Modular forms for chromatic homotopy: Supersingular congruences

Abstract

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 $i=r{p}^n$ 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 $\hat{I}{”}^t$ precisely when $1\le j\le p^n$, and admits no such representative otherwise. We prove this for all primes $p\ge 5$. 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 $V_{\ell }(\hat{I}”)/\hat{I}”$ at each supersingular point of $X_0(\ell )$ is an $(p^2-1)/12$-th root of unity.

19 pages. v2: proof significantly reorganized around supersingular discriminant rationality; sharpness of the bound and the level-2 factorization now treated in full. Comments welcome