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.