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

Abstract. Thisresearchpaperestablishesarigorousframeworkconnectingnon-Archimedean analytic geometry on Berkovich spaces with p-adic Galois representations and Iwasawa the- ory 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 cyclotomic Bloch–Kato p-adic height regulator Rp(E). Furthermore, we resolve the ordi- nary versus supersingular reduction dichotomy using Perrin-Riou’s L-invariant and demon- strate 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/Q)[p∞].