Number Theory Commons^™

Aliquot Cycles For Elliptic Curves With Complex Multiplication, Thomas Morrell

Undergraduate Theses—Unrestricted

We review the history of elliptic curves and show that it is possible to form a group law using the points on an elliptic curve over some field L. We review various methods for computing the order of this group when L is finite, including the complex multiplication method. We then define and examine the properties of elliptic pairs, lists, and cycles, which are related to the notions of amicable pairs and aliquot cycles for elliptic curves, defined by Silverman and Stange. We then use the properties of elliptic pairs to prove that aliquot cycles of length greater than two ...

Some Contributions To The Sociology Of Numbers, Robert Dawson

Journal of Humanistic Mathematics

Those who work with numbers eventually realize that they all have different personalities (the word "numbers" can of course be replaced by any number of other nouns here.) Here is one view of the issue.

An Equivariant Main Conjecture In Iwasawa Theory And The Coates-Sinnott Conjecture, Reza Taleb

Open Access Dissertations and Theses

The classical Main Conjecture (MC) in Iwasawa Theory relates values of p-adic L-function associated to 1-dimensional Artin characters over a totally real number field F to values of characteristic polynomials attached to certain Iwasawa modules. Wiles [47] proved the MC for odd primes p over arbitrary totally real base fields F and for the prime 2 over abelian totally real fields F.

An equivariant version of the MC, which combines the information for all characters of the Galois group of a relative abelian extension E/F of number fields with F totally real, was formulated and proven for odd primes ...

Brauer-Kuroda Relations For Higher Class Numbers, Adela Gherga

Open Access Dissertations and Theses

Arising from permutation representations of finite groups, Brauer-Kuroda relations are relations between Dedekind zeta functions of certain intermediate fields of a Galois extension of number fields. Let E be a totally real number field and let n ≥ 2 be an even integer. Taking s = 1 − n in the Brauer-Kuroda relations then gives a correspondence between orders of certain motivic and Galois cohomology groups. Following the works of Voevodsky and Wiles (cf. [33], [36]), we show that these relations give a direct relation on the motivic cohomology groups, allowing one to easily compute the higher class numbers, the orders of these ...