beta-family congruences and the f-invariant

Beta-elements and divided congruences

The f-invariant is an injective homomorphism from the 2-line of the Adams-Novikov spectral sequence to a group which is closely related to divided congruences of elliptic modular forms. We compute the f-invariant for two infinite families of beta-elements and explain the relation of the arithmetic of divided congruences with the Kervaire invariant one problem.

Congruences between modular forms given by the divided beta family in homotopy theory

We characterize the 2-line of the p-local Adams-Novikov spectral sequence in terms of modular forms satisfying a certain explicit congruence condition for primes p > 3. We give a similar characterization of the 1-line, reinterpreting some earlier work of A. Baker and G. Laures. These results are then used to deduce that, for l a prime which generates the p-adic units, the spectrum Q(l) detects the alpha and beta families in the stable stems.

The beta family at the prime two and modular forms of level three

We use the orientation underlying the Hirzebruch genus of level three to map the beta family at the prime p=2 into the ring of divided congruences. This procedure, which may be thought of as the elliptic greek letter beta construction, yields the f-invariants of this family.

色同伦的模形式：超奇异同余

Modular forms for chromatic homotopy: Supersingular congruences

证明了Larson在Behrens关于球稳定同伦群的Adams-Novikov谱序列中与分割β族相关的模形式同余猜想，将关键同余条件归结为模曲线超奇异点的几何陈述，对所有\(p\geq5\)的素数得证。

Explicit modular forms from the divided beta family

We compute modular forms known to arise from the order 5 generators of the 5-local Adams-Novikov spectral sequence 2-line, generalizing and contextualizing previous computations of M. Behrens and G. Laures. We exhibit analogous computations at other primes and conjecture formulas for some of the modular forms arising in this way at arbitrary primes greater than or equal to 5.