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Takuya Yamauchi

Publications and source records attributed to Takuya Yamauchi.

At least 37 records · Page 2Linked to original sources
arXiv2022

The weight reduction of mod $p$ Siegel modular forms for $GSp_4$

In this paper we investigate the (classical) weights of mod $p$ Siegel modular forms of degree 2 toward studying Serre's conjecture for $GSp_4$. We first construct various theta operators on the space of such forms a la Katz and define the theta cycles for the specific theta operators. Secondly we study the partial Hasse invariants on each Ekedahl-Oort strata and their local behaviors. This enable us to obtain a kind of weight reduction theorem for mod $p$ Siegel modular forms of degree 2 without increasing level.

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arXiv2022

Automorphy of mod 2 Galois representations associated to the quintic Dwork family and reciprocity of some quintic trinomials

In this paper, we determine mod $2$ Galois representations $\barρ_{ψ,2}:G_K:={\rm Gal}(\bar{K}/K)\longrightarrow {\rm GSp}_4(\mathbb{F}_2)$ associated to the mirror motives of rank 4 with pure weight 3 coming from the Dwork quintic family $$X^5_0+X^5_1+X^5_2+X^5_3+X^5_4-5ψX_0X_1X_2X_3X_4=0,\ ψ\in K$$ defined over a number field $K$ under the irreducibility condition of the quintic trinomial $f_ψ$ below. Applying this result, when $K=F$ is a totally real field, for some at most qaudratic totally real extension $M/F$, we prove that $\barρ_{ψ,2}|_{G_M}$ is associated to a Hilbert-Siegel modular Hecke eigen cusp form for ${\rm GSp}_4(\mathbb{A}_M)$ of parallel weight three. In the course of the proof, we observe that the image of such a mod $2$ representation is governed by reciprocity of the quintic trinomial $$f_ψ(x)=4x^5-5ψx^4+1,\ ψ\in K$$ whose decomposition field is generically of type 5-th symmetric group $S_5$. This enable us to use results on the modularity of 2-dimensional, totally odd Artin representations of ${\rm Gal}(\bar{F}/F)$ due to Shu Sasaki and several Langlands functorial lifts for Hilbert cusp forms. Then, it guarantees the existence of a desired Hilbert-Siegel modular cusp form of parallel weight three matching with the Hodge type of the compatible system in question.A twisted version is also discussed and it is related to general quintic trinomials.

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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

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

Equidistribution theorems for holomorphic Siegel cusp forms of general degree: the level aspect

We prove equidistribution theorems for a family of holomorphic Siegel cusp forms of general degree in the level aspect. Our main contribution is to estimate unipotent contributions for general degree in the geometric side of Arthur's invariant trace formula in terms of Shintani zeta functions. Several applications including the vertical Sato-Tate theorem and low-lying zeros for standard $L$-functions of holomorphic Siegel cusp forms are discussed. We also show that the "non-genuine forms" which come from non-trivial endoscopic contributions by Langlands functoriality classified by Arthur are negligible.

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arXiv2022

Isogeny graphs on superspecial abelian varieties: Eigenvalues and Connection to Bruhat-Tits buildings

We study for each fixed integer $g \ge 2$, for all primes $\ell$ and $p$ with $\ell \neq p$, finite regular directed graphs associated with the set of equivalence classes of $\ell$-marked principally polarized superspecial abelian varieties of dimension $g$ in characteristic $p$, and show that the adjacency matrices have real eigenvalues with spectral gaps independent of $p$. This implies a rapid mixing property of natural random walks on the family of isogeny graphs beyond the elliptic curve case and suggests a potential construction of the Charles-Goren-Lauter type cryptographic hash functions for abelian varieties. We give explicit lower bounds for the gaps in terms of the Kazhdan constant for the symplectic group when $g \ge 2$, and discuss optimal values in view of the theory of automorphic representations when $g=2$. As a by-product, we also show that the finite regular directed graphs constructed by Jordan-Zaytman also has the same property.

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arXiv2022

Serre weights for $GSp_4$ over totally real fields

We prove the existence of a potentially diagonalizable lift of a given automorphic mod $p$ Galois representation $\overlineρ:{\rm Gal}(\overline{F}/F)\longrightarrow {\rm GSp}_4(\overline{\mathbb{F}}_p)$ for any totally real field $F$ and any rational prime $p>2$ under the adequacy condition by using automorphic lifting techniques developed by Barnet-Lamb, Gee, Geraghty, and Taylor. As an application, when $p$ is split completely in $F$, we prove a variant of Serre's weight conjecture for $\overlineρ$. The formulation of our Serre conjecture is done by following Toby Gee's philosophy. Applying these results to the case when $F=\mathbb{Q}$ with a detailed study of potentially diagonalizable, crystalline lifts with some prescribed properties, we also define classical (naive) Serre's weights. This weight would be the minimal weight among possible classical weights in some sense which occur in candidates of holomorphic Siegel Hecke eigen cusp forms of degree 2 with levels prime to $p$. The main task is to construct a potentially ordinary automorphic lift for $\overlineρ$ by assuming only the adequacy condition. The main theorems in this paper also extend many results obtained by Barnet-Lamb, Gee and Geraghty for potentially ordinary lifts and Gee and Geraghty for companion forms.

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arXiv2022

A remark on conductor, depth and principal congruence subgroups

In this paper we study a relation between conductor, depth, and the level of principal congruence subgroups for irreducible admissible representations of ${\rm GL}_n(F)$ for a non-archimedean local field $F$ of characteristic zero. Some global and local applications are also discussed.

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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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arXiv2020

A reformulation of the Siegel series and intersection numbers

In this paper, we will explain a conceptual reformulation and inductive formula of the Siegel series. Using this, we will explain that both sides of the local intersection multiplicities of [GK93] and the Siegel series have the same inherent structures, beyond matching values. As an application, we will prove a new identity between the intersection number of two modular correspondences over Fp and the sum of the Fourier coefficients of the Siegel-Eisenstein series for Sp_4 of weight 2, which is independent of p (> 2). In addition, we will explain a description of the local intersection multiplicities of the special cycles over F_p on the supersingular locus of the `special fiber' of the Shimura varieties for GSpin(n; 2), n<=3 in terms of the Siegel series directly.

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arXiv2018

Higher level cusp forms on the exceptional group of type $E_{7}$

By using new techniques with the degenerate Whittaker functions found by Ikeda-Yamana, we construct higher level cusp form on $E_{7,3}$, called Ikeda type lift, from any Hecke cusp form whose corresponding automorphic representation has no supercuspidal local components. This generalizes the previous results on level one forms. But there are new phenomena in higher levels; first, we can handle non-trivial central characters. Second, the lift depends only on the restriction of the Hecke cusp form to $SL_2$. Hence any twist of the cusp form gives rise to the same lift. However for square free levels with the trivial central character, there is no such ambiguity.

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arXiv2018

Non-vanishing of Miyawaki type lift

Miyawaki type lifts are kinds of Langlands functorial lifts and a special case was first conjectured by Miyawaki and proved by Ikeda for Siegel cusp forms. Since then, such a lift for Hermitian modular forms was constructed by Atobe and Kojima , and for half-integral weight Siegel cusp forms by Hayashida, and we constructed Miyawaki type lift for ${\rm GSpin}(2,10)$. Recently Ikeda and Yamana generalized Ikeda type construction and accordingly did Miyawaki type lift for Hilbert cusp forms in a remarkable way. In all these works, a construction of Miyawaki type lift takes two steps as follows: First, construct Ikeda type lift on a bigger group from an elliptic cusp form, and then define a certain integral on a block diagonal element which is an analogue of pull-back formula studied by Garrett for Siegel Eisenstein series. If the integral is non-vanishing, we show that it is a Hecke eigen cusp form, and it is the Miyawaki type lift. The question of non-vanishing of the integral was left open. In this paper, we show the non-vanishing for certain special cases.

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arXiv2018

On the class numbers of the fields of the $p^n$-torsion points of elliptic curves over $\mathbb{Q}$

Let $E$ be an elliptic curve over $\mathbb{Q}$ which has multiplicative reduction at a fixed prime $p$. For each positive integer $n$ we put $K_n:=\mathbb{Q}(E[p^n])$. The aim of this paper is to extend the author's previous our results concerning with the order of the $p$-Sylow group of the ideal class group of $K_n$ to more general setting. We also modify the previous lower bound of the order and describe the new lower bound in terms of the Mordell-Weil rank of $E(\mathbb{Q})$ and the ramification related to $E$.

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arXiv2018

Equidistribution theorems for holomorphic Siegel modular forms for $GSp_4$; Hecke fields and $n$-level density

This paper is a continuation of the author's previous wotk. We supplement four results on a family of holomorphic Siegel cusp forms for $GSp_4/\mathbb{Q}$. First, we improve the result on Hecke fields. Namely, we prove that the degree of Hecke fields is unbounded on the subspace of genuine forms which do not come from functorial lift of smaller subgroups of $GSp_4$ under a conjecture in local-global compatibility and Arthur's classification for $GSp_4$. Second, we prove simultaneous vertical Sato-Tate theorem. Namely, we prove simultaneous equidistribution of Hecke eigenvalues at finitely many primes. Third, we compute the $n$-level density of degree 4 spinor $L$-functions, and thus we can distinguish the symmetry type depending on the root numbers. This is conditional on certain conjecture on root numbers. Fourth, we consider equidistribution of paramodular forms. In this case, we can prove a result on root numbers. Main tools are the equidistribution theorem in our previous work and Shin-Templier's work.

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arXiv2018

Derivatives of Eisenstein series of weight 2 and intersections of modular correspondences

We give a formula for certain values and derivatives of Siegel series and use them to compute Fourier coefficients of derivatives of the Siegel Eisenstein series of weight g/2 and genus g. When g=4, the Fourier coefficient is approximated by a certain Fourier coefficient of the central derivative of the Siegel Eisenstein series of weight 2 and genus 3, which is related to the intersection of 3 arithmetic modular correspondences. Applications include a relation between weighted averages of representation numbers of symmetric matrices.

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arXiv2016

On Prym varieties for the coverings of some singular plane curves

Let $k$ be a field of characteristic zero containing a primitive $n$-th root of unity. Let $C^0_n$ be a singular plane curve of degree $n$ over $k$ admitting an order $n$ automorphism, $n$ nodes as the singularities, and $C_n$ be its normalization. In this paper we study the factors of Prym variety $\mbox{Prym}(\widetilde{C}_n/C_n)$ associated to the double cover $\widetilde{C}_n$ of $C_n$ exactly ramified at the points obtained by the blow-up of the singularities. We provide explicit models of some algebraic curves related to the construction of $\mbox{Prym}(\widetilde{C}_n/C_n)$ as a Prym variety and determine the interesting simple factors other than elliptic curves or hyperelliptic curves with small genus which come up in $J_n$ so that the endomorphism rings contains the totally real field $\mathbb{Q}(ζ_n+ζ^{-1}_n)$.

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arXiv2016

Artin representations for $GL_n$

Let $π$ be a cuspidal automorphic representation of $GL_n(\mathbb{A}_\mathbb{Q})$ which satisfies certain reasonable assumptions such as integrality of Hecke polynomials, the existence of mod $\ell$ Galois representations attached to $π$. Under Langlands functoriality of exterior $m$-th power $\wedge^m(π)$, $m=2,...,[\frac n2]$, we will construct a unique Artin representation associated to $π$. As a corollary, we obtain that such a cuspidal representation of $GL_n(\mathbb{A}_\mathbb{Q})$ satisfies the Ramanujan conjecture. We also revisit our previous work on Artin representations associated to non-holomorphic Siegel cusp forms of weight (2,1), and show that we can associate non-holomorphic Siegel modular forms of weight $(2,1)$ to Maass forms for $GL_2/\mathbb{Q}$ and cuspidal representations of $GL_2$ over imaginary quadratic fields.

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