Consecutive Rankin-Cohen Bases, Full-Spark Periods, and Divisor-Tau Congruences
Abstract
For r≥1, let K=2r+14 and d=dim SK. We prove strict positivity for determinants of Mellin period functionals, yielding a full-spark theorem for periods in a fundamental half-range and, in particular, the mixed odd--even period independence conjecture of Xue. As a consequence, the consecutive first Rankin--Cohen brackets [E2r+10−2j,E2j+2]1, 1≤ j≤ d, form a basis of SK, and we determine the signs of all admissible ordered determinants of first Eisenstein brackets. This gives a uniform exact all-weight formula for the divisor--tau convolution Cr(n)=∑m=1n−1σ2r+1(m)τ(n−m). Reducing the same coordinate identity modulo primes, we obtain canonical prime-wise reductions for infinitely many primes in every weight and characterize sparse Ramanujan-type specializations through vanishing Cramer coordinates. Finally, for homogeneous f,g∈Q[E4,E6], we prove [f,g]1/Δ=−3456det((fE4,fE6),(gE4,gE6)), reducing the Cramer system to a one-variable coordinate problem in T=E62/E43.
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BibTeX
@misc{lam2026consecutive,
title = {{Consecutive Rankin-Cohen Bases, Full-Spark Periods, and Divisor-Tau Congruences}},
author = {Kelvin Lam},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.10052},
eprint = {2610.10052},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.10052},
}
RIS
TY - UNPB
TI - Consecutive Rankin-Cohen Bases, Full-Spark Periods, and Divisor-Tau Congruences
AU - Kelvin Lam
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.10052
UR - https://arxiv.org/abs/2610.10052
AB - For r≥1, let K=2r+14 and d=dim S_K. We prove strict positivity for determinants of Mellin period functionals, yielding a full-spark theorem for periods in a fundamental half-range and, in particular, the mixed odd--even period independence conjecture of Xue. As a consequence, the consecutive first Rankin--Cohen brackets [E_2r+10−2j,E_2j+2]_1, 1≤ j≤ d, form a basis of S_K, and we determine the signs of all admissible ordered determinants of first Eisenstein brackets. This gives a uniform exact all-weight formula for the divisor--tau convolution C_r(n)=∑_m=1^n−1σ_2r+1(m)τ(n−m). Reducing the same coordinate identity modulo primes, we obtain canonical prime-wise reductions for infinitely many primes in every weight and characterize sparse Ramanujan-type specializations through vanishing Cramer coordinates. Finally, for homogeneous f,g∈Q[E_4,E_6], we prove [f,g]_1/Δ=−3456det((f_E_4,f_E_6),(g_E_4,g_E_6)), reducing the Cramer system to a one-variable coordinate problem in T=E_6^2/E_4^3.
ER -