On Balanced Subgroups of the Multiplicative Group

Carl Pomerance, Douglas Ulmer

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

3 Scopus citations


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
Number of pages18
ISBN (Print)9781461466413
Publication statusPublished - Jan 1 2013
Externally publishedYes

Publication series

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


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