Modular forms for chromatic homotopy: Supersingular congruences

In this note, we confirm a conjecture of Larson that arises in the Adams–Novikov spectral sequence (ANSS) for the stable homotopy groups of spheres and, specifically, in Behrens’ program on explicit modular forms detecting $v_2$–periodic classes in the divided $\beta$-family. The conjecture predicts the supersingular order of the weight $12t$ form $L_2({\mathrm{\Delta }}^t)$, when $(p-1)\mid 12t,$ attached to the ${\mathrm{\Gamma }}_0(2)$ Hecke correspondence. We prove the prediction for all primes $p\ge 5$, thereby providing the precise modular input that calibrates the relevant ANSS differentials in the Behrens program and removes the last obstruction to using pure $\mathrm{\Delta }$-power across the range of indices where Hodge-scaling cancels.