Sato-Tate families in localized Bost-Connes systems: open phase, no congruence obstruction, and the symmetric power expansion
Year: 2026
Type: preprint
Abstract: For a non-CM elliptic curve over Q and a Sato-Tate interval of angles, the associated prime family has positive density, critical exponent one, and an open symmetry-broken phase, with zeta_S(beta) = (delta + o(1)) log(1/(beta-1)) unconditionally, sharpening to O(1) under any effective Sato-Tate error term. Joint equidistribution of angle and residue (open image, Newton-Thorne automorphy, Jacquet-Shalika and Shahidi non-vanishing) makes every detecting set of positive density, so the KMS state is unique on (0,1] and the simplex is full above: the opposite verdict, and the opposite mechanism, to the kernel-trapped families of Paper XXIV. The log zeta_S expansion in symmetric power L-functions is derived, with the Chebyshev coefficients in closed form and all numerics verified on y^2 = x^3 + x + 1.
Author: Chen, Ruqing
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7208810031
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