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Hidenori Katsurada

Publications and source records attributed to Hidenori Katsurada.

At least 19 recordsLinked to original sources
arXiv2026

Harder's conjecture and Hermitian automorphic forms

Let $k\ge4$ and $j\ge2$ be integers with $j$ even. Harder's conjecture predicts a congruence between a primitive elliptic cusp form of weight $2k+j-2$ and a degree-two vector-valued Siegel Hecke cusp eigenform of weight ${\det}^{k}\mathrm{Sym}^{j}$. We prove the corresponding Harder-type congruence for the spinor $L$-polynomials under explicit arithmetic hypotheses on a congruence prime and an auxiliary imaginary quadratic field. Our approach uses Hermitian automorphic forms on the quasi-split unitary group $\mathrm{U}_{2,2}$. We construct Hermitian spin lifts of Siegel cusp forms and show that the Hermitian cusp form produced by a Hermitian Klingen-Eisenstein congruence lies in the image of this lift. The key step is to prove that the Galois representation attached to this Hermitian cusp form is conjugate invariant. For this, we combine the endoscopic classification for quasi-split unitary groups with Selmer-group vanishing results. This allows us to identify the Hermitian cusp form as the spin lift of a Siegel cusp eigenform and thereby obtain the Harder-type congruence. The argument applies uniformly to both even and odd $k$.

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arXiv2026

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.

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arXiv2023

Period of the Ikeda-Miyawaki lift

In this paper, first we give a weak version of Ikeda's conjecture on the period of the Ikeda-Miyawaki lift. Next, we confirm this conjecture rigorously in some cases.

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arXiv2022

Estimates for the Fourier coefficients of the Duke-Imamoglu-Ikeda lift

Let $k$ and $n$ be positive even integers. For a Hecke eigenform $h$ in the Kohnen plus subspace of weight $k-n/2+1/2$ for $\varGamma_0(4)$, let $I_n(h)$ be the Duke-Imamoglu-Ikeda lift of $h$ to the space of cusp forms of weight $k$ for $Sp_n(\mathbb Z)$. We then give an estimate of the Fourier coefficients of $I_n(h)$. It is better than the usual Hecke bound for the Fourier coefficients of a Siegel cusp form.

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arXiv2022

Rankin-Selberg convolution for the Duke-Imamoglu-Ikeda lift

For two Hecke eigenforms $h_1$ and $h_2$ in the Kohnen plus space of half-integral weight, let $I_n(h_1)$ and $I_n(h_2)$ be the Duke-Imamoglu-Ikeda lift of $h_1$ and $h_2$, respectively, which are Siegel cusp forms with respect to $Sp_n(\ZZ)$. Moreover, let $E_{n/2+1/2}$ be the Cohen Eisenstein series of weight $n/2+1/2$. We then express the Rankin-Selberg convolution $R(s,I_n(h_1),I_n(h_2))$ of $I_n(h_1)$ and $I_n(h_2)$ in terms of a certain Dirichlet series $D(s,h_1,h_2,E_{n/2+1/2})$, which is similar to the triple convolution product of $h_1, h_2$ and $E_{n/2+1/2}$. We apply our formula to mass equidistribution for the Duke-Imamoglu-Ikeda lift assuming the holomorphy of $D(s,h_1,h_1,E_{n/2+1/2})$.

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arXiv2022

Period of the Ikeda type lift for $E_{7,3}$

In our previous work, he second and the third named authors constructed the Ikeda type lift for the exceptional group $E_{7,3}$ from an elliptic modular cusp form. In this paper, we prove an explicit formula for the period or the Petersson norm of the Ikeda type lift in terms of the product of the special values of the symmetric square $L$-function of the elliptic modular form. There are similar works done by the first author with his collaborator, but new technical inputs are required and developed to overcome some difficulties coming from the hugeness of $E_{7,3}$.

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arXiv2022

A remark on $p$-adic Siegel Eisenstein series

A generalization of Serre's $p$-adic Eisenstein series in the case of Siegel modular forms is studied and a coincidence between a $p$-adic Siegel Eisenstein series and a genus theta series associated with a quaternary quadratic form is proved.

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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.

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arXiv2022

Boundedness of denominators of special values of the L-functions for modular forms

For a cuspidal Hecke eigenform $F$ for $Sp_n(Z)$ and a Dirichlet character $χ$ let $L(s,F,χ,St)$ be the standard $L$-function of $F$ twisted by $χ$. Boecherer showed the boundedness of denominators of the algebraic part of $L(m,F,χ,St)$ at a critical point $m$ when $χ$ varies. In this paper, we give a refined version of his result We also prove a similar result for the products of Hecke $L$ functions of primitive forms for $SL_2(Z)$.

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arXiv2020

An explicit formula for the extended Gross-Keating datum of a quadratic form

In this paper, we give a formula for the extended Gross-Keating datum of a quadratic form defined over a finite extension of $\mathbb{Z}_p$ (for $p>2$) or a finite unramified extension of $\mathbb{Z}_2$. As an application, we describe an explicit formula for the Siegel series for $\mathbb{Z}_p$. We also present the details of algorithms implemented in a Mathematica package to compute the extended Gross-Keating datum and the Siegel series.

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arXiv2017

On the Gross-Keating invariant of a quadratic form over a non-archimedean local field

Let $B$ be a half-integral symmetric matrix of size $n$ defined over $\mathbb{Q}_p$. The Gross-Keating invariant of $B$ was defined by Gross and Keating, and has important applications to arithmetic geometry. But the nature of the Gross-Keating invariant was not understood very well for $n\geq 4$. In this paper, we establish basic properties of the Gross-Keating invariant of a half-integral symmetric matrix of general size over an arbitrary non-archimedean local field of characteristic zero.

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arXiv2016

On the period of the Ikeda lift for U(m,m)

Let K be an imaginary quadratic field, and x the Dirichlet character corresponding to the extension K/Q. Let m=2n or 2n+1 with n a positive integer. Let f be a primitive form of weight 2k+1 and and nebentype x, or a primitive form of weight 2k for SL(2,Z) according as m=2n, or m=2n+1. For such an f let I_m(f) be the lift of f to the space of modular forms of weight 2k+2n for the Hermitian modular group of degree m constructed by Ikeda. We then express the period of I_m(f) in terms of special values of the adjoint L-functions of f. This poves the conjecture concerning the period of the Ikeda lift proposed by Ikeda.

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arXiv2015

Koecher-Maass series of the Ikeda lift for U(m,m)

Let K be an imaginary quadratic field with discriminant -D, and x the Dirichlet character corresponding to the extension K/Q. Let m=2n or 2n+1 with n a positive integer. Let f be a primitive form of weight 2k+1 and level D with Neben character x, or a primitive form of weight 2k for SL(2,Z) according as m=2n, or m=2n+1. For such an f let I(m,f) be the lift of f to the space of Hermitian modular forms of degree m constructed by Ikeda. We then give an explicit formula of the Koecher-Maass series L(s,I(m,f)) of I(m,f). This is a generalization of [Y. Mizuno, Manuscripta Math. 119(2006), 175-216].

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arXiv2013

Ikeda's conjecture on the period of the Duke-Imamoglu-Ikeda lift

Let k and n be positive even integers. For a cuspidal Hecke eigenform h in the Kohnen plus subspace of weight k-n/2+1/2 and level 4, let I(h) be the Duke-Imamoglu-Ikeda lift of h in the space of cusp forms of weight k for Sp(n,Z), and f the primitive form of weight 2k-n for SL(2,Z) corresponding to h under the Shimura correspondence. We then express the ratio /< h, h > of the period of I(h) to that of h in terms of special values of certain L-functions of f. This proves the conjecture proposed by Ikeda concerning the period of the Duke-Imamoglu-Ikeda lift.

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arXiv2013

Koecher-Maass series of a certain half-integral weight modular form related to the Duke-Imamoglu-Ikeda lift

Let k and n be positive even integers. For a cuspidal Hecke eigenform h in the Kohnen plus space of weight k-n/2+1/2, let f be the corresponding primitive form of weight 2k-n for SL(2,Z) under the Shimura correspondence, and I(h) the Duke-Imamoglu-Ikeda lift of h to the space of cusp forms of weight k of genus n. Moreover, let FJ(I(h),1) be the first Fourier-Jacobi coefficient of I(h) and s(FJ(I(h),1)) be the cusp form in the generalized Kohnen plus space of weight k-1/2 corresponding to FJ(I(h),1) under the Ibukiyama isomorphism. We then give an explicit formula for the Koecher-Maass series of s(FJ(I(h),1)) expressed in terms of the usual L-functions of h and f.

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