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For a function field k over a finite field with F q as the field of constants, and a finite abelian group G whose exponent divides q − 1 , we study the distribution of zeta zeroes for a random G -extension of k , ordered by the degree of conductors. We prove that when the degree goes to infinity, the number of zeta zeroes lying in a prescribed arc is uniformly distributed and the variance follows a Gaussian distribution. [ABSTRACT FROM AUTHOR]

Abstract: We prove that the pair correlation of the sequence of rational numbers in the unit interval with prime denominators is Poissonian. The result is also true for rational numbers with denominators having exactly two distinct prime factors. [Copyright &y& Elsevier]

In this paper we determine all elliptic curves En:y2=x3-n2x with the smallest 2-Selmer groups Sn=Sel2(En(Q))={1} and Sn′=Sel2(En′(Q))={±1,±n}(En′:y2=x3+4n2x) based on the 2-descent method. The values of n for such curves En are described in terms of graph-theory language. It is well known that the rank of the group En(Q) for such curves En is zero, the order of its Tate-Shafarevich group is odd, and such integers n are non-congruent numbers. [Copyright &y& Elsevier]