Bernoulli numbers and the integer group determinants of C_p × C_p

@misc{panraksa2026bernoulli,
  title = {{Bernoulli numbers and the integer group determinants of $C_p \times C_p$}},
  author = {Chatchawan Panraksa},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.15262},
  eprint = {2610.15262},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.15262},
  note = {CC BY 4.0},
}

TY  - UNPB
TI  - Bernoulli numbers and the integer group determinants of C_p × C_p
AU  - Chatchawan Panraksa
PY  - 2026
DA  - 2026/10/11
PB  - arXiv
DO  - 10.48550/arxiv.2610.15262
UR  - https://arxiv.org/abs/2610.15262
AB  - For a prime p≥7, let S(C_p× C_p) be the set of integer group determinants and e(p) the least exponent for which p^e(p)Z⊆ S(C_p× C_p). Let r_p be the least integer r≥1 satisfying (2r+1)i>r(p−1) for every irregular pair (p,2i) with r| i. We prove the unconditional inclusion p^p+3+2r_pZ⊆ S(C_p× C_p). Equality e(p)=p+3+2r_p holds when Vandiver's conjecture holds for p and r_p≤(p−5)/2, hence for every prime 7≤ p<2^31. The first exception to e(p)=p+5 is p=103, where e(103)=116. A polynomial obstruction from Bernoulli numbers supplies the lower bound, and circular units give the construction. For regular primes, we characterize the entire determinant set through explicit affine systems over F_p and principality conditions in Z[ζ_p]. For regular p≥11, prime cofactors cannot occur at valuation p+3; at valuation p+4 they have natural density (p+1)/(hp^(p−3)/2), where h is the class number of Q(ζ_p). We also show that p^p+5 is the least absolute value of a nonzero p-divisible determinant for these regular primes.
ER  -