Статистика Селмерових група у фамилији елиптичких кривих придружених конгруентним бројевима

Abstract

First part of dissertation examines sumsets hA = {a1 + · · · + ah ∈ Zd : a1, . . . , ah ∈ A}, where A is a finite set in Zd. It is known that there exists a constant h0 ∈ N and a polynomial pA(X) such that pA(h) = |hA| for h ⩾ h0. However, little is known of polynomial pA and constant h0. Cone CA over the set A contains information about hA, for all h ∈ N. When A has d + 2 elements, polynomial pA and constant h0 can be explicitly described. When A has d + 3 elements, an upper bound is found for the number of elements of hA.

Second part of dissertation examines Selmer groups of elliptic curves in the congruent number family. A squarefree natural number is congruent if and only if there exists a right triangle with area n whose sides all have integer lengths. It is known that n is a congruent number if and only if elliptic curve En : y2 = x3 − n2x has nonzero rank as an algebraic group. Selmer groups of isogenies on En are interesting, because their rank is not smaller than the rank of En, so when the Selmer groups have rank zero, then the elliptic curve En also has rank zero. Elements of these Selmer groups can be represented as partitions of a particular graph, from which one may find the distribution of ranks of Selmer groups.