Elliptic and Sheaf-Theoretic Structures in the Distribution of Primes over Finite Fields

This paper investigates the distribution of prime elements over finite fields through the interplay between elliptic curves and sheaf-theoretic methods. We define new algebraic-geometric invariants of primes by interpreting them as Frobenius traces on elliptic curves defined over , and further characterize their behavior using cohomological data of associated étale sheaves. By analyzing the point counts of elliptic curves modulo , we derive a structural correspondence between the statistical distribution of primes and the arithmetic of elliptic curves, including supersingular loci and trace formulas. The use of sheaf cohomology allows us to refine classical density results and propose new congruence-based filtrations on prime sets. This fusion of arithmetic geometry and finite field analysis offers a deeper geometric interpretation of prime phenomena in positive characteristic settings.