$\mathbb{C}$-motivic modular forms

Abstract

We construct a topological model for cellular, 2-complete, stable C-motivic homotopy theory that uses no algebro-geometric foundations. We compute the Steenrod algebra in this context, and we construct a "motivic modular forms" spectrum over C.

Stable homotopy groups of spheres: from dimension 0 to 90

“…In addition, in higher stems, there are possible extensions by 2, η, and ν that are not indicated in Figure 1. See Tables 17,20, and 23 for more details about these possible extensions.…”

“…In fact, recent progress has determined that our results do not depend on this deep and difficult work. There are now purely topological constructions of homotopy categories that have identical computational properties to the cellular stable C-motivic homotopy category [20,52]. In these homotopy categories, one can obtain from first principles the fundamental computations of the cohomology of a point and of the Steenrod algebra, using only well-known classical computations.…”

“…Using a topological model of the cellular stable C-motivic homotopy category, one can construct a "motivic modular forms" spectrum mmf [20], whose motivic F 2cohomology is the quotient of the C-motivic Steenrod algebra by its subalgebra generated by Sq 1 , Sq 2 , and Sq 4 . Just as tmf plays an essential role in studies of the classical Adams spectral sequence [5,8], mmf is an essential tool for motivic computations.…”

“…(5) Deduce additional Adams differentials for the C-motivic sphere with a variety of ad hoc arguments. The most important methods are Toda bracket shuffles and comparison to the motivic modular forms spectrum mmf [20]. (6) Deduce hidden τ extensions in the C-motivic Adams spectral sequence for the sphere, using a long exact sequence in homotopy groups.…”

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

“…The C-motivic modular forms spectrum mmf . There is a C-motivic E ∞ -ring spectrum mmf that can be viewed as the analogue of the classical topological modular forms spectrum tmf [GIKR22]. The Betti realization of mmf is the classical spectrum tmf .…”