BSD Invariants and Murmurations of Elliptic Curves

This paper studies how Birch–Swinnerton-Dyer (BSD) invariants interact with murmuration patterns in elliptic curves over Q, using 3,064,705 curves from the Cremona database (conductor up to 499,998). We establish three main findings. (1) BSD invariants themselves do not murmur: real period, Tamagawa product, analytic Sha, regulator, and torsion order are monotone in conductor windows with no oscillatory behavior. (2) These invariants do modulate the shape of Frobenius trace murmurations. Within fixed rank, stratifying by Tamagawa product, analytic Sha, or real period yields significantly different murmuration profiles (p < 0.001, permutation tests with Bonferroni correction), and the differences are scale-invariant. (3) The Sha modulation is robust to simultaneous controls for L-value, real period, and conductor, meaning it captures genuinely independent information about the distribution of Frobenius traces at good primes. The effect is a pure mean shift in the trace distribution, concentrated at small primes. Analysis of low-lying L-function zeros for 2,000 curves at matched L-value reveals that curves with |Sha| ≥ 4 have systematically higher first zeros (0.627 vs. 0.606, p = 8 × 10⁻⁶), and the explicit formula connects this zero displacement to the observed murmuration modulation. This work directly addresses an open question from He–Lee–Oliver–Pozdnyakov (2024) on stratifying murmurations by invariants beyond rank.