arXiv subjects

Zicheng Qian

Publications and source records attributed to Zicheng Qian.

8 recordsLinked to original sources

arXiv2026

Splitting and making explicit the de Rham complex of the Drinfeld space

Let $p$ be a prime number, $K$ a finite extension of $\mathbb{Q}_p$ and $n$ an integer $\geq 2$. We completely and explicitly describe the global sections $Ω^\bullet$ of the de Rham complex of the Drinfeld space over $K$ in dimension $n-1$ as a complex of (duals of) locally $K$-analytic representations of $\mathrm{GL}_n(K)$. Using this description, we construct an explicit section in the derived category of (duals of) finite length admissible locally $K$-analytic representations of $\mathrm{GL}_n(K)$ to the canonical morphism of complexes $Ω^\bullet \twoheadrightarrow H^{n-1}(Ω^\bullet)[-(n-1)]$.

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arXiv2026

On generalization of Breuil--Schraen's $\mathscr{L}$-invariants to $\mathrm{GL}_n$

Let $p$ be prime number and $K$ be a $p$-adic field. We systematically compute the higher $\mathrm{Ext}$-groups between locally analytic generalized Steinberg representations (LAGS for short) of $\mathrm{GL}_n(K)$ via a new combinatorial treatment of some spectral sequences arising from the so-called Tits complex. Such spectral sequences degenerate at the second page and each $\mathrm{Ext}$-group admits a canonical filtration whose graded pieces are terms in the second page of the corresponding spectral sequence. For each pair of LAGS, we are particularly interested their $\mathrm{Ext}$-groups in the bottom two non-vanishing degrees. We write down an explicit basis for each graded piece (under the canonical filtration) of such an $\mathrm{Ext}$-group, and then describe the cup product maps between such $\mathrm{Ext}$-groups using these bases. As an application, we generalize Breuil's $\mathscr{L}$-invariants for $\mathrm{GL}_2(\mathbb{Q}_p)$ and Schraen's higher $\mathscr{L}$-invariants for $\mathrm{GL}_3(\mathbb{Q}_p)$ to $\mathrm{GL}_n(K)$. Along the way, we also establish a generalization of Bernstein--Zelevinsky geometric lemma to admissible locally analytic representations constructed by Orlik--Strauch, generalizing a result in Schraen's thesis for $\mathrm{GL}_3(\mathbb{Q}_p)$.

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arXiv2025

On $\mathrm{Ext}^{\bullet}$ between locally analytic generalized Steinberg with applications

Let $n\geq 2$ be an integer, $p$ be a prime number and $K$ be a finite extension of $\mathbb{Q}_p$. Motivated by Schraen's thesis and Gehrmann's definition of automorphic simple $\mathscr{L}$-invariants, we study the first non-vanishing extension groups between a pair of locally $K$-analytic generalized Steinberg representations of $\mathrm{GL}_n(K)$. We study subspaces of these extension groups defined by using either relative conditions with respect to Lie subalgebras of $\mathfrak{s}\mathfrak{l}_{n}$ (isomorphic to $\mathfrak{s}\mathfrak{l}_{m}$ for some $2\leq m<n$) or maps between locally $K$-analytic generalized Steinberg representations of $\mathrm{GL}_n(K)$ with different highest weights. The applications of these computations are two-fold. On one hand, we prove that a certain universal successive extension of filtered $(φ,N)$-modules can be realized as the space of homomorphisms from a suitable shift of the dual of locally $K$-analytic Steinberg representation into the de Rham complex of the Drinfeld upper-half space, generalizing one main result of Schraen's thesis from $\mathrm{GL}_{3}(\mathbb{Q}_p)$ to $\mathrm{GL}_{n}(K)$. On the other hand, we give a definition of higher $\mathscr{L}$-invariants for $\mathrm{GL}_n(K)$ (which we call Breuil-Schraen $\mathscr{L}$-invariants) and discuss its possible explicit relation to Fontaine-Mazur $\mathscr{L}$-invariants, using ideas from Breuil-Ding's higher $\mathscr{L}$-invariants for $\mathrm{GL}_{3}(\mathbb{Q}_p)$.

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arXiv2023

Moduli of Fontaine--Laffaille representations and a mod-$p$ local-global compatibility result

Let $F/F^+$ be a CM field and let $\widetilde{v}$ be a finite unramified place of $F$ above the prime $p$. Let $\overline{r}: \mathrm{Gal}(\overline{\mathbb{Q}}/F)\rightarrow \mathrm{GL}_n(\overline{\mathbb{F}}_p)$ be a continuous representation which we assume to be modular for a unitary group over $F^+$ which is compact at all real places. We prove, under Taylor--Wiles hypotheses, that the smooth $\mathrm{GL}_n(F_{\widetilde{v}})$-action on the corresponding Hecke isotypical part of the mod-$p$ cohomology with infinite level above $\widetilde{v}|_{F^+}$ determines $\overline{r}|_{\mathrm{Gal}(\overline{\mathbb{Q}}_p/F_{\widetilde{v}})}$, when this latter restriction is Fontaine--Laffaille and has a suitably generic semisimplification.

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arXiv2023

Colength one deformation rings

Let $K/\mathbf{Q}_p$ be a finite unramified extension, $\overlineρ:\mathrm{Gal}(\overline{\mathbf{Q}}_p/K)\rightarrow\mathrm{GL}_n(\overline{\mathbf{F}}_p)$ a continuous representation, and $τ$ a tame inertial type of dimension $n$. We explicitly determine, under mild regularity conditions on $τ$, the potentially crystalline deformation ring $R^{η,τ}_{\overlineρ}$ in parallel Hodge--Tate weights $η=(n-1,\cdots,1,0)$ and inertial type $τ$ when the \emph{shape} of $\overlineρ$ with respect to $τ$ has colength at most one. This has application to the modularity of a class of shadow weights in the weight part of Serre's conjecture. Along the way we make unconditional the local-global compatibility results of \cite{PQ} and further study the geometry of moduli spaces of Fontaine--Laffaille representations in terms of colength one weights.

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arXiv2021

A note on mod-$p$ local-global compatibility via Scholze's functor

We remove the semisimple condition in the mod-$p$ local-global compatibility result of arXiv:2106.10674. Namely, assuming flatness of $π_{\mathfrak{m}}^\vee$ and $\overlineσ_{\mathfrak{m}}|_{\mathrm{Gal}_{F^+_{\mathfrak{p}}}}$ being multiplicity free, we prove that $H^{n-1}_{\text{ét}}(\mathbb{P}_{\mathbb{C}_p}^{n-1}, \mathcal{F}_{π_{\mathfrak{m}}[\mathfrak{m}]})$ determines $\overlineσ_{\mathfrak{m}}|_{\mathrm{Gal}_{F^+_{\mathfrak{p}}}}$ uniquely. We give remarks on our assumptions at the end of this note.

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arXiv2019

Dilogarithm and higher $\mathscr{L}$-invariants for $\mathrm{GL}_3(\mathbb{Q}_p)$

Let $E$ be a sufficiently large finite extension of $\mathbb{Q}_p$ and $ρ_p$ be a semi-stable representation $\mathrm{Gal}(\overline{\mathbb{Q}_p}/\mathbb{Q}_p)\rightarrow\mathrm{GL}_3(E)$ with a rank two monodromy operator $N$ and a non-critical Hodge filtration. We know that $ρ_p$ has three $\mathscr{L}$-invariants. We construct a family of locally analytic representations of $\mathrm{GL}_3(\mathbb{Q}_p)$ depending on three invariants in $E$ with each of them containing the locally algebraic representation determined by $ρ_p$. When $ρ_p$ comes from an automorphic representation $π$ of $G(\mathbb{A}_{\mathbb{Q}_p})$ for a suitable unitary group $G_{/\mathbb{Q}}$, we show that there is a unique object in the above family that embeds into the associated Hecke-isotypic subspace in the completed cohomology. We recall that Breuil constructed a family of locally analytic representations depending on four invariants and proved a similar result of local-global compatibility. We prove that if a representation $Π$ in Breuil's family embeds into the completed cohomology, then it must equally embed into an object in our family determined by $Π$. This gives a purely representation theoretic necessary condition for $Π$ to embed into completed cohomology. Moreover, certain natural subquotients of each object in our family give a true complex of locally analytic representations that realizes the derived object $Σ(λ, \underline{\mathscr{L}})$ by Schraen for a unique $\underline{\mathscr{L}}$ determined by the object. Consequently, the family we construct gives a relation between the higher $\mathscr{L}$-invariants studied by Breuil and Ding and the $p$-adic dilogarithm function which appears in the construction of $Σ(λ, \underline{\mathscr{L}})$ by Schraen.

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arXiv2018

On mod $p$ local-global compatibility for $\mathrm{GL}_n(\mathbf{Q}_p)$ in the ordinary case

Let $p$ be a prime number, $n>2$ an integer, and $F$ a CM field in which $p$ splits completely. Assume that a continuous automorphic Galois representation $\overline{r}:\mathrm{Gal}(\overline{\mathbf{Q}}/F)\rightarrow\mathrm{GL}_n(\overline{\mathbf{F}}_p)$ is upper-triangular and satisfies certain genericity conditions at a place $w$ above $p$, and that every subquotient of $\overline{r}|_{\mathrm{Gal}(\overline{\mathbf{Q}}_p/F_w)}$ of dimension $>2$ is Fontaine--Laffaille generic. In this paper, we show that the isomorphism class of $\overline{r}|_{\mathrm{Gal}(\overline{\mathbf{Q}}_p/F_w)}$ is determined by $\mathrm{GL}_n(F_w)$-action on a space of mod $p$ algebraic automorphic forms cut out by the maximal ideal of a Hecke algebra associated to $\overline{r}$, assuming a weight elimination result which is a theorem of Bao V. Le Hung in his forthcoming paper~\cite{LeH}. In particular, we show that the wildly ramified part of $\overline{r}|_{\mathrm{Gal}(\overline{\mathbf{Q}}_p/F_w)}$ is determined by the action of Jacobi sum operators (seen as elements of $\mathbf{F}_p[\mathrm{GL}_n(\mathbf{F}_p)]$) on this space.

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