By Shafarevich I.R. (ed.)

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8. It has thus been established that there are precisely eighteen ﬁelds in question (see Baker [21]; Stark [236]). In 1976 Goldfeld [114] found a new approach to the topic based on the theory of elliptic curves and Gross and Zagier√[121] succeeded in this way to show that the class number h(d ) of Q( −d ) satisﬁes h(d ) (log d )1−ε for any ε > 0 where the implied constant depends only on ε and is effectively computable; thus all imaginary quadratic ﬁelds with any given class number can now be determined in principle.

1. 1 is consequently stronger. Recently some work of Matveev [174] has appeared which gives an improved form for the expression for n of the shape cn for an absolute constant c. Matveev’s articles contain a number of new elements and they constitute an important advance; for a discussion in the simplest case see the article by Nesterenko in [3, pp. 2. 15 is capable of generalisation in several directions. First one would expect a similar result for the linear form with algebraic coefﬁcients, taking B in place of B, in other words one would expect a result as in Baker [26], but the details have not been given as yet.

Some studies in this context had already been carried out in connection with the early inequalities in the subject and they had yielded expressions with log C of order n4 (see Baker [15, IV]). The motivation amongst other things was applications to classical theories furnishing estimates for the greatest prime factor of polynomials and of binary forms. After improvements by Sprindžuk and Kotov in this context (for references see [234]) the exponent was reduced, in 1975, to one of order n log n by Shorey [224].