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Naoto Dainobu

Publications and source records attributed to Naoto Dainobu.

7 recordsLinked to original sources
arXiv2026

On $p$-adic $L$-functions of elliptic curves and the ideal class groups of the division fields

Let $E$ be an elliptic curve defined over $\mathbb{Q}$ and $F$ be $\mathbb{Q}$ or an imaginary quadratic field with certain conditions. In this article, we study the ideal class group $\mathrm{Cl}(F_E)$ of the $p$-division field $F_E:=F(E[p])$ of $E$ over $F$ for an odd prime number $p$. More precisely, we investigate the non-vanishing of the $E[p]$-component in the semi-simplification of $\mathrm{Cl}(F_E)/p\mathrm{Cl}(F_E)$ as an $\mathbb{F}_p[\mathrm{Gal}(F_E/F)]$-module when $E[p]$ is an irreducible $\mathrm{Gal}(F_E/F)$-module. When the analytic rank of $E$ over $F$ is $1$, we establish a new relationship between the non-vanishing of the $E[p]$-component and the $p$-divisibility of a certain $p$-adic analytic quantity associated with $E$. The quantity is defined by the leading coefficient of the cyclotomic $p$-adic $L$-function of $E$ when $F=\mathbb{Q}$ and by that of Bertolini--Darmon--Prasanna's anticyclotomic $p$-adic $L$-function of $E$ when $F$ is the imaginary quadratic field.

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arXiv2025

On the local equivariant Tamagawa number conjecture for Tate motives

The local equivariant Tamagawa number conjecture (local ETNC) for a motive predicts a precise relationship between the local arithmetic complex and the root numbers which appear in the (conjectural) functional equations of the $L$-functions. In this paper, we prove the local ETNC for the Tate motives under a certain unramified condition at $p$. Our result gives a generalization of the previous works by Burns--Flach and Burns--Sano. Our strategy basically follows those works and builds upon the classical theory of Coleman maps and its generalization by Perrin-Riou.

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arXiv2024

Ideal class groups of division fields of elliptic curves and everywhere unramified rational points

Let $E$ be an elliptic curve over $\mathbb{Q}$, $p$ an odd prime number and $n$ a positive integer. In this article, we investigate the ideal class group $\mathrm{Cl}(\mathbb{Q}(E[p^n]))$ of the $p^n$-division field $\mathbb{Q}(E[p^n])$ of $E$. We introduce a certain subgroup $E(\mathbb{Q})_{\mathrm{ur},p^n}$ of $E(\mathbb{Q})$ and study the $p$-adic valuation of the class number $\#\mathrm{Cl}(\mathbb{Q}(E[p^n]))$. In addition, when $n = 1$, we further study $\mathrm{Cl}(\mathbb{Q}(E[p]))$ as a $\mathrm{Gal}(\mathbb{Q}(E[p])/\mathbb{Q})$- module. More precisely, we study the semi-simplification $(\mathrm{Cl}(\mathbb{Q}(E[p]))\otimes \mathbb{Z}_p)^{\mathrm{ss}}$ of $\mathrm{Cl}(\mathbb{Q}(E[p]))\otimes \mathbb{Z}_p$ as a $\mathbb{Z}_p[\mathrm{Gal}(\mathbb{Q}(E[p])/\mathbb{Q})]$-module. We obtain a lower bound of the multiplicity of the $E[p]$-component in the semi-simplification when $E[p]$ is an irreducible $\mathrm{Gal}(\mathbb{Q}(E[p])/\mathbb{Q})$-module.

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arXiv2023

Ideal class groups of number fields associated to modular Galois representations

Let $p$ be an odd prime number and $f$ a modular form. We consider the $\mathbb{F}_p$-valued Galois representation $\barρ_f$ attached to $f$ and its twist $\barρ_{f, D}$ by the quadratic character $χ_D$ corresponding to a quadratic discriminant $D$. We define $K_{f, D}$ to be the field corresponding to the kernel of $\barρ_{f, D}$. In this article, we investigate the ideal class group $\mathrm{Cl}(K_{f, D})$ of the number field $K_{f, D}$ as a $\mathrm{Gal}(K_{f, D}/\mathbb{Q})$-module. We give a condition which implies the existence of a $\mathrm{Gal}(K_{f, D}/\mathbb{Q})$-equivariant surjective homomorphism from $\mathrm{Cl}(K_{f, D})\otimes \mathbb{F}_p$ to the representation space $M_{f, D}$ of $\barρ_{f, D}$, using Bloch and Kato's Selmer group of $\barρ_{f, D}$. We also give some numerical examples where we have such surjections by calculating the central value of the $L$-function of $f$ twisted by $χ_D$ under Bloch and Kato's conjecture. Our main result in this paper is a partial generalization of the previous result of Prasad and Shekhar on elliptic curves to higher weight modular forms.

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arXiv2022

Elliptic analogue of irregular prime numbers for the $p^{n}$-division fields of the curves $y^{2} = x^{3}-(s^{4}+t^{2})x$

A prime number $p$ is said to be irregular if it divides the class number of the $p$-th cyclotomic field $\mathbb{Q}(ζ_{p}) = \mathbb{Q}(\mathbb{G}_m[p])$. In this paper, we study its elliptic analogue for the division fields of an elliptic curve. More precisely, for a prime number $p \geq 5$ and a positive integer $n$, we study the $p$-divisibility of the class number of the $p^{n}$-division field $\mathbb{Q}(E[p^{n}])$ of an elliptic curve $E$ of the form $y^{2} = x^{3}-(s^{4}+t^{2})x$. In particular, we construct a certain infinite subfamily consisting of curves with novel properties that they are of Mordell-Weil rank 1 and the class numbers of their $p^{n}$-division fields are divisible by $p^{2n}$. Moreover, we can prove that these division fields are not isomorphic to each other. In our construction, we use recent results obtained by the first author.

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arXiv2022

Ideal class groups of number fields and Bloch-Kato's Tate-Shafarevich groups for symmetric powers of elliptic curves

For an elliptic curve $E$ over $\mathbb{Q}$, putting $K=\mathbb{Q}(E[p])$ which is the $p$-th division field of $E$ for an odd prime $p$, we study the ideal class group $\mathrm{Cl}_K$ of $K$ as a $\mathrm{Gal}(K/\mathbb{Q})$-module. More precisely, for any $j$ with $1\leqslant j \leqslant p-2$, we give a condition that $\mathrm{Cl}_K\otimes \mathbb{F}_p$ has the symmetric power $\mathrm{Sym}^j E[p]$ of $E[p]$ as its quotient $\mathrm{Gal}(K/\mathbb{Q})$-module, in terms of Bloch-Kato's Tate-Shafarevich group of $\mathrm{Sym}^j V_p E$. Here $V_p E$ denotes the rational $p$-adic Tate module of $E$. This is a partial generalization of a result of Prasad and Shekhar for the case $j=1$.

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