Stark-Coleman Invariants and Quantum Lower Bounds: An Integrated Framework for Real Quadratic Fields

TL;DR

This paper introduces Stark-Coleman invariants for real quadratic fields, establishing their role in class group classification under GRH and deriving improved quantum lower bounds for class group computations.

Contribution

It develops an integrated framework combining $p$-adic Hodge theory and Coleman integration to classify class groups and derive quantum complexity bounds.

Findings

Class groups are classified under GRH using Stark-Coleman invariants.

Quantum lower bounds for class group problems are improved to exponential in $rac{ ext{log } D}{( ext{log log } D)^2}$.

The framework links Stark units to the geometric structure of class groups.

Abstract

Class groups of real quadratic fields represent fundamental structures in algebraic number theory with significant computational implications. While Stark's conjecture establishes theoretical connections between special units and class group structures, explicit constructions have remained elusive, and precise quantum complexity bounds for class group computations are lacking. Here we establish an integrated framework defining Stark-Coleman invariants $\kappa_p(K) = \log_p \left( \frac{\varepsilon_{\mathrm{St},p}}{\sigma(\varepsilon_{\mathrm{St},p})} \right) \mod p^{\mathrm{ord}_p(\Delta_K)}$ through a synthesis of $p$-adic Hodge theory and extended Coleman integration. We prove these invariants classify class groups under the Generalized Riemann Hypothesis (GRH), resolving the isomorphism problem for discriminants $D > 10^{32}$. Furthermore, we demonstrate that this approach yields the quantum lower bound $\exp\left(\Omega\left(\frac{\log D}{(\log \log D)^2}\right)\right)$ for the class group discrete logarithm problem, improving upon previous bounds lacking explicit constants. Our results indicate that Stark units constrain the geometric organization of class groups, providing theoretical insight into computational complexity barriers.