arXiv subjects

Jack Morgan Davies

Publications and source records attributed to Jack Morgan Davies.

18 recordsLinked to original sources

arXiv2026

On Galois extensions of geometric fixed point spectra

In this article, the cyclotomic Galois action on topological K-theory adjoined with a primitive $n$th root of unity and the famous $GL_1(\mathbf{Z}/n)$- and $GL_2(\mathbf{Z}/n)$-Galois actions on topological modular forms with $Γ_1(n)$- and $Γ(n)$-level structures are unified and generalised. This is done by defining a quotient stack in derived algebraic geometry parametrising constant finite abelian subgroups of $\mathbf{P}$-divisible groups, and studying how these quotient stacks and various notions of torsors interact with Galois extensions formed by taking algebraic and categorical invariants. Taking global sections then yields the titular Galois extensions on $K$-geometric fixed points, recovering and refining the well-known examples above and providing new ones. As an application, the $\infty$-category of perfect modules over a variety of $H$-equivariant ring spectra $R$ are decomposed into simple pullbacks of nonequivariant categories, leading to Mayer--Vietoris sequences for localising invariants of $R$. For example, this occurs for equivariant topological K-theory for all $p$-groups as well as any finite nonabelian simple group of order less than 500.

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arXiv2026

Comparing tempered and equivariant elliptic cohomology

Lurie and Gepner--Meier each define equivariant cohomology theories, namely tempered cohomology and equivariant elliptic cohomology, respectively, using derived algebraic geometry. We construct a natural equivalence between these theories where they overlap. Moreover, we emphasise the naturality and coherence of both these equivariant theories as well as our comparison. To demonstrate the use of this comparison, we show that the $G$-fixed points of equivariant topological modular forms is dualisable as a TMF-module for all compact Lie groups $G$ that decompose as a product of a torus and a finite group by formally reducing to an argument of Gepner--Meier.

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arXiv2026

Nonvanishing of products in $v_2$-periodic families at the prime $3$

Many products amongst $v_2$-periodic families in the stable homotopy groups of spheres are shown not to vanish and some Toda brackets are shown not to contain zero. This is done by carefully studying the action of Adams operations on topological modular forms. A crucial ingredient is Pstragowski's category of synthetic spectra which affords us the necessary freedom to work with (modified) Adams--Novikov spectral sequences.

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arXiv2026

On the derived Tate curve and global smooth Tate $K$-theory

The interplay between equivariant stable homotopy theory and spectral algebraic geometry is used to construct a derived Tate curve over $\mathrm{KU}((q))$, a lift of the classical elliptic curve of Tate over $\mathbf{Z}((q))$. Applications of both an algebro-geometric and a topological flavour follow. First, we construct a spectral algebro-geometric model for the compactification of the moduli stack of oriented elliptic curves, giving a canonical choice of holomorphic topological $q$-expansion map. Then we define globally equivariant forms of Tate $K$-theory $\mathbf{KO}((q))$ and $\mathbf{KU}((q))$, and equip them with globally equivariant meromorphic topological $q$-expansion maps from global topological modular forms. Finally, we explore $C_2$-equivariant versions of global Tate $K$-theory and connect them with $C_2$-equivariant global topological modular forms with level structures.

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arXiv2026

The descent spectral sequence for topological modular forms

We prove the Gap Theorem for the spectrum of topological modular forms $\mathrm{Tmf}$. This removes a longstanding circularity in the literature, thereby confirming the computation of $π_\ast \mathrm{tmf}$ from over two decades ago by Hopkins and Mahowald. Our approach is crucially a modern one, developing and refining many techniques in synthetic spectra.

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arXiv2026

An algebraic model for rational ultracommutative rings

Given a global equivariant ultracommutative ring spectrum $E$ and inclusion $H\hookrightarrow G$ of finite groups, one may apply geometric fixed points to the norm $N_H^G E_H \to E_G$ to obtain what we call a \emph{geometric norm} $Φ^H E \to Φ^G E$. We prove that, together with inflations, these assemble into a functor $Φ\colon\mathrm{UCom}_{\mathrm{fin}} \to \mathrm{Fun}(\mathrm{Span}(\mathcal{G},\mathcal{E},\mathcal{O}),\mathrm{CAlg})$, where $\mathrm{Span}(\mathcal{G},\mathcal{E},\mathcal{O})$ is the span category of finite connected groupoids with full backwards maps and faithful forwards maps, and that $Φ$ restricts to an equivalence between full subcategories of rational objects. Central to our construction is a refinement of geometric fixed points to a natural transformation $Φ\colon \mathrm{Sp}_\bullet\to\mathrm{Fun}(\mathrm{Orb}_\bullet^\simeq,\mathrm{Sp})$ which is compatible with restrictions and norms, and which restricts to an equivalence on full subcategories of rational objects. We explain how this may also be used to recover theorems of Barrero--Barthel--Pol--Strickland--Williamson and Wimmer on algebraic models for rational global spectra and normed $G$-commutative ring spectra respectively.

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arXiv2026

Ambidextrous global spectra and tempered cohomology

We introduce generalizations of global equivariant spectra which encode globally equivariant cohomology theories equipped with additional transfers, such as the deflation maps present in equivariant topological $K$-theory. We call these $\mathcal{Q}$-ambidextrous global spectra, where $\mathcal{Q}$ is a parameter encoding which additional transfers one allows. As our main example, we prove that the tempered cohomology theory associated with an oriented $\mathbf{P}$-divisible group, constructed by Lurie, is represented by a $π$-ambidextrous global $\mathbf{E}_\infty$ ring spectrum, encoding transfers along all relatively $π$-finite maps of global spaces. This is established by means of a general parametrized decategorification process, perhaps of independent interest, that produces $\mathcal{Q}$-ambidextrous global spectra from suitable global families of stable $\infty$-categories. By allowing $\mathcal{Q}$ to vary, we are able to coherently encode the fact that non-invertible morphisms of oriented $\mathbf{P}$-divisible groups induce maps of tempered theories that only commute with certain transfers. With these $π$-ambidextrous enhancements in hand, we explore the fundamental properties of tempered theories as equivariant stable homotopy types. We construct a well-behaved $F$-global homology theory for any $π$-finite space $F$, with good base change properties. Taking $F = \mathbf{B} H$ for a finite group $H$, this establishes general base change results for the geometric fixed points of tempered theories. We use this to compute the $H$-geometric fixed points of tempered theories, showing that they vanish for $H$ nonabelian and admit a simple algebro-geometric model when $H$ is abelian, with identifiable blueshift properties.

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arXiv2025

Affineness and reconstruction in complex-periodic geometry

Working in a generic derived algebro-geometric context, we lay the foundations for the general study of affineness and local descendability. When applied to $\mathbf{E}_\infty$ rings equipped with the fpqc topology, these foundations give an $\infty$-category of spectral stacks, a viable functor-of-points alternative to Lurie's approach to nonconnective spectral algebraic geometry. Specializing further to spectral stacks over the moduli stack of oriented formal groups, we use chromatic homotopy theory to obtain a large class of $0$-affine stacks, generalizing Mathew--Meier's famous $0$-affineness result. We introduce a spectral refinement of Hopkins' stack construction of an $\mathbf{E}_\infty$ ring, and study when it provides an inverse to the global sections of a spectral stack. We use this to show that a large class of stacks, which we call reconstructible, are naturally determined by their global sections, including moduli stacks of oriented formal groups of bounded height and the moduli stack of oriented elliptic curves.

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arXiv2025

Revisiting the $β_1$-action on the $3$-primary stable homotopy groups of spheres

Let $β_1$ be the first $3$-torsion class in the stable homotopy groups of spheres in even degree. Toda showed that $β_1^5 \neq 0$, whilst $β_1^6 = 0$. Shimomura generalised this to the $144$-periodic family generated by $β_1$, written as $\{β_{1+9s}\}_{s\geq 0}$, and showed that any $5$-fold product $\prod_5 β_{1+9s} \neq 0$, whilst all $6$-fold products $\prod_6 β_{1+9s} = 0$. In this article, we give a simple proof of these results as well as some generalisations to other $144$-periodic families. Our tools include BP-synthetic spectra, and the well-known Adams--Novikov spectral sequence for the spectrum of topological modular forms at the prime $3$ as well as its Adams operations.

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arXiv2025

On periodic families in the stable stems of height two

We discover a host of infinite periodic families in the 2-primary stable homotopy groups of spheres. We also confirm the existence of many families predicted by Hopkins--Mahowald. These families appear in nineteen different congruence classes of degrees modulo 192, seven of them consist of simple 4-torsion elements, and another four of simple 8-torsion. They all vanish in the homotopy groups of the spectrum TMF of topological modular forms, but we show that they are detected in the fixed-points of TMF with respect to an Atkin--Lehner involution. As a consequence, we confirm the existence of exotic spheres in all dimensions congruent to 72, 144, and 168 modulo 192.

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arXiv2025

Descent spectral sequences through synthetic spectra

The synthetic analogue functor $ν$ from spectra to synthetic spectra does not preserve all limits. In this paper, we give a necessary and sufficient criterion for $ν$ to preserve the global sections of a derived stack. Even when these conditions are not satisfied, our framework still yields synthetic spectra that implement the descent spectral sequence for the structure sheaf, thus placing descent spectral sequences on good footing in the $\infty$-category of synthetic spectra. As an example, we introduce a new $\mathrm{MU}$-synthetic spectrum $\mathrm{Smf}$.

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arXiv2025

A synthetic approach to detecting $v_1$-periodic families

We provide a simple proof that the unit map from the sphere spectrum to the connective image-of-$J$ spectrum $\mathrm{j}$ is surjective on homotopy groups. This is achieved using a novel $t$-structure on the category of $E$-synthetic spectra and a specific construction of $\mathbf{F}_p$- and BP-synthetic lifts of $\mathrm{j}$. These synthetic lifts then easily produce modified Adams and Adams--Novikov spectral sequences for $\mathrm{j}$ which we use the prove the above detection statement, all without ever calculating $\mathbf{F}_p$- or BP-homology nor the associated Ext groups.

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arXiv2025

Constructing and calculating Adams operations on dualisable topological modular forms

We construct Adams operations on the cohomology theory Tmf of topological modular forms; the first such stable operations on this cohomology theory. These Adams operations are then calculated on the Tmf-cohomology of spheres using a combination of descent spectral sequences and Anderson duality. Applications of these operations are then given, including constructions of connective height 2 analogues of Adams summands and image of J spectra.

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arXiv2024

On Lurie's theorem and applications

Lurie's theorem states that there exists a sheaf of ring spectra on the site of formally étale Deligne--Mumford stacks over the moduli stack of $p$-divisible groups of height $n$, which agrees with the classical Landweber exact functor theorem (LEFT) on affines. In other words, this theorem is a global, higher categorical refinement of the LEFT. In recent work, Lurie has introduced many of the ingredients one needs to prove this theorem, and in this article, we gather these ingredients together and prove Lurie's theorem. Applications of this theorem to Lubin--Tate theories, topological modular and automorphism forms, and Adams operations are also discussed.

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arXiv2024

Hecke operators on topological modular forms

The cohomology theory TMF of topological modular forms is a derived algebro-geometric interpretation of the classical ring of complex modular forms from number theory. In this article, we refine the classical Adams operations, Hecke operators, and Atkin--Lehner involutions from endomorphisms of classical modular forms to stable operators on TMF. Our algebro-geometric formulation of these operators leads to simple proofs of their many remarkable properties and computations. From these properties, we use techniques from homotopy theory to make simple number-theoretic deductions, including a rederivation of some classical congruences of Ramanujan and providing new infinite families of classical Hecke operators which satisfy Maeda's conjecture.

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arXiv2023

Uniqueness of real ring spectra up to higher homotopy

We discuss a notion of uniqueness up to $n$-homotopy and study examples from stable homotopy theory. In particular, we show that the $q$-expansion map from elliptic cohomology to topological $K$-theory is unique up to $3$-homotopy, away from the prime $2$, and that upon taking $p$-completions and $\mathbf{F}_p^\times$-homotopy fixed points, this map is uniquely defined up to $(2p-3)$-homotopy. Using this, we prove new relationships between Adams operations on connective and dualisable topological modular forms -- other applications, including a construction of a connective model of Behrens' $Q(N)$ spectra away from $2N$, will be explored elsewhere. The technical tool facilitating this uniqueness is a variant of the Goerss--Hopkins obstruction theory for real spectra, which applies to various elliptic cohomology and topological $K$-theories with a trivial complex conjugation action as well as some of their homotopy fixed points.

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arXiv2021

Elliptic cohomology is unique up to homotopy

Homotopy theory folklore tells us that the sheaf defining the cohomology theory Tmf of topological modular forms is unique up to homotopy. Here we provide a proof of this fact, although we claim no originality for the statement. This retroactively reconciles all previous constructions of Tmf.

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arXiv2020

Realising $π_\ast^e R$-algebras by global ring spectra

We approach a problem of realising algebraic objects in a certain universal equivariant stable homotopy theory; the global homotopy theory of Schwede. Specifically, for a global ring spectrum $R$, we consider which classes of ring homomorphisms $η_\ast\colonπ_\ast^e R\rightarrow S_\ast$ can be realised by a map $η\colon R\rightarrow S$ in the category of global $R$-modules, and what multiplicative structures can be placed on $S$. If $η_\ast$ witnesses $S_\ast$ as a projective $π_\ast^e R$-module, then such an $η$ exists as a map between homotopy commutative global $R$-algebras. If $η_\ast$ is in addition étale or $S_0$ is a $\mathbb{Q}$-algebra, then $η$ can be upgraded to a map of $\mathbb{E}_\infty$-global $R$-algebras or a map of $\mathbb{G}_\infty$-$R$-algebras, respectively. Various global spectra and $\mathbb{E}_\infty$-global ring spectra are then obtained from classical homotopy theoretic and algebraic constructions, with a controllable global homotopy type.

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