On balanced subgroups of the multiplicative group

2012/04/30 by Carl Pomerance, Pomerance, Carl, Douglas Ulmer +1 · 1 citation
Mathematics · #11N37 (Primary) 11G05 (Secondary) #Algebraic Geometry and Number Theory #Analytic Number Theory Research #FOS: Mathematics #Limits and Structures in Graph Theory #Number Theory (math.NT) #math.NT #msc:11G05 #msc:11N37

paper · pdf · doi:10.48550/arxiv.1204.6705

14 pages

arxiv created 2012/04/30 · openalex publication_date 2012/04/30 · arxiv updated 2012/05/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A subgroup H of G=(Z/dZ)^* is called balanced if every coset of H is evenly distributed between the lower and upper halves of G, 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.

Citations

• The average rank of elliptic curves I

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