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Abstract: Recently, the weight distributions of the duals of the cyclic codes with two zeros have been obtained for several cases in Ma et al. (2011) , Ding et al. (2011) , Wang et al. (2011) . In this paper we provide a slightly different approach toward the general problem and use it to solve one more special case. We make extensive use of standard tools in number theory such as characters of finite fields, the Gauss sums and the Jacobi sums to transform the problem of finding the weight distribution into a problem of evaluating certain character sums over finite fields, which on the special case is related with counting the number of points on some elliptic curves over finite fields. Other cases are also possible by this method. [Copyright &y& Elsevier]

Abstract: We prove the existence of the pair correlation measure associated to torsion points on the real locus of an elliptic curve E and provide an explicit formula for the limiting pair correlation function. [Copyright &y& Elsevier]

Abstract: We study the distribution of the size of the Selmer groups arising from a 2-isogeny and its dual 2-isogeny for quadratic twists of elliptic curves with full 2-torsion points in . We show that one of these Selmer groups is almost always bounded, while the 2-rank of the other follows a Gaussian distribution. This provides us with a small Tate–Shafarevich group and a large Tate–Shafarevich group. When combined with a result obtained by Yu [G. Yu, On the quadratic twists of a family of elliptic curves, Mathematika 52 (1–2) (2005) 139–154 (2006)], this shows that the mean value of the 2-rank of the large Tate–Shafarevich group for square-free positive integers n less than X is , as . [Copyright &y& Elsevier]

Abstract: We investigate local spacing problems along curves via smooth maps. Moreover, we provide explicit formulas for the nearest neighbor spacing distribution function of torsion points on elliptic curves over , and of rational points on the unit circle. [Copyright &y& Elsevier]