Tunnell-type criteria for variants of the congruent number problem
Abstract
We study the θ-congruent number problem for cosθ=±3/5 and ±4/5 using the generalized theta series construction of Sirolli--Tornaría. We describe its specialization to newforms of weight $2$ over ℚ with nontrivial square-free odd part of the level, and explain the reduction of quadratic twists to odd fundamental discriminants. The same construction gives an effective procedure for every θ-congruent number problem with nonzero rational cosine. For the four angles, we construct explicit forms of weight 3/2 whose Fourier coefficients determine the central L-values of the associated elliptic curves. This gives Tunnell-type criteria for every positive square-free integer: a nonzero coefficient implies non-θ-congruence unconditionally, and the converse holds assuming the Birch--Swinnerton-Dyer conjecture. We also prove unconditional non-θ-congruence for primes in explicit arithmetic progressions.
Related
- Higher Reciprocity, Cassels Pairings, and Selmer Towers for the 3/5 Congruent Number Problem
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- Prescribed Abscissae on Congruent-Number Curves over Simplest Cubic Fields
- Representation Defects and Cassels Pairings for Congruent Number Curves: Rédei Symbols and Governing Fields
- The number of touching pairs of congruent sphere packings in Euclidean 3-space
BibTeX
@misc{im2026tunnelltype,
title = {{Tunnell-type criteria for variants of the congruent number problem}},
author = {Bo-Hae Im and Minseo Shin},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.19085},
eprint = {2609.19085},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.19085},
}
RIS
TY - UNPB
TI - Tunnell-type criteria for variants of the congruent number problem
AU - Bo-Hae Im
AU - Minseo Shin
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.19085
UR - https://arxiv.org/abs/2609.19085
AB - We study the θ-congruent number problem for cosθ=±3/5 and ±4/5 using the generalized theta series construction of Sirolli--Tornaría. We describe its specialization to newforms of weight $2$ over ℚ with nontrivial square-free odd part of the level, and explain the reduction of quadratic twists to odd fundamental discriminants. The same construction gives an effective procedure for every θ-congruent number problem with nonzero rational cosine. For the four angles, we construct explicit forms of weight 3/2 whose Fourier coefficients determine the central L-values of the associated elliptic curves. This gives Tunnell-type criteria for every positive square-free integer: a nonzero coefficient implies non-θ-congruence unconditionally, and the converse holds assuming the Birch--Swinnerton-Dyer conjecture. We also prove unconditional non-θ-congruence for primes in explicit arithmetic progressions.
ER -