Morse Topology, Zero Products, and Elliptic Prime Crossings
Abstract
We study how the noncentral zeros of an elliptic-curve L-function enter the Birch--Swinnerton-Dyer (BSD) leading coefficient and the prime-scale crossings of an elliptic Chebyshev race. Under the elliptic-curve Riemann hypothesis (ECRH), the completed L-function yields an exact Hadamard decomposition, and the zero statistics JE and VE satisfy the positive identity JE=VE/8+ΔE. For seventeen rank-one isogeny classes of conductor $50700$, the first zero contributes on average 82.4% of ΔE,20, while independent edge values reveal a discrete 35/36 zero-count split beyond height $20$. For Haar SO(2N+1) we prove a fixed-dimensional hard-edge law PrJN>y∼ CNe−3y/2. A finite zero truncation also defines a Morse function on a phase torus, whose level sets and coarea density provide a topological model for crossing statistics. Motivated by false Chebyshev primes, we define elliptic crossing primes through the midpoint and half-jump of a logarithmically weighted Frobenius race. A four-stage conductor-$50700$ computation shows agreement at the level of ensemble profiles, while omitted-zero noise prevents resolution of individual shrinking-window events at height $20$. The experiment therefore identifies a quantitative resolution barrier rather than a failure of the prime-sampled transfer.
Related
- Topology, Morse Index, and Area of Minimal Surfaces in Three-Manifolds
- Morse index, topology, and ends of minimal surfaces with noncompact free boundary
- Radial Nodal Dirichlet Solutions of Singular Elliptic Equations: Global Branches, Endpoint Asymptotics, and Morse Indices
- Average twin prime conjecture for elliptic curves in arithmetic progressions
- On the algebraic and analytic ranks of the twin-prime elliptic curve y2=x(x−2)(x−p)
BibTeX
@misc{planat2026morse,
title = {{Morse Topology, Zero Products, and Elliptic Prime Crossings}},
author = {Michel Planat},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.27962},
eprint = {2609.27962},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.27962},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Morse Topology, Zero Products, and Elliptic Prime Crossings
AU - Michel Planat
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.27962
UR - https://arxiv.org/abs/2609.27962
AB - We study how the noncentral zeros of an elliptic-curve L-function enter the Birch--Swinnerton-Dyer (BSD) leading coefficient and the prime-scale crossings of an elliptic Chebyshev race. Under the elliptic-curve Riemann hypothesis (ECRH), the completed L-function yields an exact Hadamard decomposition, and the zero statistics J_E and V_E satisfy the positive identity J_E=V_E/8+Δ_E. For seventeen rank-one isogeny classes of conductor $50700$, the first zero contributes on average 82.4% of Δ_E,20, while independent edge values reveal a discrete 35/36 zero-count split beyond height $20$. For Haar SO(2N+1) we prove a fixed-dimensional hard-edge law PrJ_N>y∼ C_Ne^−3y/2. A finite zero truncation also defines a Morse function on a phase torus, whose level sets and coarea density provide a topological model for crossing statistics. Motivated by false Chebyshev primes, we define elliptic crossing primes through the midpoint and half-jump of a logarithmically weighted Frobenius race. A four-stage conductor-$50700$ computation shows agreement at the level of ensemble profiles, while omitted-zero noise prevents resolution of individual shrinking-window events at height $20$. The experiment therefore identifies a quantitative resolution barrier rather than a failure of the prime-sampled transfer.
ER -