Searcharxiv⌕ Search

arXiv subjects

Tadashi Ochiai

Publications and source records attributed to Tadashi Ochiai.

14 recordsLinked to original sources
arXiv2026

Multi-variable admissible distributions

The theory of admissible distributions over a weight-space of one-variable was studied by Amice--Vélu and played important roles in the cyclotomic Iwasawa theory of non-ordinary p-adic Galois representations. In this article, we discuss the multi-variable generalization of the theory of admissible distributions over a weight-space of several variables. As an application, we construct a two-variable non-ordinary p-adic L-function which interpolates the special values of Rankin-Selberg L-functions.

math.NT↗
arXiv2025

On the $p$-adic $L$-function and Iwasawa Main Conjecture for an Artin motive over a CM field

For an algebraic Hecke character defined on a CM field $F$ of degree $2d$, Katz constructed a $p$-adic $L$-function of $d+1+δ_{F,p}$ variables in his innovative paper published in 1978, where $δ_{F,p}$ denotes the Leopoldt defect for $F$ and $p$. In the present article, we generalise the result of Katz under several technical conditions (containing the absolute unramifiedness of $F$ at $p$), and construct a $p$-adic Artin $L$-function of $d+1+δ_{F,p}$ variables, which interpolates critical values of the Artin $L$-function associated to a $p$-unramified Artin representation of the absolute Galois group $G_F$. Our construction is an analogue over a CM field of Greenberg's construction over a totally real field, but there appear new difficulties which do not matter in Greenberg's case.

math.NT↗
arXiv2025

Kato explicit reciprocity law for Siegel modular forms of weight $(3, 3)$

We extend Kato explicit reciprocity law, in the version written by Scholl, for a modular curve to a product of two modular curves. By embedding the product of two modular curves in the Siegel threefold, we deduce an explicit reciprocity law for the unique critical twist of the $p$-adic Galois representation attached to cuspidal Siegel modular forms of weight $(3,3)$.

math.NT↗
arXiv2023

A formal model of Coleman families and applications to Iwasawa invariants

For a given Coleman family of modular forms, we construct a formal modeland prove the existence of a family of Galois representations associated to the Colemanfamily. As an application, we study the variations of Iwasawa $λ$- and $μ$-invariants of dualfine (strict) Selmer groups over the cyclotomic Zp-extension of Q in Coleman families ofmodular forms. This generalizes an earlier work of Jha and Sujatha for Hida families.

math.NT↗
arXiv2020

Coleman Map in Coleman Families

In this paper, we aimed at constructing a two-variable Coleman map for a given $p$-adic family of eigen cuspforms with a fixed non-zero slope (Coleman family). A Coleman map is a machinary which transforms a hypothetical $p$-adic family of zeta elements to a $p$-adic $L$-function. The result would be a non-ordinary generalization of a two-variable Coleman map for a given Hida deformation obtained by the second-named author.

math.NT↗
arXiv2020

Control theorem and functional equation of Selmer groups over $p$-adic Lie extensions

Let us consider a $p$-adic Lie extension of a number field $K$ which fits into the setting of non-commutative Iwasawa theory formulated by Coates-Fukaya-Kato-Sujatha-Venjakob. For the first main result, we will prove the control theorem of Selmer group associated to a motive, which generalizes previous results by the second author and Greenberg. For the second main result, we prove the functional equation of the dual Selmer groups, which generalizes previous results by Greenberg, Perrin-Riou and Zabradi. Note that our proof of the functional equation is different from the proof of Zabradi even in the case where the Selmer group is associated to an elliptic curve. We also discuss the functional equation for the analytic $p$-adic $L$-functions and check the compatibility with the functional equation of the dual Selmer groups.

math.NT↗
arXiv2019

Endoscopic congruences modulo adjoint $L$-values for $\mathrm{GSp}(4)$

We establish the existence of congruences between a fixed endoscopic cuspidal automorphic representation $Π$ of $\mathrm{GSp}(4)$ of square-free conductor and stable cuspidal automorphic representations of the same level and weight modulo certain prime factors of the value at $1$ of the adjoint $L$-function of $Π$ normalized by a suitable period.

math.NT↗
arXiv2019

Iwasawa Main Conjecture for $p$-adic families of elliptic modular cuspforms

In this article, we discuss Iwasawa Main Conjecture for $p$-adic families of elliptic modular cuspforms. After the overview on the situation of the ordinary case of Hida family, we will introduce a Coleman map for Coleman family for the non-ordinary case (Coleman family) which was obtained as a joint work with Filippo Nuccio [NO16] and we give some results on Iwasawa Theory for Coleman families as applications of . First, we give a construction of a two-variable $p$-adic $L$-function for a Coleman family thanks to the ingredients given in [NO16] (Theorem 5.1). Combining this result and Coleman map obtained in [NO16], we also construct Beilinson-Kato Euler systems over a Coleman family (Thorem 5.3). Finally we formulate Iwasawa Main conjecture for a Coleman family and prove the half of Iwasawa Main Conjecture (Theorem 5.6).

math.NT↗
arXiv2019

Main conjectures for higher rank nearly ordinary families -- I

In this article, we present the first half of our project on the Iwasawa theory of higher rank Galois deformations over deformations rings of arbitrary dimension. We develop a theory of Coleman maps for a very general class of coefficient rings, devise a dimension reduction procedure for locally restricted Euler systems and finally, put these into use in order to prove a divisibility in a 3-variable main conjecture for nearly ordinary families of Rankin-Selberg convolutions.

math.NT↗
arXiv2017

Specialization Method in Krull Dimension two and Euler System Theory over Normal Deformation Rings

The aim of this article is to establish the specialization method on characteristic ideals for finitely generated torsion modules over a complete local normal domain R that is module-finite over $O[[x_1, ..., x_d]]$, where $O$ is the ring of integers of a finite extension of the field of p-adic integers $Q_p$. The specialization method is a technique that recovers the information on the characteristic ideal $char_R(M)$ from $char_{R/I}(M/IM)$, where I varies in a certain family of nonzero principal ideals of R. As applications, we prove Euler system bound over Cohen-Macaulay normal domains by combining the main results in an earlier article of the first named author and then we prove one of divisibilities of the Iwasawa main conjecture for two-variable Hida deformations generalizing the main theorem obtained in an article of the first named author.

math.NT↗
arXiv2015

On twists of modules over non-commutative Iwasawa algebras

It is well known that, for any finitely generated torsion module M over the Iwasawa algebra Z_p [[Γ ]], where Γ is isomorphic to Z_p, there exists a continuous p-adic character ρ of Γ such that, for every open subgroup U of Γ, the group of U-coinvariants M(ρ)_U is finite; here M( ρ) denotes the twist of M by ρ. This twisting lemma was already applied to study various arithmetic properties of Selmer groups and Galois cohomologies over a cyclotomic tower by Greenberg and Perrin-Riou. We prove a non commutative generalization of this twisting lemma replacing torsion modules over Z_p [[ Γ ]] by certain torsion modules over Z_p [[G]] with more general p-adic Lie group G.

math.NT↗
arXiv2015

Bertini theorem for normality on local rings in mixed characteristic (applications to characteristic ideals)

In this article, we prove a strong version of local Bertini theorem for normality on local rings in mixed characteristic. The main result asserts that a generic hyperplane section of a normal, Cohen-Macaulay, and complete local domain of dimension at least 3 is normal. Applications include the study of characteristic ideals attached to torsion modules over Noetherian normal domains, which is fundamental in the study of Euler system theory over normal domains and Iwasawa main conjectures.

math.NT↗
arXiv2007

On the Selmer groups of abelian varieties over function fields of characteristic p>0

In this paper, we study a (p-adic) geometric analogue for abelian varieties over a function field of characteristic p of the cyclotomic Iwasawa theory and the non-commutative Iwasawa theory for abelian varieties over a number field initiated by Mazur and Coates respectively. We will prove some analogue of the principal results obtained in the case over a number field and we study new phenomena which did not happen in the case of number field case. We propose also a conjecture which might be considered as a counterpart of the principal conjecture in the case over a number field. \par This is a preprint which is distributed since 2005 which is still in the process of submision. Following a recent modification of some technical mistakes in the previous version of the paper as well as an amelioration of the presentation of the paper, we decide wider distribution via the archive.

math.NT↗