On Balanced Subgroups of the Multiplicative Group

Carl Pomerance, Douglas Ulmer

Research output: Chapter in Book/Report/Conference proceedingConference contribution

3 Scopus citations

Abstract

A subgroup H of (ℤ/dℤ)× is called balanced if every coset of H is evenly distributed between the lower and upper halves of (ℤ/dℤ)×, i.e., has equal numbers of elements with representatives in (0,d/2) and (d/2,d). This notion has applications to ranks of elliptic curves. We give a simple criterion in terms of characters for a subgroup H to be balanced, and for a fixed integer p, we study the distribution of integers d such that the cyclic subgroup of (ℤ/dℤ)× generated by p is balanced.

Original languageEnglish (US)
Title of host publicationNumber Theory and Related Fields
Subtitle of host publicationIn Memory of Alf van der Poorten
PublisherSpringer New York LLC
Pages253-270
Number of pages18
ISBN (Print)9781461466413
DOIs
Publication statusPublished - Jan 1 2013
Externally publishedYes

Publication series

NameSpringer Proceedings in Mathematics and Statistics
Volume43
ISSN (Print)2194-1009
ISSN (Electronic)2194-1017

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ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Pomerance, C., & Ulmer, D. (2013). On Balanced Subgroups of the Multiplicative Group. In Number Theory and Related Fields: In Memory of Alf van der Poorten (pp. 253-270). (Springer Proceedings in Mathematics and Statistics; Vol. 43). Springer New York LLC. https://doi.org/10.1007/978-1-4614-6642-0_14