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Meng Fai Lim

Publications and source records attributed to Meng Fai Lim.

At least 19 recordsLinked to original sources
arXiv2026

On capitulations of even $K$-groups and pseudo-null submodules in $\mathbb{Z}_p^d$-extensions

The capitulations of ideals in $\mathbb{Z}_p^d$-extensions and pseudo-null submodules of the classical Iwasawa modules are closely related as evidenced in the works of Ozaki and Fujii. In this paper, we investigate the analogous situation for the even $K$-groups. As an application, we obtain a new sufficient condition for existence of non-trivial pseudo-null submodules in the classical Iwasawa modules.

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arXiv2026

On Iwasawa theory of abelian varieties over $\mathbb{Z}_p^2$-extension with applications to Diophantine stability and integally Diophantine extensions

We present certain results on the Iwasawa theory of an abelian variety with potentially good ordinary reduction at all primes above $p$. These are then applied to study Diophantine stability and integally Diophantine extensions. Along the way, we also obtain some results pertaining to Mazur growth conjecture which refine previous results of Gajek-Leonard, Hatley, Kundu and Lei. Finally, we extend our investigation to the case of an elliptic curve with good supersingular reduction at the prime $p$ and make a similar analysis.

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arXiv2026

Algebraic functional equation for big Galois representations over multiple $\mathbb{Z}_p$-extensions

We present a general approach to establish algebraic functional equations for big Galois representations over multiple $\mathbb{Z}_p$-extensions. Our result is formulated in both Selmer group and Selmer complex settings, and encompasses a broad range of Iwasawa-theoretic scenarios. In particular, our result applies to the triple product of Hida families in both balanced and unbalanced cases, as well as the half-ordinary Rankin-Selberg universal deformations recently studied by the first named author and Loeffler. Our result also significantly generalizes many previously known cases of algebraic functional equations and answers a question of Greenberg.

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arXiv2025

Non-commutative Iwasawa theory of abelian varieties over global function fields

Let $A$ be an abelian variety defined over a global function field $F$, and let $p$ be a prime distinct from the characteristic of $F$. Let $F_\infty$ be a $p$-adic Lie extension of $F$ that contains the cyclotomic $\mathbb{Z}_p$-extension $F^{\mathrm{cyc}}$ of $F$. In this paper, we investigate the structure of the $p$-primary Selmer group $\mathrm{Sel}(A/F_\infty)$ of $A$ over $F_\infty$. We prove the $\mathfrak{M}_H(G)$-conjecture for $A/F_\infty$. Furthermore, we show that both the $μ$-invariant of the Pontryagin dual of the Selmer group $\mathrm{Sel}(A/F^\mathrm{cyc})$ and the generalised $μ$-invariant of the Pontryagin dual of the Selmer group $\mathrm{Sel}(A/F_\infty)$ are zero, therby proving Mazur's conjecture for $A/F$. We then relate the order of vanishing of the characteristic elements, evaluated at Artin representations, to the corank of the Selmer group of the corresponding twist of $A$ over the base field $F$. Assuming the finiteness of the Tate-Shafarevich group, we establish that this corank equals the order of vanishing of the $L$-function of $A/F$ at $s=1$. Finally, we extend a theorem of Sechi - originally proved for elliptic curves without complex multiplication - to abelian varieties over global function fields. This is achieved by adapting the notion of generalised Euler characteristic, introduced by Zerbes for elliptic curves over number fields. This new invariant allows us, via Akashi series, to relate the generalised Euler characteristic of $\mathrm{Sel}(A/F_\infty)$ to the Euler characteristic of $\mathrm{Sel}(A/F^{\mathrm{cyc}})$.

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arXiv2025

On even $K$-groups of rings of integers of real abelian fields

We present approaches for calculating the precise orders of the algebraic $K$-groups $K_{4n-2}(\mathcal{O}_K)$ for a totally real abelian $K$. Along the way, we also establish a formula connecting the order of $K_{4n-2}(\mathcal{O}_E)$ of a totally real $p$-elementary field $E$ to its intermediate cyclic $p$-degree fields. Additionally, we provide a compiled list of values for these $K$-groups.

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arXiv2025

On the $p$-divisibility of even $K$-groups of the ring of integers of a cyclotomic field

Let $k$ be a given positive odd integer and $p$ an odd prime. In this paper, we shall give a sufficient condition when a prime $p$ divides the order of the groups $K_{2k}(\mathbb{Z}[ζ_m+ζ_m^{-1}])$ and $K_{2k}(\mathbb{Z}[ζ_m])$, where $ζ_m$ is a primitive $m$th root of unity. When $F$ is a $p$-extension contained in $\mathbb{Q}(ζ_l)$ for some prime $l$, we also establish a necessary and sufficient condition for the order of $K_{2(p-2)}(\mathcal{O}_F)$ to be divisible by $p$.

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arXiv2025

On even $K$-groups over $p$-adic Lie extensions of global function fields

Let $p$ be a fixed prime number, and $F$ a global function field of characteristic not equal to $p$. In this paper, we shall study the growth of the Sylow $p$-subgroups of the even $K$-groups in a $p$-adic Lie extension of $F$, where the $p$-adic Lie extension is assumed to contain the cyclotomic $\mathbb{Z}_p$-extension of $F$. We also establish a duality between the direct limit and inverse limit of the even $K$-groups.

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arXiv2024

On characteristic elements modulo $p$ in non-commutative Iwasawa theory

Coates, Fukaya, Kato, Sujatha and Venjakob come up with a procedure of attaching suitable characteristic element to Selmer groups defined over a non-commutative $p$-adic Lie extension, which is subsequently refined by Burns and Venjakob. By their construction, these characteristic elements are realized as elements in an appropriate localized $K_1$-group. In this paper, we will introduce a notion of modulo $p$ for these elements and study some of their properties. As an application, we study the Greenberg Selmer group of a tensor product of modular forms, where $p$ is an Eisenstein prime for one of these forms.

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arXiv2024

On fine Mordell-Weil groups over $\mathbb{Z}_p$-extensions of an imaginary quadratic field

Let $E$ be an elliptic curve over $\mathbb{Q}$. Greenberg has posed a question whether the structure of the fine Selmer group over the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$ can be described by cyclotomic polynomials in a certain precise manner. A recent work of Lei has made progress on this problem by proving that the fine Mordell-Weil group (in the sense of Wuthrich) does have this required property. The goal of this paper is study the analogous question of Greenberg over various $\mathbb{Z}_p$-extensions of an imaginary quadratic field $F$. In particular, when the elliptic curve has complex multiplication by the ring of integers of the imaginary quadratic field, we obtain analogous results of Lei over the cyclotomic $\mathbb{Z}_p$-extension and anti-cyclotomic $\mathbb{Z}_p$-extension of $F$. In the event that the elliptic curve has good ordinary reduction at the prime $p$, we further obtain a result over the $\mathbb{Z}_p$-extension of $F$ unramified outside precisely one of the prime of $F$ above $p$. Finally, we study the situation of an elliptic curve over the anticyclotomic $\mathbb{Z}_p$-extension under the generalized Heegner hypothesis. Along the way, we establish an analogous result for the BDP-Selmer group. This latter result is then applied to obtain a relation between the BDP $p$-adic $L$-function and the Mordell-Weil rank growth in the anticyclotomic $\mathbb{Z}_p$-extension which may be of independent interest.

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arXiv2024

On pseudo-nullity of fine Mordell-Weil group

Let $E$ be an elliptic curve defined over $\mathbb{Q}$ with good ordinary reduction at a prime $p\geq 5$, and let $F$ be an imaginary quadratic field. Under appropriate assumptions, we show that the Pontryagin dual of the fine Mordell-Weil group of $E$ over the $\mathbb{Z}_p^2$-extension of $F$ is pseudo-null as a module over the Iwasawa algebra of the group $\mathbb{Z}_p^2$.

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arXiv2023

Structure of fine Selmer groups over $\mathbb{Z}_p$-extensions

This paper is concerned with the study of the fine Selmer group of an abelian variety over a $\mathbb{Z}_p$-extension which is not necessarily cyclotomic. It has been conjectured that these fine Selmer groups are always torsion over $\mathbb{Z}_p[[Γ]]$, where $Γ$ is the Galois group of the $\mathbb{Z}_p$-extension in question. In this paper, we shall provide several strong evidences towards this conjecture. Namely, we show that the conjectural torsionness is consistent with the pseudo-nullity conjecture of Coates-Sujatha. We also show that if the conjecture is known for the cyclotomic $\mathbb{Z}_p$-extension, then it holds for almost all $\mathbb{Z}_p$-extensions. We then carry out a similar study for the fine Selmer group of an elliptic modular form. When the modular forms are ordinary and come from a Hida family, we relate the torsionness of the fine Selmer groups of the specialization. This latter result allows us to show that the conjectural torsionness in certain cases is consistent with the growth number conjecture of Mazur. Finally, we end with some speculations on the torsionness of fine Selmer groups over an arbitrary $p$-adic Lie extension.

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arXiv2023

On the structure of even $K$-groups of rings of algebraic integers

In this paper, we describe the higher even $K$-groups of the ring of integers of a number field in terms of class groups of an appropriate extension of the number field in question. This is a natural extension of the previous collective works of Browkin, Keune and Kolster, where they considered the case of $K_2$. We then revisit the Kummer's criterion of totally real fields as generalized by Greenberg and Kida. In particular, we give an algebraic $K$-theoretical formulation of this criterion which we will prove using the algebraic $K$-theoretical results developed here.

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arXiv2022

Norm principle for even K-groups of number fields

We investigate the norm maps of algebraic even $K$-groups of finite extensions of number fields. Namely, we show that they are surjective in most situations. In the event that they are not surjective, we give a criterion in determining when an element in the even $K$-group of the base field comes from a norm of an element from the even $K$-groups of the extension field. This latter criterion is only reliant on the real primes of the base field.

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arXiv2022

On the codescent of étale wild kernels in $p$-adic Lie extensions

Let $F$ be a number field and $p$ an odd prime. We estimate the kernels and cokernels of the codescent maps of the étale wild kernels over various $p$-adic Lie extensions. For this, we propose a novel approach of viewing the étale wild kernel as an appropriate fine Selmer group in the sense of Coates-Sujatha. This viewpoint reduces the problem to a control theorem of the said fine Selmer groups, which in turn allows us to employ the strategies developed by Mazur and Greenberg. As applications of our estimates on the kernels and cokernels of the codescent maps, we establish asymptotic growth formulas for the étale wild kernels in the various said $p$-adic Lie extensions. We then relate these growth formulas to the Greenberg's conjecture (and its noncommutative analogue). Finally, we shall give some examples to illustrate our results.

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