arXiv subjects

Kwok-Wing Tsoi

Publications and source records attributed to Kwok-Wing Tsoi.

6 recordsLinked to original sources

arXiv2026

Specialisations of the Burungale-Castella-Skinner main conjecture to $\mathbb Z_p$-lines

Let $p>3$ be a prime, $E/\mathbb Q$ be an elliptic curve and $K$ an imaginary quadratic field satisfying the hypotheses of Burungale-Castella-Skinner, and let $L/K$ be the unique $\mathbb{Z}_p^2$-extension. In this note, by combining their integral two-variable main conjecture with the specialisation formula of the first-named author, we obtain a characteristic-ideal identity over every $\mathbb{Z}_p$-line in $L/K$, involving an explicit local factor. With the characteristic ideal of a non-torsion module defined to be zero, this identity also applies when the two-variable Perrin-Riou element specialises to zero. We call such lines exceptional and prove that only finitely many occur. We also prove that the cyclotomic line is non-exceptional with trivial local factor, and that the anticyclotomic line is exceptional. Finally, we bound the number of exceptional lines by the cyclotomic augmentation order and prove that if the $p$-primary Tate-Shafarevich group over $K$ is finite and the cyclotomic $p$-adic height pairing is non-degenerate, then this number is at most ${\rm rank}(E(K))$. In particular, when $E(K)$ has rank one, the anticyclotomic line is the unique exceptional line.

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arXiv2026

Iwasawa Main Conjecture for ordinary semistable elliptic curves over global function fields

Let $A$ be an ordinary elliptic curve over a global function field $K$ of characteristic $p$, assumed semistable at every place, and let $L/K$ be a $\mathbb{Z}_p^d$-extension ramified only at finitely many places where $A$ has ordinary reduction. Building on the framework of [Tan26] (arXiv:2603.10576), we prove the Iwasawa Main Conjecture for $A$ over $L$, subject to a technical $μ$-invariant hypothesis that is already detected after specialization to the unramified $\mathbb{Z}_p$-extension. The principal new input is a `$χ$-formula' that compares appropriate $χ$-isotypic characteristic ideals of Selmer modules with the corresponding specializations of the $p$-adic $L$-function. Finally, to show that our $μ$-hypothesis is non-vacuous, we prove, for $p>3$, that the hypothesis holds on a Zariski open dense locus in the moduli of semistable elliptic curves.

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arXiv2024

On a Birch and Swinnerton-Dyer type conjecture for the Hasse-Weil-Artin $L$-functions in characteristic $p>0$

Given an abelian variety $A$ over a global function field $K$ of characteristic $p>0$ and an irreducible complex continuous representation $ψ$ of the absolute Galois group of $K$, we obtain a BSD-type formula for the leading term of Hasse--Weil--Artin $L$-function for $(A,ψ)$ at $s=1$ under certain technical hypotheses. The formula we obtain can be applied quite generally; for example, it can be applied to the $p$-part of the leading term even when $ψ$ is weakly wildly ramified at some place under additional hypotheses. Our result is the function field analogue of the work of D. Burns and D. Macias Castillo, built upon the work on the equivariant refinement of the BSD conjecture by D. Burns, M. Kakde and the first-named author. To handle the $p$-part of the leading term, we need the Riemann--Roch theorem for equivariant vector bundles on a curve over a finite field generalising the work of S. Nakajima, B. Köck, and H. Fischbacher-Weitz and B. Köck, which is of independent interest.

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arXiv2024

The $μ$-invariant change for abelian varieties over finite $p$-extensions of global fields

We extend the work of Lai, Longhi, Suzuki, the first two authors and study the change of $μ$-invariants, with respect to a finite Galois p-extension $K'/K$, of an ordinary abelian variety $A$ over a $\mathbb{Z}_p^d$-extension of global fields $L/K$ that ramifies at a finite number of places at which $A$ has ordinary reductions. In characteristic $p>0$, we obtain an explicit bound for the size $δ_v$ of the local Galois cohomology of the Mordell-Weil group of $A$ with respect to a $p$-extension ramified at a supersingular place $v$. Next, in all characteristics, we describe the asymptotic growth of $δ_v$ along a multiple $\mathbb{Z}_p$-extension $L/K$ and provide a lower bound for the change of $μ$-invariants of $A$ from the tower $L/K$ to the tower $LK'/K'$. Finally, we present numerical evidence supporting these results.

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arXiv2022

On non-abelian higher special elements of $p$-adic representations

We develop a theory of `non-abelian higher special elements' in the non-commutative exterior powers of the Galois cohomology of $p$-adic representations. We explore their relation to the theory of organising matrices and thus to the Galois module structure of Selmer modules. In concrete applications, we relate our general theory to the formulation of refined conjectures of Birch and Swinnerton-Dyer type and to the Galois structure of Tate-Shafarevich and Selmer groups of abelian varieties.

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arXiv2018

On higher special elements of $p$-adic representations

As a natural generalization of the notion of `higher rank Euler system', we develop a theory of `higher special elements' in the exterior power biduals of the Galois cohomology of $p$-adic representations. We show, in particular, that such elements encode detailed information about the structure of Galois cohomology groups and are related by families of congruences involving natural height pairings on cohomology. As a first concrete application of the approach, we use it to refine, and extend, a variety of existing results and conjectures concerning the values of derivatives of Dirichlet $L$-series.

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