Optimal explicit-formula bounds for analytic ranks: the limiting radix kernel
- 1Independent researcher (MEGALODON Research)
Abstract
For an elliptic curve E over Q and a window [-L,L], the radix kernel of Hackfort (2026) is the test function that is optimal for each curve, within a finite-dimensional space, for bounding the analytic rank of E under GRH by the explicit formula. Assuming GRH, we prove that the optimum over all admissible test functions with this support exists, is unique, equals the reciprocal of the value at the origin of the reproducing kernel of the Hilbert space defined by the Weil quadratic form, and is the limit of the finite-dimensional radix bounds, with an exact error identity. By Akhiezer's factorisation theorem, no continuous test function g with support in [-2L,2L], non-negative Fourier transform and positive value of the transform at 0 gives a smaller bound. The key step is a coercivity estimate for the Weil form, obtained by writing it as a truncated Wiener-Hopf form with a logarithmically growing symbol.
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BibTeX
@misc{hackfort2026optimal,
title = {{Optimal explicit-formula bounds for analytic ranks: the limiting radix kernel}},
author = {Sabrina Hackfort},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23649057},
url = {https://zenodo.org/records/23649057},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Optimal explicit-formula bounds for analytic ranks: the limiting radix kernel
AU - Sabrina Hackfort
PY - 2026
DA - 2026/10/25
PB - Zenodo
DO - 10.5281/zenodo.23649057
UR - https://zenodo.org/records/23649057
AB - For an elliptic curve E over Q and a window [-L,L], the radix kernel of Hackfort (2026) is the test function that is optimal for each curve, within a finite-dimensional space, for bounding the analytic rank of E under GRH by the explicit formula. Assuming GRH, we prove that the optimum over all admissible test functions with this support exists, is unique, equals the reciprocal of the value at the origin of the reproducing kernel of the Hilbert space defined by the Weil quadratic form, and is the limit of the finite-dimensional radix bounds, with an exact error identity. By Akhiezer's factorisation theorem, no continuous test function g with support in [-2L,2L], non-negative Fourier transform and positive value of the transform at 0 gives a smaller bound. The key step is a coercivity estimate for the Weil form, obtained by writing it as a truncated Wiener-Hopf form with a logarithmically growing symbol.
ER -