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Abstract:
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First part of dissertation examines sumsets hA = {a1 + · · · + ah ∈ Z
d
:
a1, . . . , ah ∈ A}, where A is a finite set in Z
d
. 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 con gruent 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 : y
2 = x
3 − n
2x 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. |