The $\mathbb C$-motivic Adams-Novikov spectral sequence for topological modular forms

TL;DR

This paper studies the algebraic structure of the $oldsymbol{ ext{C}}$-motivic Adams-Novikov spectral sequence for modular forms spectra, resolving a key detail about the multiplicative structure of $ ext{tmf}$ homotopy groups.

Contribution

It provides a detailed algebraic analysis of the $oldsymbol{ ext{C}}$-motivic Adams-Novikov spectral sequence for $ ext{mmf}$ and $ ext{tmf}$, clarifying the multiplicative structure of $ ext{tmf}$ homotopy groups.

Findings

Resolved a previously unknown aspect of $ ext{tmf}$ homotopy groups' multiplicative structure.

Applied algebraic techniques to analyze the spectral sequence for $ ext{mmf}$ and $ ext{tmf}.

Enhanced understanding of the algebraic properties of topological modular forms spectra.

Abstract

We analyze the $\mathbb{C}$-motivic (and classical) Adams-Novikov spectral sequence for the $\mathbb{C}$-motivic modular forms spectrum $\mathit{mmf}$ (and for the classical topological modular forms spectrum $\mathit{tmf}$). We primarily use purely algebraic techniques, with a few exceptions. Along the way, we settle a previously unresolved detail about the multiplicative structure of the homotopy groups of $\mathit{tmf}$.