A uniform degree bound for modular rank-two Nahm sums

Abstract

Let $F$ be a rank-two symmetrizable Nahm sum, with any symmetrizer $\operatorname{diag}(d_1,d_2)$, 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 $[K:\mathbb{Q}]\le 24$. 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 $[K:\mathbb{Q}]\le 2$. For complementary Nahm coordinates, the bound is $2$ when $m\ne 1$ and $4$ when $m=1$, using the first two asymptotic corrections in the latter case. Several steps of the proofs rely on exact computation.