Modular forms for chromatic homotopy: Supersingular congruences

Summary · qwen2.5:32b

The article proves Larson's conjecture within Behrens' program, providing a criterion for when modular forms attached to the divided beta family in the Adams-Novikov spectral sequence can be represented by powers of the discriminant form $\Delta^t$, specifically for primes $p \geq 5$. The proof relies on showing that for any prime $\ell \neq p$, the value of the modular function $V_\ell(\Delta)/\Delta$ at each supersingular point of $X_0(\ell)$ is a $(p^2-1)/12$-th root of unity.

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Exploring the relationship between modular forms and chromatic homotopy, specifically focusing on supersingular congruences in Behrens' program. Learn about the criteria for when a modular form can be represented by a pure power of the discriminant modular form.

Excerpt

arXiv:2509.16175v2 Announce Type: replace 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 $i=rp^{n}$ 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.