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The arithmetic of cyclic subgroups / by Zebediah Engberg.

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Author	Engberg, Zebediah E.
Title	The arithmetic of cyclic subgroups / by Zebediah Engberg.
Date	2014
Abstract	In this thesis, we consider several problems relating to cyclic subgroups of the group [mathematical equation]. Each element of [mathematical equation] has a unique representative in one of the two intervals [mathematical equation] and [mathematical equation]. A subgroup H of [mathematical equation] is balanced if every coset of H intersects these two intervals equally. Such subgroups have repercussion for ranks of elliptic curves over function fields. We investigate several statistical questions about balanced subgroups, including the following: For a fixed integer g ? 0,1, how frequently is the cyclic subgroup <g mod n> balanced? We prove a conjecture of Pomerance and Ulmer that this distribution is essentially determined by two special families of balanced subgroups. One of the deeper analytic tools we require is a prime number theorem of Grantham which counts primes in an arithmetic progression splitting completely in a given number field. The behavior of the cyclic subgroup <2 mod p> as p ranges over odd primes is closely connected to the arithmetic of the Mersenne numbers. Let [mathematical equation], the reciprocal sum of the primes dividing the nth Mersenne number [mathematical equation]. Erd?s showed that f(n) < log log log n +C for some constant C. Apart from the exact value of C, it is easy to prove that this inequality is tight. (Although it would be more interesting to understand the maximal order of [mathematical equation], the function f(n) is more tractable, albeit still difficult.) We give a conditional answer to Erd?s's question on the exact value of C. Moreover, we show that Erd?s's theorem is true when the Mersenne number [mathematical equation] is replaced with the nth Fibonacci number. Finally, we consider values of the order of the subgroup <2 mod p>.
Description	vii, 164 leaves : ill.
Format	application/pdf
College	Dartmouth College. Department of Mathematics.
Issuing Body	Dartmouth Dissertations are made available through the Dartmouth Digital Library.
Degree	Thesis (Ph.D.)--Dartmouth College.
Copyright	All rights reserved. No part of this book may be reproduced in any form by any electronic or mechanical means without permission in writing from the author.
DOI	http://dx.doi.org/10.1349/ddlp.1251

Title	Page 1
Full Text	THE ARITHMETIC OF CYCLIC SUBGROUPS A Thesis Submitted to the Faculty in partial ful llment of the requirements for the degree of Doctor of Philosophy in Mathematics by Zebediah Engberg DARTMOUTH COLLEGE Hanover, New Hampshire May 6, 2014 Examining Committee: Carl Pomerance, Chair Thomas R. Shemanske John Voight Florian Luca Cameron L. Stewart F. Jon Kull Dean of Graduate Studies

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The arithmetic of cyclic subgroups / by Zebediah Engberg.

The arithmetic of cyclic subgroups / by Zebediah Engberg.