p-adic Analytic Methods and Berkovich Monge-Amp`ere Geometry on the Birch and Swinnerton-Dyer Conjecture

Abstract

This research paper establishes a rigorous framework connecting non-Archimedean ana- lytic geometry on Berkovich spaces with p-adic Galois representations and Iwasawa theory to address structural aspects of the Birch and Swinnerton-Dyer (BSD) Conjecture for elliptic curves over Q. By constructing a Push-forward Monge-Amp`ere measure on the canonical dual simplicial complex Σ of the strictly semi-stable compactification X , we prove a precise comparison identity relating the Berkovich Monge-Amp`ere density νber(T,x0) to the cyclo- tomic Bloch-Kato p-adic height regulator Rp(E). Furthermore, we resolve the ordinary versus supersingular reduction dichotomy using Perrin-Riou’s L-invariant and demonstrate mass conservation on the Berkovich skeleton. Conditioned on the p-adic height non-degeneracy hypothesis and the Iwasawa Main Conjecture for TpE, we establish the finite corank property of the p-adic Selmer group and the conditional finiteness of the p-primary Tate-Shafarevich group Sha(E)[p∞]. : Declaration of Generative AI and AI-Assisted Technologies (including Claude and Gemini) : AI Use Declaration: Generative AI tools were used during manuscript preparation exclusively for language polishing, LaTeX syntax formatting, and assisting in the structuring of Lean 4 tactic workflows. All mathematical proofs, derivations, and underlying theoretical concepts were formulated, checked, and validated independently by the author, who holds sole responsibility for the work.