PreprintCC BY 4.0arXiv
How many points does an affine algebraic set have in residue classes modulo n ?
Abstract
We show that for every uniform family of affine algebraic sets, there is a formula in the parameters of the family and in n, expressing the cardinality of the set in the ring ℤ/n ℤ. In particular, we construct such a formula for the elliptic curves in Weierstrass normal form. These formulas are arithmetic terms: fixed finite compositions of additions, multiplications, subtractions, divisions with remainder and integer exponentiations.
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BibTeX
@misc{prunescu2026many,
title = {{How many points does an affine algebraic set have in residue classes modulo n ?}},
author = {Mihai Prunescu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.22049},
eprint = {2608.22049},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.22049},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - How many points does an affine algebraic set have in residue classes modulo n ?
AU - Mihai Prunescu
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2608.22049
UR - https://arxiv.org/abs/2608.22049
AB - We show that for every uniform family of affine algebraic sets, there is a formula in the parameters of the family and in n, expressing the cardinality of the set in the ring ℤ/n ℤ. In particular, we construct such a formula for the elliptic curves in Weierstrass normal form. These formulas are arithmetic terms: fixed finite compositions of additions, multiplications, subtractions, divisions with remainder and integer exponentiations.
ER -