Contributions to the Theory of Transcendental Numbers by Gregory V. Chudnovsky
By Gregory V. Chudnovsky
This quantity includes a set of papers dedicated basically to transcendental quantity concept and diophantine approximations written by means of the writer. lots of the fabrics integrated during this quantity are English translations of the author's Russian manuscripts, broadly rewritten and taken fullyyt modern. those papers and different papers integrated during this quantity have been on hand to experts in manuscript shape, yet this can be the 1st time that they've been accumulated and released. although the sooner papers were preserved within the shape within which they have been ready at the beginning, the quantity is geared up in the sort of approach as to mirror contemporary development and to permit readers to stick with fresh advancements within the box. As an introductory advisor to the quantity, the writer integrated an increased and up to date textual content of his invited handle on his paintings at the idea of transcendental numbers to the 1978 overseas Congress of Mathematicians in Helsinki. The appendix features a paper at the extremality of convinced multidimensional manifolds ready via A. I. Vinogradov and the writer in 1976. Chudnovsky bought a MacArthur beginning Fellowship in 1981.
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4. The only difference is in the choice of the auxiliary function. We choose as an auxiliary function the following one: PROOF OF THEOREM F(Z) = L-\ L-\ 2 2 c-\,M 7 7r(X,+A 2 /)z A , = 0 X? = 0 The coefficients Cx x , being polynomials in em, are determined by Siegel's lemma from the following system of linear equations: F^\zx + z2i) = 0: a = 0 , . . ,X 0 . 9. 29) | P(e") | > exp(-a/ln 3 ), where the constant c depends, in general, on c. 29) is, in fact, very close to a trivial upper bound. 29) can be also proved for numbers of the form ema, where a E A, a £ iQ.
Cambridge Philos. Soc. 77 (1975), 499-513; 79 (1976), 55-70. Part III, Proc. London Math. Soc. 33 (1976), 549-564. M5. , The transcendence of certain quasi-periods associated with Abelian functions in two variables, Compositio Math. 35 (1977), 239-258. M6. , Diophantine approximations and lattices with complex multiplication, Invent. Math. 45(1978), 61-82. M7. , On quasi-periods of Abelian functions with complex multiplication, Bull. Soc. Math. France Mem. No. 2 (1980), 55-68. M8. D. Masser and G.
5,7]. 9an be algebraic numbers linearly independent over K. , p(an))\>H(Pr^)d(Pr where c 24 (a) > 0 depends only on [K(g 2, g 3 , a , , . . ,a„):Q] provided that loglogi/(P)>c 2 5 (P) 2 , 1 + 1 . Similar results are true for Abelian varieties of CM-type of an arbitrary dimension, cf. [C19]. We present one such result in dimension two. Let A be an Abelian variety of CM-type and dimension two, defined over Q with Abelian functions Ax(z)9 A2(z)9 B(z)9 where Ax(z)9 A2(z) are Abelian functions of A, algebraically independent over C and regular with B(z) at z — 0 [M5, C22].