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In previous work, the authors have each introduced methods for studying the 2-line of the p-local Adams-Novikov spectral sequence in terms of the arithmetic of modular forms. We give the precise relationship between the congruences of…

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…

We compute the Adams-Novikov E_2-term of a spectrum Q(2) constructed by Behrens. The homotopy groups of Q(2) are closely tied to the 3-primary stable homotopy groups of spheres; in particular, they are conjectured to detect the homotopy…

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…

In this note, we confirm a conjecture of Larson that arises in the Adams--Novikov spectral sequence (ANSS) for the stable homotopy groups of spheres and, specifically, in Behrens' program on explicit modular forms detecting $v_2$--periodic…

We compute the mod $2$ homology of the spectrum $\mathrm{tmf}$ of topological modular forms by proving a 2-local equivalence $\mathrm{tmf} \wedge DA(1) \simeq \mathrm{tmf}_1(3) \simeq BP\left \langle 2\right\rangle$, where $DA(1)$ is an…

We study the motivic Adams-Novikov spectral sequence at an odd prime $l$ over the base fields $\mathbb{C}$ and $\mathbb{R}$. This spectral sequence converges to the stable motivic homotopy groups of the $l$-completed motivic sphere…

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,…

We use the structure of the homotopy groups of the connective spectrum tmf of topological modular forms and the elliptic and Adams-Novikov spectral sequences to compute the homotopy groups of the non-connective version, Tmf, of that…

This paper contains a complete computation of the homotopy ring of the spectrum of topological modular forms constructed by Hopkins and Miller. The computation is done away from 6, and at the (interesting) primes 2 and 3 separately, and in…

We study the Adams-Novikov spectral sequence in $\mathbb{F}_p$-synthetic spectra, computing the synthetic analogs of $\mathrm{BP}$ and its cooperations to identify the synthetic Adams-Novikov $\mathrm{E}_2$-page, computed in a range with a…

We show that the spectral sequence converging to the stable homotopy groups of spheres, induced by the Betti realization of the slice tower for the motivic sphere spectrum, agrees with the Adams-Novikov spectral sequence, after a suitable…

In this paper we constructs a new nontrivial family in the stable homotopy groups of spheres $\pi_{p^nq+2pq+q-3}S$ which is of order $p$ and is represented by $k_0h_{n} \in Ext_A^{3,p^nq+2pq+q}(\mathbb{Z}_p,\mathbb{Z}_p)$ in the Adams…

We describe some periodic structure in the cohomology of the moduli stack of 1-dimensional formal group laws, also known as the $E_2$-page of the classical Adams--Novikov spectral sequence. This structure is distinct from the familiar…

We present a detailed analysis of 2-complete stable homotopy groups, both in the classical context and in the motivic context over C. We use the motivic May spectral sequence to compute the cohomology of the motivic Steenrod algebra over C…

For groups of prime order, equivariant stable maps between equivariant representation spheres are investigated using the Borel cohomology Adams spectral sequence. Features of the equivariant stable homotopy category, such as stability and…

Let $p$ be an odd prime. Denote a Sylow $p$-subgroup of $GL_2(\mathbb{Z}/p^n)$ and $SL_2(\mathbb{Z}/p^n)$ by $S_p(n,GL)$ and $S_p(n,SL)$ respectively. The theory of stable elements tells us that the mod-$p$ cohomology of a finite group is…

We show that $v_2^9$ is a permanent cycle in the 3-primary Adams-Novikov spectral sequence computing $\pi_*(S/(3,v_1^8))$, and use this to conclude that the families $\beta_{9t+3/i}$ for $i=1,2$, $\beta_{9t+6/i}$ for $i=1,2,3$,…

We establish a homotopy-theoretic description of the homology of stable moduli spaces of $(2n+1)$-dimensional manifold triads $(N, \partial^h N, \partial^v N)$ with fixed $\partial^v N$, whenever $n \geq 3$ and $(N, \partial^h N)$ is…

In the stable homotopy groups $\pi_{q(p^n+p^m+1)-3}(S)$ of the sphere spectrum $S$ localized at the prime $p$ greater than three, J. Lin constructed an essential family $\xi_{m,n}$ for $n \geq m + 2 >5$. In this paper, the authors show that…