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Ki-Seng Tan

Publications and source records attributed to Ki-Seng Tan.

At least 19 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 $\mu$-invariant hypothesis that is already detected after specialization to the unramified $\mathbb{Z}_p$-extension. The principal new input is a `$\chi$-formula' that compares appropriate $\chi$-isotypic characteristic ideals of Selmer modules with the corresponding specializations of the $p$-adic $L$-function. Finally, to show that our $\mu$-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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arXiv2026

$p$-adic $L$-functions for elliptic curves over global function fields

We introduce a $p$-adic $L$-function $\mathscr L_{A/L}$ associated to each ordinary elliptic curve $A$ over a global function field $K$ of characteristic $p$ together with a $\mathbb{Z}_{p}^{d}$-extension $L/K$, $d=0$ allowed, unramified outside a finite set of places where $A$ has ordinary (good ordinary or multiplicative) reductions. This $\mathscr L_{A/L}$ is characterized by its interpolation of the special values of twisted Hasse-Weil $L$-functions. We show that it satisfies the desired functional equation, specialization formula, and restriction formula in connection with the characteristic ideal of the dual $p^\infty$-Selmer group of $A/L$. The Iwasawa main conjecture having $\mathscr{L}_{A / L}$ as the analytic side is proven in several cases. In the $d\geq 3$ case, the conjecture holds for $A/L$ if and only if it holds for all intermediate $\mathbb{Z}_p^2$-extensions $L'/K$ belonging to a given non-empty Zariski open subset of the Grassmannian $\mathrm{Gr}(d-2,d)(\mathbb{Z}_p)$. Recently, subject to a technical $\mu$-invariant hypothesis, if $A/K$ has semistable reduction everywhere, the Iwasawa main conjecture is proven for $A$ over $L$ \cite{ttt26}.

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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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arXiv2020

On the $μ$-invariants of abelian varieties over function fields of positive characteristic

Let $A$ be an abelian variety over a global function field $K$ of characteristic $p$. We study the $μ$-invariant appearing in the Iwasawa theory of $A$ over the unramified $\mathbb{Z}_p$-extension of $K$. Ulmer suggests that this invariant is equal to what he calls the dimension of the Tate-Shafarevich group of $A$ and that it is indeed the dimension of some canonically defined group scheme. Our first result is to verify his suggestions. He also gives a formula for the dimension of the Tate-Shafarevich group (which is now the $μ$-invariant) in terms of other quantities including the Faltings height of $A$ and Frobenius slopes of the numerator of the Hasse-Weil $L$-function of $A / K$ assuming the conjectural Birch-Swinnerton-Dyer formula. Our next result is to prove this $μ$-invariant formula unconditionally for Jacobians and for semistable abelian varieties. Finally, we show that the "$μ=0$" locus of the moduli of isomorphism classes of minimal elliptic surfaces endowed with a section and with fixed large enough Euler characteristic is a dense open subset.

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arXiv2014

Pontryagin duality for Iwasawa modules and abelian varieties

We prove a functional equation for two projective systems of finite abelian $p$-groups, $\{\fa_n\}$ and $\{\fb_n\}$, endowed with an action of $\ZZ_p^d$ such that $\fa_n$ can be identified with the Pontryagin dual of $\fb_n$ for all $n$. Let $K$ be a global field. Let $L$ be a $\ZZ_p^d$-extension of $K$ ($d\geq 1$), unramified outside a finite set of places. Let $A$ be an abelian variety over $K$. We prove an algebraic functional equation for the Pontryagin dual of the Selmer group of $A$.

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arXiv2013

On the Iwasawa Main conjecture of abelian varieties over function fields

We study a geometric analogue of the Iwasawa Main Conjecture for abelian varieties in the two following cases: constant ordinary abelian varieties over $Z_p^d$-extensions of function fields ($d\geq 1$) ramified at a finite set of places, and semistable abelian varieties over the arithmetic $Z_p$-extension of a function field. One of the tools we use in our proof is a pseudo-isomorphism relating the duals of the Selmer groups of $A$ and its dual abelian variety $A^t$. This holds as well over number fields and is a consequence of a quite general algebraic functional equation.

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arXiv2013

Selmer groups over $\Z_p^d$-extensions

Consider an abelian variety $A$ defined over a global field $K$ and let $L/K$ be a $\Z_p^d$-extension, unramified outside a finite set of places of $K$, with $\Gal(L/K)=Γ$. Let $Λ(Γ):=\Z_p[[Γ]]$ denote the Iwasawa algebra. In this paper, we study how the characteristic ideal of the $Λ(Γ)$-module $X_L$, the dual $p$-primary Selmer group, varies when $L/K$ is replaced by a intermediate $\Z_p^e$-extension.

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arXiv2012

On the Hasse principle for finite group schemes over global function fields

Let K be a global function field of positive characteristic p and let M be a (commutative) finite and flat K-group scheme. We show that the kernel of the canonical localization map H^{1}(K,M)\to\prod_{all v}H^{1}(K_{v},M) in flat (fppf) cohomology can be computed solely in terms of Galois cohomology. We then give applications to the case where M is the kernel of multiplication by p^{m} on an abelian variety defined over K.

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arXiv2007

On the m-torsion Subgroup of the Brauer Group of a Global Field

In this note, we give a short proof of the existence of certain abelian extension over a given global field $K$. This result implies that for every positive integer $m$, there exists an abelian extension $L/K$ of exponent $m$ such that the $m$-torsion subgroup of $\Br(K)$ equals $\Br(L/K)$.

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arXiv2007

On Iwasawa Theory over Function Fields

Let $k_{\infty}$ be a $\Z_p^d$-extension of a global function field $k$ of characteristic $p$. Let $\Cl_{k_{\infty},p}$ be the $p$ completion of the class group of $k_{\infty}$. We prove that the characteristic ideal of the Galois module $\Cl_{k_{\infty},p}$ is generated by the Stickelberger element of Gross which calculates the special values of $L$ functions.

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