Sato-Tate Distributions and Moment Sequences in Elliptic Curves — E8 Intelligence Research

FINDING: Sato-Tate distributions govern the statistical equidistribution of Frobenius traces of elliptic curves, with moment sequences encoding deep arithmetic-geometric structure. MATH: For an elliptic curve \(E/Q\) without CM, the normalized trace \(a_p/(2√p)\) is equidistributed in \([-1,1]\) with respect to the Sato-Tate measure \(μST = 2/π√1-x^2\,dx\). Its moments are \(∫₋₁¹ x²ᵏ dμST = 1/k+1{2k}{k}\) (Catalan numbers), and odd moments vanish. For CM curves, the measure is \(1/π{dx}{√1-x^2}\) (arcsine law), with moments \({2k}{k}/4^k\). Sutherland's work extends this to moment sequences of Sato-Tate groups (finite subgroups of \(USp(4)\), \(SU(2)\), etc.), where moments are integer-valued and classify the group up to conjugacy. CONNECTION: The Catalan numbers \(C_k = 1/k+1{2k}{k}\) appear — these are the moments of the semicircle law, w Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com