Searcharxiv⌕ Search

arXiv subjects

Hiraku Atobe

Publications and source records attributed to Hiraku Atobe.

At least 19 recordsLinked to original sources
arXiv2026

Endoscopic transfer and the wavefront upper bound conjecture

We verify the local analogue of Jiang's conjecture for the upper bound of the geometric wavefront sets of Arthur type representations of split classical $p$-adic groups with $p\gg 0$, under a certain condition. As a consequence, we also obtain the upper bound conjecture of Kim and the second author, and Hazeltine--Liu--Lo--Shahidi, under the same assumptions. The proof uses Waldspurger's work on the endoscopic transfer supplemented by results of Konno and Varma, as well as the wavefront set computations in the unipotent case by Mason-Brown--Okada and the second author.

math.RT↗
arXiv2025

Unitary dual of $p$-adic split $\mathrm{SO}_{2n+1}$ and $\mathrm{Sp}_{2n}$: The good parity case (and slightly beyond)

Let $F$ be a $p$-adic field, and let $G$ be either the split special orthogonal group $\mathrm{SO}_{2n+1}(F)$ or the symplectic group $\mathrm{Sp}_{2n}(F)$, with $n \geq 0$. We prove that a smooth irreducible representation of good parity of $G$ is unitary if and only if it is of Arthur type. Combined with the algorithms of the first author or Hazeltine-Liu-Lo for detecting Arthur type representations, our result leads to an explicit algorithm for checking the unitarity of any given irreducible representation of good parity. Finally, we determine the set of unitary representations that may appear as local components of the discrete automorphic spectrum.

math.RT↗
arXiv2024

Local Intertwining Relations and Co-tempered $A$-packets of Classical Groups

The local intertwining relation is an identity that gives precise information about the action of normalized intertwining operators on parabolically induced representations. We prove several instances of the local intertwining relation for quasi-split classical groups and the twisted general linear group, as they are required in the inductive proof of the endoscopic classification for quasi-split classical groups due to Arthur and Mok. In addition, we construct the co-tempered local $A$-packets by Aubert duality and verify their key properties by purely local means, which provide the seed cases needed as an input to the inductive proof. Together with further technical results that we establish, this makes the endoscopic classification conditional only on the validity of the twisted weighted fundamental lemma.

math.NT↗
arXiv2023

Local newforms for generic representations of unramified even unitary groups I: Even conductor case

In this paper, we define compact open subgroups of quasi-split even unitary groups for each even non-negative integers, and establish the theory of local newforms for irreducible tempered generic representations with a certain condition on the central characters. To do this, we use the local Gan-Gross-Prasad conjecture, the local Rankin-Selberg integrals, and the local theta correspondence.

math.RT↗
arXiv2023

Harder's conjecture II

Let $f$ be a primitive form of weight $2k+j-2$ for $SL_2(Z)$, and let $P$ be a prime ideal of the Hecke field of $f$. We denote by $Sp_m(Z)$ the Siegel modular group of degree $m$. Suppose that $k$ is congruent to $0$ modulo $4$, $j$ is congruent to $0$ modulo $4$, and that $P$ divides the algebraic part of $L(k+j,f)$. Put ${\bf k}=(k+j/2,k+j/2,j/2+4,j/2+4)$. Then under certain easily checkable conditions, we prove that there exists a Hecke eigenform $F$ in the space of modular forms of weight $(k+j,k)$ for $Sp_2(Z)$ such that $[I_2(f)]^{\bf k}$ is congruent to $A^{(I)}_4(F)$ modulo $P$. Here, $[I_2(f)]^{\bf k}$ is the Klingen-Eisenstein lift of the Saito-Kurokawa lift $I_2(f)$ of $f$ to the space of modular forms of weight ${\bf k}$ for $Sp_4(Z)$, and $A^{(I)}_4(F)$ is a certain lift of $F$ to the space of cusp forms of weight ${\bf k}$ for $Sp_4(Z)$. As an application, we prove Harder's conjecture on the congruence between the Hecke eigenvalues of $F$ and some quantities related to the Hecke eigenvalues of $f$. This version gives proofs of Lemmas 7.2 and 7.3 and Corollaries 7.4 and 7.5 in the paper arXiv:2306.07582v2.

math.NT↗
arXiv2022

An analogue of ladder representations for classical groups

In this paper, we introduce a notion of ladder representations for split odd special orthogonal groups and symplectic groups over a non-archimedean local field of characteristic zero. This is a natural class in the admissible dual which contains both strongly positive discrete series representations and irreducible representations with irreducible A-parameters. We compute Jacquet modules and the Aubert duals of ladder representations, and we establish a formula to describing ladder representations in terms of linear combinations of standard modules.

math.RT↗
arXiv2022

Local newforms for the general linear groups over a non-archimedean local field

In [12], Jacquet--Piatetskii-Shapiro--Shalika defined a family of compact open subgroups of $p$-adic general linear groups indexed by non-negative integers, and established the theory of local newforms for irreducible generic representations. In this paper, we extend their results to all irreducible representations. To do this, we define a new family of compact open subgroups indexed by certain tuples of non-negative integers. For the proof, we introduce the Rankin--Selberg integrals for Speh representations.

math.NT↗
arXiv2022

Harder's conjecture I

Let $f$ be a primitive form with respect to $SL_2(Z)$. Then we propose a conjecture on the congruence between the Klingen-Eisenstein lift of the Duke-Imamoglu-Ikeda lift of $f$ and a certain lift of a vector valued Hecke eigenform with respect to $Sp_2(Z)$. This conjecture implies Harder's conjecture. We prove the above conjecture in some cases.

math.NT↗
arXiv2022

Construction of local A-packets

In this paper, we reformulate Moeglin's explicit construction of local A-packets of split odd special orthogonal groups and symplectic groups. By this reformulation together with results of the previous paper with Minguez, we can compute the A-packets explicitly. Also, we give a non-vanishing criterion of our parametrization, and an algorithm to compute certain derivatives. Finally, we prove a formula for the Aubert duals of irreducible representations of Arthur type.

math.RT↗
arXiv2022

On the socles of certain parabolically induced representations of $p$-adic classical groups

In this paper, we consider representations of $p$-adic classical groups parabolically induced from the products of shifted Speh representations and unitary representations of Arthur type of good parity. We describe how to compute the socles (the maximal semisimple subrepresentations) of these representations. As a consequence, we can determine whether these representations are reducible or not. In particular, our results produce many unitary representations, which are called complementary series.

math.RT↗
arXiv2022

The set of local A-packets containing a given representation

In this paper, we give an algorithm to determine all local A-packets containing a given irreducible representation of a p-adic classical group. Especially, we can determine whether a given irreducible representation is of Arthur type or not.

math.RT↗
arXiv2020

The explicit Zelevinsky-Aubert duality

In this paper, we give an explicit computable algorithm for the Zelevinsky-Aubert dual of irreducible representations of $p$-adic symplectic and odd special orthogonal groups. To do this, we establish explicit formulas for certain derivatives and socles. We also give a combinatorial criterion for the irreducibility of certain parabolically induced representations.

math.RT↗
arXiv2020

On an algorithm to compute derivatives

In this paper, we complete Jantzen's algorithm to compute the highest derivatives of irreducible representations of $p$-adic odd special orthogonal groups or symplectic groups. As an application, we give some examples of the Langlands data of the Aubert duals of irreducible representations, which are in the integral reducibility case.

math.RT↗
arXiv2018

Jacquet modules and local Langlands correspondence

In this paper, we explicitly compute the semisimplifications of all Jacquet modules of irreducible representations with generic L-parameters of p-adic split odd special orthogonal groups or symplectic groups. Our computation represents them in terms of linear combinations of standard modules with rational coefficients. The main ingredient of this computation is to apply Moeglin's explicit construction of local A-packets to tempered L-packets.

math.RT↗
arXiv2018

Applications of Arthur's multiplicity formula to Siegel modular forms

We give two applications of Arthur's multiplicity formula to Siegel modular forms. The one is a lifting theorem for vector valued Siegel modular forms, which contains Miyawaki's conjectures and Ibukiyama's conjectures. The other is the strong multiplicity one theorem for Siegel modular forms of scalar weights and level one.

math.NT↗
arXiv2018

A theory of Miyawaki liftings: The Hilbert-Siegel case

The Miyawaki liftings are defined by the pullbacks of Ikeda liftings. Recently, Ikeda and Yamana extended the theory of Ikeda liftings. In this paper, using their results, we establish a theory of Miyawaki liftings, both locally and globally. In the local theory, we describe the Miyawaki liftings for almost tempered unitary representations explicitly. In the global theory, we discuss the non-vanishing of the Miyawaki liftings using seesaw identities and the global Gan-Gross-Prasad conjecture. As an application of local Miyawaki liftings, we prove a new case of the local Gan-Gross-Prasad conjecture.

math.NT↗