The Hecke--Lerch Correspondence II: Imaginary Quadratic Fields and the Analytic Construction of Class Field Theory

Part II of "The Hecke-Lerch Correspondence" SeriesContext:This is the second paper in a four-part series establishing the Hecke-Lerch Correspondence—an explicit analytic framework connecting the Lerch transcendent to the arithmetic of global fields. While Part I established the correspondence for $\mathbb{Q}$ (recovering the Kronecker-Weber theorem), this paper extends the theory to imaginary quadratic fields.Abstract:This paper establishes the Hecke-Lerch correspondence for imaginary quadratic fields, extending the analytic construction of abelian class field theory from the rational numbers to quadratic imaginary extensions.For a quadratic field $K = \mathbb{Q}(\sqrt{-d})$, we define a generalized Lerch transcendent $\Phi_K(t,s,a)$, construct its reciprocal generating function $G_{s,a}(t) = 1/[(-1)^{s-1}(s-1)!\Phi_K(t,s,a)]$, and prove that the coefficients $b_n^{(s,a)}$ decompose as finite linear combinations of Hecke character sums. The Mellin transform of $G_{s,a}(e^{-t})$ recovers a linear combination of Hecke L-functions, yielding a direct analytic bridge between Lerch series and class field theory.We verify the correspondence for $K = \mathbb{Q}(i)$, thereby proving a conjecture from earlier work, and interpret the arithmetic of the coefficients in terms of Frobenius splitting in ray class fields.Series Links:Part I: The Rational Case and Explicit Class Field TheoryPart III: The Archimedean Completion and Gamma FactorsPart IV: The Global Operator and the Riemann Hypothesis