arXiv subjects

Chun-Yin Hui

Publications and source records attributed to Chun-Yin Hui.

5 recordsLinked to original sources

arXiv2026

On coefficients, potentially abelian quotients, and residual irreducibility of compatible systems

Let $\{ρ_λ:G_K\rightarrow GL_n(\overline E_λ)\}$ be a semisimple E-rational compatible system of a number field K. In a first step, building upon the theory of pseudocharacters [Ro96],[Ch14], we attach to each $ρ_λ$ an algebraic monodromy group $G_λ$ defined over $E_λ$ and also prove that the compatible system can be descended to a strongly E'-rational compatible system $\{ρ_{λ'}: G_K\rightarrow GL_n(E'_{λ'})\}$ for some finite extension E'/E. Secondly, we demonstrate that the maximal potentially abelian quotient of $G_λ$ is independent of $λ$ in a strong sense. Finally, as an application, we generalize a result of Patrikis--Snowden--Wiles on residual irreducibility of compatible systems.

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arXiv2025

Rectangular representations and $\lambda$-independence of algebraic monodromy groups

Let $\mathfrak g$ be a complex semisimple Lie algebra. We define what it means for a finite dimensional representation of $\mathfrak g$ to be rectangular and completely classify faithful rectangular representations. As an application, we obtain new $\lambda$-independence results on the algebraic monodromy groups of compatible systems of $\lambda$-adic Galois representations of number fields.

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arXiv2025

On distribution of supersingular primes of abelian varieties and K3 surfaces

Let X be an abelian variety or a K3 surface defined over a number field K. We prove that the density of the supersingular primes of X is zero if X is non-CM. By applying an effective Chebotarev density theorem of Serre, we obtain asymptotic upper bounds of the counting function for these supersingular primes.

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arXiv2025

Monodromy and irreducibility of type $A_1$ automorphic Galois representations

Let $K$ be a totally real field and $π$ be a regular algebraic polarized cuspidal automorphic representation of $\mathrm{GL}_n(\mathbb A_K)$. Let $\{ρ_{π,λ}:\mathrm{Gal}_K\to\mathrm{GL}_n(\overline E_λ)\}_λ$ be the compatible system of Galois representations attached to $π$ and denote by $\mathbf G_λ$ the algebraic monodromy group of $ρ_{π,λ}$. Suppose there exists $λ_0$ such that (a) $ρ_{π,λ_0}$ is irreducible; (b) $\mathbf G_{λ_0}$ is connected and of type $A_1$; and (c) the tautological representation of $\mathbf G_{λ_0}$ is of a certain type. We prove that $\bullet$ $\mathbf G_{λ,\mathbb C}\subset\mathrm{GL}_{n, \mathbb C}$ is independent of $λ$; $\bullet$ $ρ_{π,λ}$ is irreducible for all $λ$, and residually irreducible for almost all $λ$. Moreover, if $K=\mathbb Q$ or $n$ is odd, we prove that the same conclusions hold without the assumption that $π$ is polarized. We also prove that if $K=\mathbb Q$, then the compatible system $\{ρ_{π,λ}\}_λ$ is constructed from certain two-dimensional modular compatible systems up to twist.

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arXiv2024

Weak abelian direct summands and irreducibility of Galois representations

Let $ρ_\ell$ be a semisimple $\ell$-adic representation of a number field $K$ that is unramified almost everywhere. We introduce a new notion called weak abelian direct summands of $ρ_\ell$ and completely characterize them, for example, if the algebraic monodromy of $ρ_\ell$ is connected. If $ρ_\ell$ is in addition $E$-rational for some number field $E$, we prove that the weak abelian direct summands are locally algebraic (and thus de Rham). We also show that the weak abelian parts of a connected semisimple Serre compatible system form again such a system. Using our results on weak abelian direct summands, when $K$ is totally real and $ρ_\ell$ is the three-dimensional $\ell$-adic representation attached to a regular algebraic cuspidal automorphic, not necessarily polarizable representation $π$ of $\mathrm{GL}_3(\mathbb{A}_K)$ together with an isomorphism $\mathbb{C}\simeq \overline{\mathbb{Q}}_\ell$, we prove that $ρ_\ell$ is irreducible. We deduce in this case also some $\ell$-adic Hodge theoretic properties of $ρ_\ell$ if $\ell$ belongs to a Dirichlet density one set of primes.

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