied by There are functions algebraic number field itself. ^i'^? 1) When an algebraic number f^(z),f 2 (z) satisfy an algebraic IK . 2 (Schneider-Lang [29]). satisfy an algebraic differential at most [3K:IJ] (p^ + p 2 ) Let f^zhf^z) equation over K . as above Then there are complex numbers 6 such that f.

Exp(-c 1 log H(£) for algebraic f o (z) , 0 £ and and f 2 (0) if It is possible to prove 2 ) c. 3) depending only on f^(z) , d(£) . We want to prove a more strong transcendence measure: |f 9 (0) - Cl > H U ) for an algebraic £ and 3 : H(C) ^ c. c 3 > 0 , c. > 0 (1r 4) depending on f^(z) , 20 f 2 (z) and d(C) . 4) a 'normal1 number. of f~(9) . converting with the property Our aim is to establish conditions on the differential equations satisfied by 'normality1 of $2^®^ f,. 5) We will work below only with Fuchsian linear differential equations.

These results are particular cases of our general one-dimensional and multidimensional criteria of normality that incorporate meromorphic functions of finite order of growth satisfying algebraic differential equations and ( G , O functions. We give a brief indication of all necessary steps and the outline of the proof. We do not present all proofs in all of the cases since later in the course of the paper we give the complete proof of a much more difficult theorem on the normal measure of algebraic independence of two numbers arising from elliptic function theory.