Modular forms for chromatic homotopy: Supersingular congruences

We prove a conjecture of Larson in Behrens' program on congruences of modular forms attached to the divided beta family in the Adams--Novikov spectral sequence for the stable homotopy groups of spheres.

The conjecture gives a sharp criterion for when the modular form associated to a divided beta element can be represented by a pure power of the discriminant modular form.

Writing $i=rp^{n}$ with $(r,p)=1$ and $t=i(p^2-1)/12$, Larson's conjecture asserts that the Behrens form $f_{i/j}$ (which is well defined modulo $p$) may be taken to be the pure power $\Delta^{t}$ precisely when $1\le j\le p^{n}$, and admits no such representative otherwise.

We prove this for all primes $p\ge5$.

The proof reduces the decisive congruence condition to a geometric statement on supersingular points of modular curves.

Namely, that for every prime $\ell\ne p$, the value of the modular function $V_\ell(\Delta)/\Delta$ at each supersingular point of $X_0(\ell)$ is an $(p^2-1)/12$-th root of unity.