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 $L_2(\Delta^t)$, when $(p-1)\mid 12t,$ attached to the $\Gamma_0(2)$ Hecke correspondence. We prove the prediction for all primes $p\ge5$, thereby providing the precise modular input that calibrates the relevant ANSS differentials in the Behrens program and removes the last obstruction to using pure $\Delta$-power across the range of indices where Hodge-scaling cancels.