arXiv subjects

Yukako Kezuka

Publications and source records attributed to Yukako Kezuka.

8 recordsLinked to original sources

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 $\mathrm{Sel}(A/F^{\mathrm{cyc}})$.

math.NT↗

arXiv2025

Non-vanishing of central $L$-values of the Gross family of elliptic curve

We prove non-vanishing theorems for the central values of $L$-series of quadratic twists of the Gross elliptic curve with complex multiplication by the imaginary quadratic field $\mathbb{Q}(\sqrt{-q})$, where $q$ is any prime congruent to $7$ modulo $8$. This completes the non-vanishing theorems proven by Coates and the second author in which the primes $q$ were taken to be congruent to $7$ modulo $16$. From this, we obtain the finiteness of the Mordell-Weil group and the Tate-Shafarevich group for these curves. For a prime $\mathfrak{P}$ lying above the prime $2$, we also prove a converse theorem in the rank $0$ case and the $\mathfrak{P}$-part of the Birch-Swinnerton-Dyer conjecture for the higher-dimensional abelian varieties obtained by restriction of scalars.

math.NT↗

arXiv2024

On central $L$-values and the growth of the $3$-part of the Tate-Shafarevich group

Given any cube-free integer $λ>0$, we study the $3$-adic valuation of the algebraic part of the central $L$-value of the elliptic curve $$X^3+Y^3=λZ^3.$$ We give a lower bound in terms of the number of distinct prime factors of $λ$, which, in the case $3$ divides $λ$, also depends on the power of $3$ in $λ$. This extends an earlier result of the author in which it was assumed that $3$ is coprime to $λ$. We also study the $3$-part of the Tate-Shafarevich group for these curves and show that the lower bound is as expected from the conjecture of Birch and Swinnerton-Dyer, taking into account also the growth of the Tate-Shafarevich group.

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arXiv2021

Tamagawa number divisibility of central $L$-values of twists of the Fermat elliptic curve

Given any integer $N>1$ prime to $3$, we denote by $C_N$ the elliptic curve $x^3+y^3=N$. We first study the $3$-adic valuation of the algebraic part of the value of the Hasse-Weil $L$-function $L(C_N,s)$ of $C_N$ over $\mathbb{Q}$ at $s=1$, and we exhibit a relation between the $3$-part of its Tate-Shafarevich group and the number of distinct prime divisors of $N$ which are inert in the imaginary quadratic field $K=\mathbb{Q}(\sqrt{-3})$. In the case where $L(C_N,1)\neq 0$ and $N$ is a product of split primes in $K$, we show that the order of the Tate-Shafarevich group as predicted by the conjecture of Birch and Swinnerton-Dyer is a perfect square.

math.NT↗

arXiv2021

On the main conjecture of Iwasawa theory for certain non-cyclotomic $\mathbb{Z}_p$-extensions

Let $K=\mathbb{Q}(\sqrt{-q})$, where $q$ is any prime number congruent to $7$ modulo $8$, with ring of integers $\mathcal{O}$ and Hilbert class field $H$. Suppose $p\nmid [H:K]$ is a prime number which splits in $K$, say $p\mathcal{O}=\mathfrak{p}\mathfrak{p}^*$. Let $H_\infty=HK_\infty$ where $K_\infty$ is the unique $\mathbb{Z}_p$-extension of $K$ unramified outside $\mathfrak{p}$. Write $M(H_\infty)$ for the maximal abelian $p$-extension of $H_\infty$ unramified outside the primes above $\mathfrak{p}$, and set $X(H_\infty)=\mathrm{Gal}(M(H_\infty)/H_\infty)$. In this paper, we establish the main conjecture of Iwasawa theory for the Iwasawa module $X(H_\infty)$. As a consequence, we have that if $X(H_\infty)=0$, the relevant $L$-values are $\mathfrak{p}$-adic units. In addition, the main conjecture for $X(H_\infty)$ has implications toward (a) the BSD Conjecture for a class of CM elliptic curves; (b) weak $\mathfrak{p}$-adic Leopoldt conjecture.

math.NT↗

arXiv2021

A classical family of elliptic curves having rank one and the $2$-primary part of their Tate-Shafarevich group non-trivial

We study elliptic curves of the form $x^3+y^3=2p$ and $x^3+y^3=2p^2$ where $p$ is any odd prime satisfying $p\equiv 2\bmod 9$ or $p\equiv 5\bmod 9$. We first show that the $3$-part of the Birch-Swinnerton-Dyer conjecture holds for these curves. Then we relate their $2$-Selmer group to the $2$-rank of the ideal class group of $\mathbb{Q}(\sqrt[3]{p})$ to obtain some examples of elliptic curves with rank one and non-trivial $2$-part of the Tate-Shafarevich group.

math.NT↗

arXiv2018

Analogues of Iwasawa's $μ=0$ conjecture and the weak Leopoldt conjecture for a non-cyclotomic $\mathbb{Z}_2$-extension

Let $K = \mathbb{Q}(\sqrt{-q})$, where $q$ is any prime number congruent to $7$ modulo $8$, and let $\mathcal{O}$ be the ring of integers of $K$. The prime $2$ splits in $K$, say $2\mathcal{O} = \mathfrak{p} \mathfrak{p}^\ast$, and there is a unique $\mathbb{Z}_2$-extension $K_\infty$ of $K$, which is unramified outside $\mathfrak{p}$. Let $H$ be the Hilbert class field of $K$, and write $H_\infty = HK_\infty$. Let $M(H_\infty)$ be the maximal abelian $2$-extension of $H_\infty$, which is unramified outside the primes above $\mathfrak{p}$, and put $X(H_\infty) = \mathrm{Gal}(M(H_\infty)/H_\infty)$. We prove that $X(H_\infty)$ is always a finitely generated $\mathbb{Z}_2$-module, by an elliptic analogue of Sinnott's cyclotomic argument. We then use this result to prove for the first time the weak $\mathfrak{p}$-adic Leopoldt conjecture for the compositum $J_\infty$ of $K_\infty$ with arbitrary quadratic extensions $J$ of $H$. We also prove some new cases of the finite generation of the Mordell-Weil group $E(J_\infty)$ modulo torsion of certain elliptic curves $E$ with complex multiplication by $\mathcal{O}$.

math.NT↗

arXiv2017

On the $p$-part of the Birch-Swinnerton-Dyer conjecture for elliptic curves with complex multiplication by the ring of integers of $\mathbb{Q}(\sqrt{-3})$

We study infinite families of quadratic and cubic twists of the elliptic curve $E = X_0(27)$. For the family of quadratic twists, we establish a lower bound for the $2$-adic valuation of the algebraic part of the value of the complex $L$-series at $s=1$, and, for the family of cubic twists, we establish a lower bound for the $3$-adic valuation of the algebraic part of the same $L$-value. We show that our lower bounds are precisely those predicted by the celebrated conjecture of Birch and Swinnerton-Dyer.

math.NT↗