R-Motivic Modular Forms Spectrum

• R-motivic modular forms spectrum is a motivic refinement of classical tmf, encoding modular forms in real algebraic geometry through a specialized Steenrod algebra.
• It employs computational tools like the ρ-Bockstein and Adams spectral sequences to resolve hidden extensions and quantify τ- and ρ-torsion phenomena.
• The spectrum bridges real, complex, and classical homotopy theories, offering new methods for analyzing orientability and motivic stable stems.

The spectrum of $\mathbb{R}$-motivic modular forms, frequently denoted $\mathrm{mmf}^{\mathbb{R}}$ or similar, is a candidate for the universal object in the stable $\mathbb{R}$-motivic homotopy category whose cohomology reflects the structure of modular forms in the context of real algebraic geometry. This spectrum is built to have mod 2 motivic cohomology $\mathcal{A}^{\mathbb{R}}\sslash \mathcal{A}^{\mathbb{R}}(2)$, analogous to classical $\mathrm{tmf}$ and its cellular and C-motivic analogues. Its construction synthesizes $\mathbb{R}$-motivic Steenrod algebra computations, descent-theoretic approaches, synthetic spectra methods, and explicit manipulations of the $\rho$-Bockstein and Adams spectral sequences, providing a model for computations of motivic stable stems, orientation phenomena, and realization questions.

1. Algebraic and Homotopical Foundations

The $\mathbb{R}$-motivic modular forms spectrum is conceived as a motivic refinement of topological modular forms, leveraging the structure of the $\mathbb{R}$-motivic Steenrod algebra and its subalgebras. The key algebraic object is $\mathcal{A}^{\mathbb{R}}$, the bigraded Steenrod algebra over $\mathbb{R}$, with commutative $\mathbb{F}_2$-coefficient ring $\mathcal{M}_2 = \mathbb{F}_2[\tau, \rho]$ (where $|\tau| = (0, 1)$ and $|\rho| = (1, 1)$), and structural operations such as $Sq^1$, $Sq^2$, $Sq^4$.

The relevant subalgebra is $$\mathcal{A}^{\mathbb{R}}(2) = \langle Sq^1, Sq^2, Sq^4 \rangle$$. The Ext groups $$\mathrm{Ext}_{\mathcal{A}^{\mathbb{R}}(2)}(\mathcal{M}_2, \mathcal{M}_2)$$ form the $E_2$-term in the Adams spectral sequence for a hypothetical $\mathbb{R}$-motivic modular forms spectrum, as computed in detail via the $\rho$-Bockstein spectral sequence in (Emming, 14 Sep 2025).

The construction of $\mathrm{mmf}^{\mathbb{R}}$ is achieved by "lifting" equivariant and classical models to the motivic setting over $Spec(\mathbb{R})$. The process typically starts with a $C_2$-equivariant refinement of $\mathrm{tmf}$, utilizing the Tate diagram and then applying descent machinery via an adjunction between $\mathbb{R}$-motivic and $C_2$-equivariant stable homotopy categories (Ricka, 2017). Extension of scalars from the $C_2$-equivariant to the motivic Steenrod algebra yields the required cohomological properties.

Orientation results are intrinsic: rational motivic spectra are $\mathrm{HQ}$-modules (the rational motivic cohomology spectrum) if and only if they are orientable, if the involution $(–1)$ is trivial, if the Hopf map $\eta$ vanishes, and if the spectrum satisfies étale descent, among other equivalent conditions (Hoyois, 22 Oct 2024).

2. $\rho$-Bockstein Spectral Sequence and Cohomology of $\mathcal{A}(2)$

The $\rho$-Bockstein spectral sequence is the central computational device for probing $\mathbb{R}$-motivic cohomology of $\mathcal{A}(2)$ (Emming, 14 Sep 2025). Starting from an $E_1$-page formed from the $\mathbb{C}$-motivic Ext groups (i.e., where $\rho = 0$), the spectral sequence filters by powers of $\rho$ to capture the real phenomena. Nontrivial differentials ($d_2$ up to $d_{10}$) and their effect on indecomposable classes are fully catalogued. The spectral sequence abuts to explicit Ext groups with significant $\rho$- and $\tau$-torsion, not present in classical settings.

Hidden extensions in the final cohomology—especially those by periodicity operators such as $\tau^8$, $h_0$, $h_1$, $h_2$—are resolved using comparison with the $\rho$-localization (essentially the $\mathbb{C}$-motivic case) and investigations of quotients and exact sequences. These hidden extensions have implications for multiplicative structures in $\mathrm{mmf}^{\mathbb{R}}$ and the Adams differentials in the spectral sequence for the spectrum.

The complete $E_2$-page as computed provides the input for the Adams spectral sequence converging to $\pi_*\mathrm{mmf}^{\mathbb{R}}$, and this calculation is essential groundwork for understanding motivic stable homotopy groups over $\mathbb{R}$ (the motivic "stems") and for making structural comparisons to classical variants (Emming, 14 Sep 2025).

3. Adams Spectral Sequence and Topological Implications

The Adams spectral sequence for $\mathrm{mmf}^{\mathbb{R}}$ has $E_2$-term $$\mathrm{Ext}_{\mathcal{A}^{\mathbb{R}}(2)}(\mathcal{M}_2, \mathcal{M}_2)$$, and the differentials—especially $d_2$—are calculated using lifting arguments from the motivic Bockstein spectral sequence and comparisons with the $\mathbb{C}$-motivic case (Emming, 14 Sep 2025). The degree of $d_2$ is $(-1,2,0)$ in triple grading (stem, Adams filtration, internal coweight).

Many Adams differentials are realized by comparison: the map $p$ that sends classes in the $\rho$-torsion-free part of the Ext group is an isomorphism to the corresponding complex-motivic Ext group up to predictable degree shift. Consequently, known formulas for $d_2$ in the complex-motivic case inform the $\mathbb{R}$-motivic calculations. For instance,

$$d_2(u) = h_1^2 c, \qquad d_2(\tau^6 a) = \tau^6 P h_2,$$

reflect the influence of periodicity operators and the interaction with the motivic parameter $\rho$.

Charts, tables, and the precise structure of hidden extensions organize the passage from cohomology of $\mathcal{A}(2)$ to the spectrum-level homotopy groups. This analytic machinery is expected to make direct contact with the computation of the $\mathbb{R}$-motivic modular forms spectrum, its role as a detecting object for stable stems, and, via comparison, the classical stable stems.

4. Real and Complex Comparisons

A key methodological principle is comparison with the $\mathbb{C}$-motivic context, where setting $\rho=0$ yields the $\mathbb{C}$-motivic theory as a limiting case. Many $\rho$-free phenomena can be detected in both contexts, and the Adams differentials "lift" accordingly.

Conversely, when further specializing by setting $\tau=1$, one recovers classical topological computations, thereby demonstrating that the $\mathbb{R}$-motivic modular forms spectrum interpolates between the real, complex, and classical worlds, each with their own torsion and extension phenomena.

This approach illuminates which phenomena are genuinely real-motivic and which are inherited from the complex or classical settings. For example, certain exotic $\rho$- or $\tau$-torsion classes have no classical analog and are visible only in the real-motivic context; these inform and govern the extra structure in motivic stable stems and modular forms.

5. Implications for Stable Homotopy Theory and Further Directions

The explicit calculation of $$\mathrm{Ext}_{\mathcal{A}(2)}(\mathcal{M}_2,\mathcal{M}_2)$$ and the Adams spectral sequence for $\mathrm{mmf}^{\mathbb{R}}$ provide essential machinery for approaching:

• the computation of $\mathbb{R}$-motivic stable stems,
• classification and realization problems for motivic spectra endowed with modular forms cohomology,
• comparisons with classical, $C_2$-equivariant, and complex-motivic settings,
• further analysis of the motivic sphere spectrum (including rational splittings and orientation phenomena (Hoyois, 22 Oct 2024)),
• and the transfer or deformation techniques (e.g., Galois descent, Artin-Tate categories, deformation of module categories via period elements such as $\tau$ (Burklund et al., 2020)).

The fine structure of periodic families, detached by sparse and non-arithmetic degree progressions as in (Isaksen et al., 22 Apr 2025), presents new challenges for detection and classification of motivic modular forms in the real case, in contrast to the v$_1$-periodicity dominating classical and complex-motivic regimes.

Potential future work involves:

• extending the spectral calculations to higher subalgebras ($\mathcal{A}^{\mathbb{R}}(n)$ for $n>2$),
• constructing and validating $\mathbb{R}$-motivic versions of known type-2 spectra (e.g., Bhattacharya-Egger spectrum (Bhattacharya et al., 2021)),
• exploiting synthetic spectra methods and descent spectral sequence analogues (Carrick et al., 2 Dec 2024), and
• exploring explicit geometric/framed models (Druzhinin, 2018).

6. Key Formulas and Structures

Some representative formulas and tables organizing the computational work include:

Structure	Notation	Main Formula / Feature
$\mathbb{R}$-motivic coefficient	$\mathcal{M}_2$	$\mathbb{F}_2[\tau, \rho]$
Steenrod algebra subalgebra	$\mathcal{A}(2)$	$Sq^1, Sq^2, Sq^4$
Periodicity operator	$\tau$	(period element, often $\tau^8$)
$\rho$-Bockstein differential	$d_k$	$d_2(u) = h_1^2 c$, $d_2(\tau^6 a) = \tau^6 P h_2$
Adams Spectral Sequence degrees	$(-1,2,0)$	degree of $d_2$ differential

The careful resolution of hidden extensions, identification of all nontrivial differentials on indecomposables, and classification of $\rho$- and $\tau$-torsion are pivotal for the internal algebraic structure of $\mathrm{mmf}^{\mathbb{R}}$ (Emming, 14 Sep 2025).

7. Conclusion

The $\mathbb{R}$-motivic modular forms spectrum encapsulates a motivic refinement of modular forms, designed to reflect both the algebraic and topological intricacies of real algebraic geometry and homotopy theory. Its cohomology, far richer than its classical and complex analogues, demands deep spectral sequence analysis, $\rho$-Bockstein techniques, and careful attention to hidden extensions and periodicity phenomena. As the input for the Adams spectral sequence computing motivic stable homotopy groups, it serves as a foundational object for further paper in the interplay between motivic, equivariant, and classical stable homotopy categories and their associated invariants.

The methods and explicit computations developed in (Emming, 14 Sep 2025) not only clarify the internal structure of $\mathrm{mmf}^{\mathbb{R}}$ but also lay the groundwork for broader comparisons, orientation criteria, deformation-theoretic approaches, and modular forms detection in real motivic homotopy theory, thus advancing the algebraic and geometric foundation for future research on motivic modular forms.