Consecutive Rankin-Cohen Bases, Full-Spark Periods, and Divisor-Tau Congruences

Source abstract

For $r\ge1$, 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 $[E_{2r+10-2j},E_{2j+2}]_1$, $1\le j\le d$, 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 $C_r(n)=\sum_{m=1}^{n-1}σ_{2r+1}(m)τ(n-m)$. 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 $f,g\in\mathbf Q[E_4,E_6]$, we prove $[f,g]_1/Δ=-3456\det((f_{E_4},f_{E_6}),(g_{E_4},g_{E_6}))$, reducing the Cramer system to a one-variable coordinate problem in $T=E_6^2/E_4^3$.