### 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 language | English (US) |
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Title of host publication | Number Theory and Related Fields |

Subtitle of host publication | In Memory of Alf van der Poorten |

Publisher | Springer New York LLC |

Pages | 253-270 |

Number of pages | 18 |

ISBN (Print) | 9781461466413 |

DOIs | |

State | Published - Jan 1 2013 |

Externally published | Yes |

### Publication series

Name | Springer Proceedings in Mathematics and Statistics |
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Volume | 43 |

ISSN (Print) | 2194-1009 |

ISSN (Electronic) | 2194-1017 |

### Fingerprint

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*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

**On Balanced Subgroups of the Multiplicative Group.** / Pomerance, Carl; Ulmer, Douglas.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Number Theory and Related Fields: In Memory of Alf van der Poorten.*Springer Proceedings in Mathematics and Statistics, vol. 43, Springer New York LLC, pp. 253-270. https://doi.org/10.1007/978-1-4614-6642-0_14

}

TY - GEN

T1 - On Balanced Subgroups of the Multiplicative Group

AU - Pomerance, Carl

AU - Ulmer, Douglas

PY - 2013/1/1

Y1 - 2013/1/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=84883411751&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84883411751&partnerID=8YFLogxK

U2 - 10.1007/978-1-4614-6642-0_14

DO - 10.1007/978-1-4614-6642-0_14

M3 - Conference contribution

AN - SCOPUS:84883411751

SN - 9781461466413

T3 - Springer Proceedings in Mathematics and Statistics

SP - 253

EP - 270

BT - Number Theory and Related Fields

PB - Springer New York LLC

ER -