Analyzing Time Complexity in Primality Testing via p-adic Unit Conditions and Smooth Models of Elliptic Curves <div> </div>

This paper introduces a novel approach to analyzing the time complexity of deterministic primality testing algorithms by leveraging -adic unit conditions and the geometry of smooth models of elliptic curves. By characterizing input integers through their proximity to powers of primes in a -adic setting, and applying reduction techniques on elliptic curves over finite fields, we construct an algorithmic framework that filters candidate primes via congruence and torsion conditions. Furthermore, we formalize a stratification of complexity classes according to the arithmetic of formal groups and Néron models. The proposed method bridges local p-adic behavior with global algebraic invariants, offering refined bounds and optimization pathways for prime identification. The resulting complexity analysis is not only theoretical but supports practical enhancements in algorithmic number theory.