Localized GL_2 Bost-Connes systems: the equilibrium range and the automorphic obstruction range are disjoint, by exactly one

Abstract

For GL_2 the semigroup crossed product is unavailable (the unit group SL_2(Z) is infinite, by the unit-group exclusion of Paper X) and one must pass to the Connes-Marcolli Hecke quotient, whose local factor is the Solomon zeta of M_2(Z_p), computed by the sublattice count sigma(p^n). The obstruction series obeys the GL_1-type dictionary with only the standard L-function (the symmetric square term is spurious, being quadratic in the trace), and the equilibrium range (1 + beta_c, infinity) is disjoint from the obstruction range (0, beta_c] by a gap of width exactly one, the same multiplicity-norm mismatch as Paper XII.