2025/09/19 by Ken Ono, Ono, Ken · 1 voice
Mathematics · #Algebraic structures and combinatorial models #Algebraic Geometry and Number Theory #Advanced Algebra and Geometry
paper · pdf · doi:10.48550/arxiv.2509.16175
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=rpn with (r,p)=1 and t=i(p2-1)/12, Larson's conjecture asserts that the Behrens form fi/j (which is well defined modulo p) may be taken to be the pure power Δt precisely when 1≤ j≤ pn, and admits no such representative otherwise. We prove this for all primes p≥5. The proof reduces the decisive congruence condition to a geometric statement on supersingular points of modular curves. Namely, that for every prime ℓ≠ p, the value of the modular function V_ℓ(Δ)/Δ at each supersingular point of X0(ℓ) is an (p2-1)/12-th root of unity.