BSD Invariants and Murmurations of Elliptic Curves

Abstract

We investigate the interaction between Birch and Swinnerton-Dyer (BSD) invariants and the murmuration phenomenon for elliptic curves over the rational numbers. Our study, based on a dataset of 3,064,705 curves from the Cremona database with conductor up to 499,998, yields three results. First, the BSD invariants themselves - real period, Tamagawa product, analytic order of the Tate-Shafarevich group, regulator, and torsion order - do not exhibit murmuration-type oscillations when averaged in sliding conductor windows. Second, these invariants modulate the shape of the standard Frobenius trace murmurations: within a fixed rank, curves stratified by Tamagawa product, analytic order of the Tate-Shafarevich group, or real period display significantly different murmuration profiles, with p-values less than 0.001 against permutation null models, and these differences are scale-invariant across conductor ranges. Third, the Tate-Shafarevich group modulation survives controlling simultaneously for the L-value at 1, the real period, and the conductor, establishing that the order of the Tate-Shafarevich group encodes information about the distribution of Frobenius traces at good primes that is not captured by any other standard BSD invariant. We further show that this modulation is a pure mean shift in the Frobenius trace distribution - variance, skewness, and kurtosis are identical between Tate-Shafarevich group strata - and that it concentrates at small primes. Computing low-lying L-function zeros for 2,000 curves at fixed L-value, we find that curves with Tate-Shafarevich group order at least four have systematically different low-lying zero distributions, with the first zero displaced higher and subsequent zeros more tightly packed. The explicit formula connects this zero displacement to the observed murmuration modulation, consistent with the zero distribution acting as a mediating mechanism.

Figures (11)

• Figure 1: Mean $a_p$ for rank-0 curves stratified by Tamagawa product. Curves with $\prod_{p \mid N} c_p = 1$ (blue) have lower-amplitude murmurations than curves with $\prod_{p \mid N} c_p \geq 5$ (orange). Conductor range $[10{,}000,\, 50{,}000]$.
• Figure 2: Mean $a_p$ for rank-0 curves stratified by analytic $\Sha$. Curves with $|\Sha| \geq 4$ (orange) exhibit a qualitatively different murmuration shape from $|\Sha| = 1$ curves (blue).
• Figure 3: Scale invariance of the Tamagawa and $\Sha$ modulations across four conductor windows. The qualitative shape of the difference persists; quantitative amplitude decays as $N^{-1/4}$.
• Figure 4: The Tamagawa modulation with $\omega(N)$ controlled. At every fixed number of prime factors of the conductor, $\prod_{p \mid N} c_p = 1$ and $\prod_{p \mid N} c_p \geq 5$ curves have different murmuration profiles.
• Figure 5: The $\Sha$ modulation with $L$-value controlled. Top: $|\Sha| = 1$ vs. $|\Sha| \geq 4$ within a fixed $L$-value band. Bottom: difference in mean $a_p$. The effect is stronger after $L$-value control than without it.

Theorems & Definitions (2)

• Remark 1
• Remark 2