Walsh eta-units at squarefree level-integral structure and Fricke values

Abstract

For squarefree $M>1$, we determine the quotient of the principal cuspidal lattice of $X_0(M)$ by the sum of its primitive Atkin-Lehner character sublattices. Its invariant factors are explicit at every level. The quotient is a 2-group, and with $k\ge 3$ prime divisors its exponent is exactly $2^{k-2}$, the sharp universal power expressing every modular unit, modulo constants, through primitive sign units. We construct mixed generators and operator actions and relate the quotient to cuspidal class numbers and integral Eisenstein periods. At the Fricke point, singleton exponent sums determine the prime factors of every normalized unit value; the remaining factor is an algebraic unit. When the quadratic order has class group of exponent at most two, Kronecker's limit formula expresses the even sign values as explicit rational powers of real quadratic fundamental units. The Fricke-fixed coefficient identity for $1/\pi$ leads to a strictly monotone equation with rigorous truncation certificates. For reciprocal sign products we prove a colored-partition asymptotic and a congruence modulo 5. The classical eta representation and Kronecker-Pell results are explicitly identified as inputs.

arXiv comment22 pages