TY - GEN

T1 - On Balanced Subgroups of the Multiplicative Group

AU - Pomerance, Carl

AU - Ulmer, Douglas

N1 - Funding Information: The authors gratefully thank the referee for some useful suggestions. In addition, CP acknowledges partial support from NSF grant DMS-1001180.

PY - 2013

Y1 - 2013

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.

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

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

M3 - Conference contribution

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 -