Boolean Walsh Eta Units and Eisenstein Bases For Squarefree Levels

Abstract

Let $M>1$ be squarefree and let $D(M)$ be its Boolean divisor cube. To each Boolean character $\chi_T$ we attach the eta-quotient \[ R_T^{(M)}(\tau)=\prod_{d\mid M}\eta(d\tau)^{\chi_T(d)}, \qquad \chi_T(d)=(-1)^{|T\cap\supp(d)|}. \] At squarefree level, the finite Fourier transform on the divisor cube simultaneously diagonalizes the squarefree Ligozat cusp-order matrix, Fricke complementation, Atkin--Lehner action on cusp labels, and the constant-term map for logarithmic Eisenstein series. In particular, for $T\ne\varnothing$, \[ \ord_{1/c}R_T^{(M)} =\frac{\Lambda_T^{(M)}}{24}\chi_T(c), \qquad \Lambda_T^{(M)}= \prod_{p\in T}(p-1) \prod_{\substack{p\mid M\\ p\notin T}}(p+1), \] and the forms $D\log R_T^{(M)}$ form a Walsh basis of the Eisenstein subspace of $M_2(\Gamma_0(M))$. The structural theorem also determines explicit Fricke constants, Atkin--Lehner eigenvalues, good-prime Hecke eigenvalues, local $U_p$ triangular blocks, a simultaneous bad-prime eigenbasis, and the indices of two explicit principal cuspidal divisor sublattices inside the formal degree-zero cusp-divisor lattice. As an application we specialize to the Heegner prime product \[ N=2\cdot3\cdot7\cdot11\cdot19\cdot43\cdot67\cdot163. \] The first Boolean boundary gives the eta-normalized Heegner-coloured partition product, while the top Walsh character gives the M\"obius eta-unit identity \[ D\log R_{\mathcal P}^{(N)}(\tau)=40415760- \sum_{n\ge1}\sigma_1(n^\perp)q^n. \] The same application gives algebraic modular-unit relations for the reciprocal partition product and exact Fricke-fixed logarithmic derivative identities for $1/\pi$, interpreted through the modular completion of $E_2$ and through accelerated paired products.