On balanced subgroups of the multiplicative group

TL;DR

This paper introduces a criterion for identifying balanced subgroups in the multiplicative group modulo d, explores their distribution, and discusses implications for elliptic curve ranks.

Contribution

It provides a character-based criterion for balanced subgroups and analyzes the distribution of such subgroups generated by a fixed integer p.

Findings

Criterion for balanced subgroups using characters

Distribution patterns of balanced subgroups generated by p

Applications to elliptic curve rank analysis

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.