Integer group determinants for the elementary abelian group of order 49

@misc{panraksa2026integer,
  title = {{Integer group determinants for the elementary abelian group of order 49}},
  author = {Chatchawan Panraksa},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.09542},
  eprint = {2609.09542},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.09542},
  note = {CC BY 4.0},
}

TY  - UNPB
TI  - Integer group determinants for the elementary abelian group of order 49
AU  - Chatchawan Panraksa
PY  - 2026
DA  - 2026/09/08
PB  - arXiv
DO  - 10.48550/arxiv.2609.09542
UR  - https://arxiv.org/abs/2609.09542
AB  - Let G=C_7× C_7, and let S(G) denote the set of integer values of its group determinant. We determine S(G) by classifying the values divisible by $7$; the coprime values are already known. Every nonzero divisible value has valuation at least $10$, and every multiple of 7^12 occurs. At valuations $10$ and $11$, we give necessary and sufficient conditions on the cofactor in terms of two additive invariants of ideals in ℤ[ζ_7]. The first condition is a bounded signed sum of prime-ideal invariants. The second requires a prime ideal with nonzero invariant pair whose norm divides the cofactor. Neither cofactor set is a union of congruence classes modulo any positive integer. The least positive divisible value is 43·7^10, and the least positive value of valuation $11$ is 8·7^11. The proof combines integral reconstruction from character values with a calculation of the global-unit image modulo $7$. We conclude by identifying the additional local conditions and realization problems that arise at primes at least $11$.
ER  -