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Lennart Meier

Publications and source records attributed to Lennart Meier.

At least 19 recordsLinked to original sources
arXiv2025

A new approach to (3+1)-dimensional TQFTs via topological modular forms

In this paper, we present a construction toward a new type of TQFTs at the crossroads of low-dimensional topology, algebraic geometry, physics, and homotopy theory. It assigns TMF-modules to closed 3-manifolds and maps of TMF-modules to 4-dimensional cobordisms. This is a mathematical proposal for one of the simplest examples in a family of ${\pi}_*({\rm TMF})$-valued invariants of 4-manifolds which are expected to arise from 6-dimensional superconformal field theories. As part of the construction, we define TMF-modules associated with symmetric bilinear forms, using (spectral) derived algebraic geometry. The invariant of unimodular bilinear forms takes values in ${\pi}_*({\rm TMF})$, conjecturally generalizing the theta function of a lattice. We discuss gluing properties of the invariants. We also demonstrate some interesting physics applications of the TMF-modules such as distinguishing phases of quantum field theories in various dimensions.

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arXiv2025

Topological Jacobi Forms

As a generalization of the ring spectrum of topological modular forms, we construct a graded ring spectrum of topological Jacobi forms, $\operatorname{TJF}_*$. This is constructed as the global sections of a sheaf of $E_\infty$-ring spectra on the stacky universal elliptic curve using circle-equivariant $\operatorname{TMF}$. Complete calculations of its homotopy at odd primes and partial results at $p=2$ are given.

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arXiv2024

A stratification of the equivariant slice filtration

In this paper, we construct a stratification tower for the equivariant slice filtration. This tower stratifies the slice spectral sequence of a $G$-spectrum $X$ into distinct regions. Within each of these regions, the differentials are determined by the localized slice spectral sequences, which compute the geometric fixed points along with their associated residue group actions. Consequently, the stratification tower offers an inductive method of understanding the entirety of the equivariant slice spectral sequence of $X$ by examining each of its distinct stratification regions.

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arXiv2024

Transchromatic phenomena in the equivariant slice spectral sequence

In this paper, we prove a transchromatic phenomenon for Hill--Hopkins--Ravenel and Lubin--Tate theories. This establishes a direct relationship between the equivariant slice spectral sequences of height-$h$ and height-$(h/2)$ theories. As applications of this transchromatic phenomenon, we prove periodicity and vanishing line results for these theories.

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arXiv2023

Purity in chromatically localized algebraic $K$-theory

We prove a purity property in telescopically localized algebraic $K$-theory of ring spectra: For $n\geq 1$, the $T(n)$-localization of $K(R)$ only depends on the $T(0)\oplus \dots \oplus T(n)$-localization of $R$. This complements a classical result of Waldhausen in rational $K$-theory. Combining our result with work of Clausen--Mathew--Naumann--Noel, one finds that $L_{T(n)}K(R)$ in fact only depends on the $T(n-1)\oplus T(n)$-localization of $R$, again for $n \geq 1$. As consequences, we deduce several vanishing results for telescopically localized $K$-theory, as well as an equivalence between $K(R)$ and $\mathrm{TC}(τ_{\geq 0} R)$ after $T(n)$-localization for $n\geq 2$.

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arXiv2023

Invariant prime ideals in equivariant Lazard rings

Let $A$ be an abelian compact Lie group. In this paper we compute the spectrum of invariant prime ideals of the $A$-equivariant Lazard ring, or equivalently the spectrum of points of the moduli stack of $A$-equivariant formal groups. We further show that this spectrum is homeomorphic to the Balmer spectrum of compact $A$-spectra, with the comparison map induced by equivariant complex bordism homology.

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arXiv2023

On equivariant topological modular forms

Following ideas of Lurie, we give in this article a general construction of equivariant elliptic cohomology without restriction to characteristic zero. Specializing to the universal elliptic curve we obtain in particular equivariant spectra of topological modular forms. We compute the fixed points of these spectra for the circle group and more generally for tori.

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arXiv2023

The localized slice spectral sequence, norms of Real bordism, and the Segal conjecture

In this paper, we introduce the localized slice spectral sequence, a variant of the equivariant slice spectral sequence that computes geometric fixed points equipped with residue group actions. We prove convergence and recovery theorems for the localized slice spectral sequence and use it to analyze the norms of the Real bordism spectrum. As a consequence, we relate the Real bordism spectrum and its norms to a form of the $C_2$-Segal conjecture. We compute the localized slice spectral sequence of the $C_4$-norm of $BP_\mathbb{R}$ in a range and show that the Hill--Hopkins--Ravenel slice differentials is in one-to-one correspondence with a family of Tate differentials for $N_1^2 H{\mathbb{F}}_2$.

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arXiv2022

Picard sheaves, local Brauer groups, and topological modular forms

We prove that the Brauer group of TMF is isomorphic to the Brauer group of the derived moduli stack of elliptic curves. Then, we compute the local Brauer group, i.e., the subgroup of the Brauer group of elements trivialized by some étale cover of the moduli stack, up to a finite 2-torsion group.

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arXiv2022

Additive decompositions for rings of modular forms

We study rings of integral modular forms for congruence subgroups as modules over the ring of integral modular forms for the full modular group. In many cases these modules are free or decompose at least into well-understood pieces. We apply this to characterize which rings of modular forms are Cohen--Macaulay and to prove finite generation results. These theorems are based on decomposition results about vector bundles on the compactified moduli stack of elliptic curves.

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arXiv2022

Topological modular forms with level structure: decompositions and duality

Topological modular forms with level structure were introduced in full generality by Hill and Lawson. We show that these decompose additively in many cases into a few simple pieces and give an application to equivariant $TMF$. Furthermore, we show which $Tmf_1(n)$ are self-Anderson dual up to a shift, both with and without their natural $C_2$-action.

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arXiv2020

The Brauer group of the moduli stack of elliptic curves

We compute the Brauer group of the moduli stack of elliptic curves over the integers, localizations of the integers, finite fields of odd characteristic, and algebraically closed fields of characteristic not $2$. The methods involved include the use of the parameter space of Legendre curves and the moduli stack of curves with full (naive) level $2$ structure, the study of the descent spectral sequence in étale cohomology and the Leray spectral sequence in fppf cohomology, the computation of the group cohomology of $S_3$ in a certain integral representation, the classification of cubic Galois extensions of the field of rational numbers, the computation of Hilbert symbols in the ramified case for the primes $2$ and $3$, and finding $p$-adic elliptic curves with specified properties.

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arXiv2018

Rings of modular forms and a splitting of $TMF_0(7)$

Among topological modular forms with level structure, $TMF_0(7)$ at the prime $3$ is the first example that had not been understood yet. We provide a splitting of $TMF_0(7)$ at the prime 3 as $TMF$-module into two shifted copies of $TMF$ and two shifted copies of $TMF_1(2)$. This gives evidence to a much more general splitting conjecture. Along the way, we develop several new results on the algebraic side. For example, we show the normality of rings of modular forms of level $n$ and introduce cubical versions of moduli stacks of elliptic curves with level structure.

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arXiv2017

Monadicity of the Bousfield-Kuhn functor

We consider the localization of the $\infty$-category of spaces at the $v_n$-periodic equivalences, the case $n=0$ being rational homotopy theory. We prove that this localization is for $n\geq 1$ equivalent to algebras over a certain monad on the $\infty$-category of $T(n)$-local spectra. This monad is built from the Bousfield--Kuhn functor.

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arXiv2017

(Topological) modular forms with level structures: decompositions and duality

The present article studies decompositions of vector bundles on the moduli stack of elliptic curves that are pushforwards of vector bundles on moduli of elliptic curves with level structure. These imply decomposition results for rings of modular forms and also for topological modular forms. We give explicit formulas for these decompositions and also apply them to equivariant topological modular forms. Moreover, we study the dualizing sheaf on $\overline{\mathcal{M}}_1(n)$ and characterize the numbers $n$ such that $Tmf_1(n)$ is Anderson self-dual.

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arXiv2017

Gorenstein duality for Real spectra

Following Hu and Kriz, we study the $C_2$-spectra $BP\mathbb{R}\langle n \rangle$ and $E\mathbb{R}(n)$ that refine the usual truncated Brown-Peterson and the Johnson-Wilson spectra. In particular, we show that they satisfy Gorenstein duality with a representation grading shift and identify their Anderson duals. We also compute the associated local cohomology spectral sequence in the cases $n=1$ and $2$.

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