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Donghyeok Lim

Publications and source records attributed to Donghyeok Lim.

9 recordsLinked to original sources
arXiv2026

Local Minkowski units in non-abelian extensions with cyclic Sylow $p$-subgroups

We establish a criterion for the existence of a local Minkowski unit at $p$ that applies to all Galois extensions with Galois group isomorphic to the direct product of a non-$p$-group and a cyclic $p$-group. As applications, we construct non-abelian extensions admitting a local Minkowski unit at $p$ under various ramification conditions and analyze the Iwasawa module structure of units in $\mathbb{Z}_p$-extensions of number fields. We also extend our study to the case where the group of $p$-power roots of unity is nontrivial.

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arXiv2026

Construction Of Non-Odd Galois Representations With Large Image

In this work, we construct Galois representations with large image that are not GL r -odd (or, equivalently, not regular at infinity). More precisely, for every prime p {\v e} 3, every integer r {\v e} 2, and every integer a P rp1 'p'1q r q{2, r '2s satisfying a '' r pmod 2q, we construct a continuous Galois representation $\rho$\,: G Q __ GL r pQ p q whose image is commensurable with GL r pZ p q and satisfies |trp$\rho$pcqq| '' a, where c denotes a complex conjugation. We also obtain analogous results for the orthogonal group O r pQ p q.

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arXiv2026

Tame Galois Groups, Linking Numbers and Mildness

Let $p$ be an odd prime and let $S$ be a set of tame primes. We denote by $G_S$ the Galois group of the maximal pro-$p$ extension of $\mathbb{Q}$ unramified outside $S$. We prove that for every finite set of tame primes $S_0$ with $|S_0|\geq 2$, there exists a set $S_1$ consisting of two tame primes such that $G_{S_0\cup S_1}$ has cohomological dimension $2$. This refines a result of Labute. More generally, we establish an analogous result for number fields not containing a primitive $p$-th root of unity, under a suitable splitting condition. Our approach answers a question of Labute, from his seminal paper on mild groups, and combines weighted Zassenhaus filtrations, graph-theoretic methods, and Koch-type presentations. As an application, we solve several cohomological Galois inverse problems with prescribed ramification and splitting. We also provide numerical examples and statistics.

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arXiv2025

Massey products and unipotent extensions with restricted ramification

We fix a prime p and construct new cases of pro-p extensions of number fields with restricted ramification and splitting, whose Galois groups decompose as coproducts of pro-p absolute Galois groups of local fields. As a consequence, these pro-p extensions satisfy the strong Massey vanishing property and thus admit large unipotent quotients.

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arXiv2024

On Krull-Schmidt decompositions of unit groups of number fields

We prove that the Krull-Schmidt decomposition of the Galois module of the $p$-adic completion of algebraic units is controlled by the primes that are ramified in the Galois extension and the $S$-ideal class group. We also compute explicit upper bounds for the number of possible Galois module structures of algebraic units when the Galois group is cyclic of order $p^{2}$ or $p^{3}$.

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arXiv2024

The finitude of tamely ramified pro-$p$ extensions of number fields with cyclic $p$-class groups

Let $p$ be an odd prime and $F$ be a number field whose $p$-class group is cyclic. Let $F_{\{\mathfrak{q}\}}$ be the maximal pro-$p$ extension of $F$ which is unramified outside a single non-$p$-adic prime ideal $\mathfrak{q}$ of $F$. In this work, we study the finitude of the Galois group $G_{\{\mathfrak{q}\}}(F)$ of $F_{\{\mathfrak{q}\}}$ over $F$. We prove that $G_{\{\mathfrak{q}\}}(F)$ is finite for the majority of $\mathfrak{q}$'s such that the generator rank of $G_{\{\mathfrak{q}\}}(F)$ is two, provided that for $p = 3$, $F$ is not a complex quartic field containing the primitive third roots of unity.

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arXiv2023

On the existence of Minkowski units

We investigate the Galois structure of algebraic units in cyclic extensions of number fields and thereby obtain strong new results on the existence of independent Minkowski $S$-units.

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arXiv2023

On the Galois structure of units in totally real $p$-rational number fields

The theory of factor-equivalence of integral lattices establishes a far-reaching relationship between the Galois module structure of the unit group of the ring of integers of a number field and its arithmetic. For a number field $K$ that is Galois over $\mathbb{Q}$ or an imaginary quadratic field, we prove a necessary and sufficient condition on the quotients of class numbers of subfields of $K$, for the quotient $E_{K}$ of the unit group of the ring of integers of $K$ modulo the subgroup of roots of unity to be factor equivalent to the standard cyclic Galois module. Using strong arithmetic properties of totally real $p$-rational number fields, we prove that the non-abelian $p$-rational $p$-extensions of $\mathbb{Q}$ do not admit Minkowski units, thereby extending a result of Burns to non-abelian number fields. We also study the relative Galois module structure of $E_{L}$ for varying Galois extensions $L/F$ of totally real $p$-rational number fields whose Galois groups are isomorphic to a fixed finite group $G$. In that case, we prove that there exists a finite set $\Omega$ of $\mathbb{Z}_p[G]$-lattices such that for every $L$, $\mathbb{Z}_{p} \otimes_{\mathbb{Z}} E_{L}$ is factor equivalent to $\mathbb{Z}_{p}[G]^{n} \oplus X$ as $\mathbb{Z}_p[G]$-lattices for some $X \in \Omega$ and an integer $n \geq 0$.

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