{"ID":23029917,"CreatedAt":"2026-09-17T05:12:50.116335254Z","UpdatedAt":"2026-09-17T05:12:50.116335254Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2609.19085","arxiv_id":"2609.19085","title":"Tunnell-type criteria for variants of the congruent number problem","abstract":"We study the $θ$-congruent number problem for $\\cosθ=\\pm3/5$ and $\\pm4/5$ using the generalized theta series construction of Sirolli--Tornaría. We describe its specialization to newforms of weight $2$ over $\\mathbb Q$ 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.","short_abstract":"We study the $θ$-congruent number problem for $\\cosθ=\\pm3/5$ and $\\pm4/5$ using the generalized theta series construction of Sirolli--Tornaría. We describe its specialization to newforms of weight $2$ over $\\mathbb Q$ with nontrivial square-free odd part of the level, and explain the reduction of quadratic twists to od...","url_abs":"https://arxiv.org/abs/2609.19085","url_pdf":"https://arxiv.org/pdf/2609.19085v1","authors":"[\"Bo-Hae Im\",\"Minseo Shin\"]","published":"2026-09-16T17:25:30Z","proceeding":"math.NT","tasks":"[\"math.NT\"]","methods":"[]","has_code":false}