A uniform degree bound for modular rank-two Nahm sums
Abstract
Let F be a rank-two symmetrizable Nahm sum, with any symmetrizer diag(d1,d2), a positive-definite rational matrix and rational shifts, and let K be the field generated by its Nahm point; write m=d2/d1. We prove that if qcF is modular, then [K:ℚ]≤ 24. Modularity forces the higher coefficients of the logarithmic radial asymptotic expansion to vanish; we study the resulting polynomial equations through their Newton polygons and a pole-order analysis along the determinant conic, and for m=1 show that their common components occur only on one explicit family, whose Nahm point is rational. For equal row sums, Bloch-group torsion gives [K:ℚ]≤ 2. For complementary Nahm coordinates, the bound is $2$ when m≠ 1 and $4$ when m=1, using the first two asymptotic corrections in the latter case. Several steps of the proofs rely on exact computation.
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BibTeX
@misc{hakimoglubrown2026uniform,
title = {{A uniform degree bound for modular rank-two Nahm sums}},
author = {Cetin Hakimoglu-Brown},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.02743},
eprint = {2610.02743},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.02743},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A uniform degree bound for modular rank-two Nahm sums
AU - Cetin Hakimoglu-Brown
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2610.02743
UR - https://arxiv.org/abs/2610.02743
AB - Let F be a rank-two symmetrizable Nahm sum, with any symmetrizer diag(d_1,d_2), a positive-definite rational matrix and rational shifts, and let K be the field generated by its Nahm point; write m=d_2/d_1. We prove that if q^cF is modular, then [K:ℚ]≤ 24. Modularity forces the higher coefficients of the logarithmic radial asymptotic expansion to vanish; we study the resulting polynomial equations through their Newton polygons and a pole-order analysis along the determinant conic, and for m=1 show that their common components occur only on one explicit family, whose Nahm point is rational. For equal row sums, Bloch-group torsion gives [K:ℚ]≤ 2. For complementary Nahm coordinates, the bound is $2$ when m≠ 1 and $4$ when m=1, using the first two asymptotic corrections in the latter case. Several steps of the proofs rely on exact computation.
ER -