arXiv subjects

Ningchuan Zhang

Publications and source records attributed to Ningchuan Zhang.

7 recordsLinked to original sources

arXiv2026

Picard groups of quotient ring spectra

We develop tools to study Picard groups of quotients of ring spectra by a finitely generated ideal, which we use to show that $\mathrm{Pic}(\mathrm{E}_n/I) = \mathbb{Z}/2$, where $\mathrm{E}_n$ is a Lubin--Tate theory and $I$ is an ideal generated by suitable powers of a regular sequence. We apply this to obtain spectral sequences computing Picard groups of $\mathrm{K}(n)$-local generalized Moore algebras, and make some preliminary computations including the height $1$ case.

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arXiv2025

The inverse limit topology and profinite descent on Picard groups in $K(n)$-local homotopy theory

In this paper, we study profinite descent theory for Picard groups in $K(n)$-local homotopy theory through their inverse limit topology. Building upon Burklund's result on the multiplicative structures of generalized Moore spectra, we prove that the module category over a $K(n)$-local commutative ring spectrum is equivalent to the limit of its base changes by a tower of generalized Moore spectra of type $n$. As a result, the $K(n)$-local Picard groups are endowed with a natural inverse limit topology. This topology allows us to identify the entire $E_1$ and $E_2$-pages of a descent spectral sequence for Picard spaces of $K(n)$-local profinite Galois extensions. Our main examples are $K(n)$-local Picard groups of homotopy fixed points $E_n^{hG}$ of the Morava $E$-theory $E_n$ for all closed subgroups $G$ of the Morava stabilizer group $\mathbb{G}_n$. The $G=\mathbb{G}_n$ case has been studied by Heard and Mor. At height $1$, we compute Picard groups of $E_1^{hG}$ for all closed subgroups $G$ of $\mathbb{G}_1=\mathbb{Z}_p^\times$ at all primes as a Mackey functor.

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arXiv2024

Congruences of Eisenstein series of level $Γ_1(N)$ via Dieudonné theory of formal groups

In this paper, we give a new explanation of congruences of Eisenstein series of level $Γ_1(N)$ and character $χ$. Our approach is based on Katz's algebro-geometric explanation of $p$-adic congruences of normalized Eisenstein series $E_{2k}$ of level $1$. One crucial step in our argument is to reformulate a Riemann-Hilbert correspondence in Katz's explanation in terms of Dieudonné theory of height $1$ formal $A$-modules and their finite subgroup schemes. We give a generalization of this Riemann-Hilbert correspondence in terms of formal groups of height greater than $1$.

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arXiv2024

Equivariant algebraic $\mathrm{K}$-theory and Artin $L$-functions

In this paper, we generalize the Quillen-Lichtenbaum Conjecture relating special values of Dedekind zeta functions to algebraic $\mathrm{K}$-groups. The former has been settled by Rost-Voevodsky up to the Iwasawa Main Conjecture. Our generalization extends the scope of this conjecture to Artin $L$-functions of Galois representations of finite, function, and totally real number fields. The statement of this conjecture relates norms of the special values of these $L$-functions to sizes of equivariant algebraic $\mathrm{K}$-groups with coefficients in an equivariant Moore spectrum attached to a Galois representation. We prove this conjecture in many cases, integrally, except up to a possible factor of powers of $2$ in the non-abelian and totally real number field case. In the finite field case, we further determine the group structures of their equivariant algebraic $\mathrm{K}$-groups with coefficients in Galois representations. At heart, our method lifts the Möbius inversion formula for factorizations of zeta functions as a product of $L$-functions, to the $E_1$-page of an equivariant spectral sequence converging to equivariant algebraic $\mathrm{K}$-groups. Additionally, the spectral Mackey functor structure on equivariant $\mathrm{K}$-theory allows us to incorporate certain ramified extensions that appear in these $L$-functions.

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arXiv2023

Exotic Picard groups and chromatic vanishing via the Gross-Hopkins duality

In this paper, we study the exotic $K(h)$-local Picard groups $κ_h$ when $2p-1=h^2$ and the homological Chromatic Vanishing Conjecture when $p-1$ does not divide $h$. The main idea is to use the Gross-Hopkins duality to relate both questions to certain Greek letter element computations in chromatic homotopy theory. Classical results of Miller-Ravenel-Wilson then imply that an exotic element at height $3$ and prime $5$ is not detected by the type-$2$ complex $V(1)$. For the homological Vanishing Conjecture, we prove it holds modulo the invariant prime ideal $I_{h-1}$. We further show that this special case of the Vanishing Conjecture implies the exotic Picard group $κ_h$ is zero at height $3$ and prime $5$. Both results can be thought of as a first step towards proving the vanishing of $κ_3$ at prime $5$.

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arXiv2023

Borel's rank theorem for Artin $L$-functions

Borel's rank theorem identifies the ranks of algebraic $K$-groups of the ring of integers of a number field with the orders of vanishing of the Dedekind zeta function attached to the field. Following the work of Gross, we establish a version of this theorem for Artin $L$-functions by considering equivariant algebraic $K$-groups of number fields with coefficients in rational Galois representations. This construction involves twisting algebraic $K$-theory spectra with rational equivariant Moore spectra. We further discuss integral equivariant Moore spectra attached to Galois representations and their potential applications in $L$-functions.

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arXiv2022

Analogs of Dirichlet $L$-functions in chromatic homotopy theory

The relation between Eisenstein series and the $J$-homomorphism is an important topic in chromatic homotopy theory at height $1$. Both sides are related to the special values of the Riemann $ζ$-function. Number theorists have studied the twistings of the Riemann $ζ$-functions and Eisenstein series by Dirichlet characters. Motivated by the Dirichlet equivariance of these Eisenstein series, we introduce the Dirichlet $J$-spectra in this paper. The homotopy groups of the Dirichlet $J$-spectra are related to the special values of the Dirichlet $L$-functions. Moreover, we find Brown-Comenetz duals of the Dirichlet $J$-spectra, whose formulas resemble functional equations of the corresponding Dirichlet $L$-functions. In this sense, the Dirichlet $J$-spectra we constructed are analogs of Dirichlet $L$-functions in chromatic homotopy theory.

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