250 problems in elementary number theory by Waclaw Sierpinski

By Waclaw Sierpinski

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Suppose that 0, is in a(X,) and 0, 2 0, @,(h(b)) > 0 ; then, by making V smaller if necessary, we may assume that defines a tempered positive measure on Xu with support contained in We proved the above results in [4]-I1 under the following assumption: let X* denote the projective space obtained from X by adding a hyperplane H, and f' the rational function on X X which extends f ; then the assumption is that a tame resolution h*: Y #-,XX'of ((fX),,H,) over k, exists ; this assumption is always satisfied if char (k,) = 0.

It is my pleasure to express my gratitude to E. Horikawa who guided me to his theory of deformations of varieties carrying divisors ([3]) which was very helpful in the first part of this study. I am also grateful to T. Shioda and S. Iitaka for their valuable conversations with me in connection with this problem. Notations and conventions In addition to the basic notation introduced in use the following notations and conventions. 1) Available at the Univ. of Tokyo. 5 2, we shall also frequently FROBESICS CORRESPONDENCES For each 0 5 in 5 n and an R,-algebra A, (resp.

To check this, take any differential El of the first kind. By the same reason as above, there exists a differential 7, such that ~ ( 7 = ~ )El. Put v1 = E nbn(Q)tnQdte. Then since 6, is of the first kind, we have b-,,-,(Q) = 0 for all i 2 0 ; hence pl - dwQ is finite at Q for some rational function wQ. Obviously, we can choose WQ in such a way that (we) belongs to the adele ring of the function field of C. Since /(Kc - pD) = 0, we can find a rational function w such that w - (wQ)>- -pD for the adele (wQ).

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