A p-adic Conservation Law for BSD Invariants under Isogeny: Quantization, Compensation, and Sha Propagation

We discover a p-adic conservation law governing the behavior of Birch and Swinnerton-Dyer (BSD) invariants under rational isogeny, and develop a complete computational theory of its consequences. For a degree-p isogeny φ: E →E′ between elliptic curves over Q with rk(E) = r, we decompose the BSD denominator D(E) = Ω·Reg· cℓ/|E(Q)tors|2 into a local part F and a global part Reg, and define a= vp(F′/F), b= vp(Reg′ /Reg). We prove computationally that a+ b is a constant depending only on (r,p): specifically, a+ b=−2 when r<rcrit(p) and a+ b= 0 when r≥rcrit(p), where rcrit(2) = 3 and rcrit(p) = 2 for all odd primes p. Using the full Cremona database (2,483,649 curves, N ≤500,000), we establish: (1) |III|· D is constant within every isogeny class (545,784 classes, zero exceptions); (2) the D-ratio Dmax/Dmin is always a perfect square (36,587 cases verified); (3) a Selmer saturation mech- anism: the 2-Selmer saturation rate increases from 28% at rank 0 to 100% at rank ≥2, explaining why |III|= 1 for all 6,680 curves of rank ≥3 and |III|∈{1,4}for all 274,346 rank-2 curves; (4) the probability that vp(|III|) varies under isogeny follows approximately ∼1/p4; (5) the 21 rank-2 exceptions all have ω(N) ≥5, E(Q)[2] ̸= 0, and 2-Selmer rank 5; (6) quantitative corrections to the Delaunay heuristic, which overestimates Pr(p ||III|) by factors of 5–60×depending on rank. These results reveal a rigid arithmetic structure govern- ing BSD invariants under isogeny, with the Selmer saturation mechanism as the underlying cause of Sha vanishing at high rank.