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, 4for 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.
Tao Rui (Fri,) studied this question.