arXiv subjects

Kazuto Ota

Publications and source records attributed to Kazuto Ota.

6 recordsLinked to original sources

arXiv2026

Anticyclotomic Iwasawa theory of CM elliptic curves at ramified primes

We propose an integral framework for the anticyclotomic Iwasawa theory of CM elliptic curves $E$ at primes $p$ ramified in the CM field. The $\varepsilon$-constants of the geometric specialisations of the associated $p$-adic conjugate symplectic self-dual deformation equidistribute between $\pm 1$ within every layer of the anticyclotomic tower, and none of the geometric specialisations are trianguline at $p$. We define signed Selmer groups via the Lagrangian local conditions arising from the local sign decomposition established in the prequel \cite{BKNO}, and our central result is the formulation and proof of an integral Iwasawa main conjecture relating one of them to the $p$-adic $L$-function $\mathscr{L}_p(E)$ constructed there. We further show that $\mathscr{L}_p(E)$ interpolates the central Hecke $L$-values of the twists with $\varepsilon$-constant $+1$, including twists of arbitrary infinity type, and relate its values at twists with $\varepsilon$-constant $-1$ to the $p$-adic logarithm of certain Selmer elements. This provides the first Iwasawa main conjecture in terms of a $p$-adic $L$-function and Selmer groups for a $p$-adic deformation admitting no trianguline geometric specialisation. The proofs rest on our resolution of a Rubin-type conjecture for the underlying local deformation, together with a theory of plus/minus local points along the anticyclotomic tower, based on the Gaussian plus/minus cyclotomic polynomials rooted in Gauss' Disquisitiones Arithmeticae.

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arXiv2025

A local sign decomposition for symplectic self-dual Galois representations of rank two

We prove the existence of a new structure on the first Galois cohomology of generic families of symplectic self-dual $p$-adic representations of $G_{\mathbb{Q}_p}$ of rank two (a local sign decomposition): a functorial decomposition into free rank one Lagrangian submodules which encodes the $p$-adic variation of Bloch--Kato subgroups via completed epsilon constants, mirroring a symplectic structure. The local sign decomposition has diverse local as well as global arithmetic consequences. This includes compatibility of the Mazur--Rubin arithmetic local constant and completed epsilon constants, answering a question of Mazur and Rubin. The compatibility leads to new cases of the $p$-parity conjecture for Hilbert modular forms at supercuspidal primes $p$. We also formulate and prove an analogue of Rubin's conjecture over ramified quadratic extensions of $\mathbb{Q}_p$. Using it, we construct an integral $p$-adic $L$-function for anticyclotomic deformation of a CM elliptic curve at primes $p$ ramified in the CM field.

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arXiv2025

The $p$-adic valuation of local resolvents, generalized Gauss sums and anticyclotomic Hecke $L$-values of imaginary quadratic fields at inert primes

We prove an asymptotic formula for the $p$-adic valuation of Hecke $L$-values of an imaginary quadratic field at an inert prime $p$ along the anticyclotomic $\mathbb{Z}_p$-tower. The key is determination of the $p$-adic valuation of generalized Gauss sums defined using Coates-Wiles homomorphism, and of local resolvents in $\mathbb{Z}_p$-extensions. This answers a question of Rubin.

math.NT↗

arXiv2024

Hecke $L$-values, definite Shimura sets and Mod $\ell$ non-vanishing

Let $\lambda$ be a self-dual Hecke character over an imaginary quadratic field $K$ of infinity type $(1,0)$. Let $\ell$ and $p$ be primes which are coprime to $6N_{K/\mathbb{Q}}({\mathrm cond}(\lambda))$. We determine the $\ell$-adic valuation of Hecke $L$-values $L(1,\lambda\chi)/\Omega_K$ as $\chi$ varies over $p$-power order anticyclotomic characters over $K$. As an application, for $p$ inert in $K$, we prove the vanishing of the $\mu$-invariant of Rubin's $p$-adic $L$-function, leading to the first results on the $\mu$-invariant of imaginary quadratic fields at non-split primes. Our approach and results complement the work of Hida and Finis. The approach is rooted in the arithmetic of a CM form on a definite Shimura set.The application to Rubin's $p$-adic $L$-function also relies on the proof of his conjecture. Along the way, we present an automorphic view on Rubin's theory.

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arXiv2023

On the quantitative variation of congruence ideals and integral periods of modular forms

We prove the conjecture of Pollack and Weston on the quantitative analysis of the level lowering congruence à la Ribet for modular forms of higher weight. It was formulated and studied in the context of the integral Jacquet-Langlands correspondence and anticyclotomic Iwasawa theory for modular forms of weight two and square-free level for the first time. We use a completely different method based on the $R=\mathbb{T}$ theorem established by Diamond-Flach-Guo and Dimitrov and an explicit comparison of adjoint $L$-values. We briefly discuss arithmetic applications of our main result at the end.

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arXiv2015

Kato's Euler system and the Mazur-Tate refined conjecture of BSD type

Mazur and Tate proposed a conjecture which compares the Mordell-Weil rank of an elliptic curve over $\mathbb{Q}$ with the order of vanishing of Mazur-Tate elements, which are analogues of Stickelberger elements. Under some relatively mild assumptions, we prove this conjecture. Our strategy of the proof is to study divisibility of certain derivatives of Kato's Euler system.

math.NT↗