arXiv subjects

Takuya Yamauchi

Publications and source records attributed to Takuya Yamauchi.

47 records · Page 3Linked to original sources

arXiv2016

An equidistribution theorem for holomorphic Siegel modular forms for $GSp_4$

We prove an equidistribution theorem for a family of holomorphic Siegel cusp forms for $GSp_4/\mathbb{Q}$ in various aspects. A main tool is Arthur's invariant trace formula. While Shin and Shin-Templier used Euler-Poincaré functions at infinity in the formula, we use a pseudo-coefficient of a holomorphic discrete series to extract holomorphic Siegel cusp forms. Then the non-semisimple contributions arise from the geometric side, and this provides new second main terms $A, B_1$ in the main theorem which have not been studied and a mysterious second term $B_2$ also appears in the second main term coming from the semisimple elements. Furthermore our explicit study enables us to treat more general aspects in the weight. We also give several applications including the vertical Sato-Tate theorem, the unboundedness of Hecke fields and low-lying zeros for degree 4 spinor $L$-functions and degree 5 standard $L$-functions of holomorphic Siegel cusp forms.

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arXiv2015

Miyawaki type lift for $GSpin(2,10)$

Let $\frak T_2$ (resp. $\mathfrak{T}$) be the Hermitian symmetric domain of $Spin(2,10)$ (resp. $E_{7,3}$). In the previous work, we constructed holomorphic cusp forms on $\mathfrak{T}$ from elliptic cusp forms with respect to $SL_2(\mathbb{Z})$. By using such cusp forms we construct holomorphic cusp forms on $\mathfrak{T}_2$ which are similar to Miyawaki lift in symplectic groups established by T. Ikeda.

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arXiv2015

Cusp forms for exceptional group of type $E_{7}$

Let $\bf{G}$ be the connected reductive group of type $E_{7,3}$ over $\mathbb{Q}$ and $\mathfrak{T}$ be the corresponding symmetric domain in $\mathbb{C}^{27}$. Let $Γ=\bf{G}(\mathbb{Z})$ be the arithmetic subgroup defined by Baily. In this paper, for any positive integer $k\ge 10$, we will construct a (non-zero) holomorphic cusp form on $\mathfrak{T}$ of weight $2k$ with respect to $Γ$ from a Hecke cusp form in $S_{2k-8}(SL_2(\mathbb{Z}))$. This lift is an analogue of Ikeda's construction.

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arXiv2015

Ikeda type construction of cusp forms

This is a survey of results on the construction of holomorphic cusp forms on tube domains originally initiated by Ikeda. Besides a survey it includes conjectures and possible applications of our work.

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arXiv2015

On intermediate Jacobians of cubic threefolds admitting an automorphism of order five

Let $k$ be a field of characteristic zero containing a primitive fifth root of unity. Let $X/k$ be a smooth cubic threefold with an automorphism of order five, then we observe that over a finite extension of the field actually the dihedral group $D_5$ is a subgroup of ${\rm Aut}(X)$. We find that the intermediate Jacobian $J(X)$ of $X$ is isogenous to the product of an elliptic curve $E$ and the self-product of an abelian surface $B$ with real multiplication by $\mathbb{Q}(\sqrt{5})$. We give explicit models of some algebraic curves related to the construction of $J(X)$ as a Prym variety. This includes a two parameter family of curves of genus 2 whose Jacobians are isogenous to the abelian surfaces mentioned as above.

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arXiv2015

Artin representations for GSp_4 attached to real analytic Siegel cusp forms of weight (2,1)

Let $F$ be a vector-valued real analytic Siegel cusp eigenform of weight $(2,1)$ with the eigenvalues $-\frac 5{12}$ and 0 for the two generators of the center of the algebra consisting of all $Sp_4(\R)$-invariant differential operators on the Siegel upper half plane of degree 2. Under the assumptions (1) the validity of the transfer of automorphic representations of $GSp_4$ to $GL_4$; (2) the existence of mod $\ell$ Galois representation attached to $F$ and its lift to characteristic zero; (3) rationality of the space consisting of any such $F$; and (4) the integrality of Hecke polynomials of $F$, we construct a unique Artin representation of type $GSp_4$ associated to $F$. Several examples which satisfy these assumptions are given by using various transfers and automorphic descent.

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arXiv2014

On some Siegel threefold related to the tangent cone of the Fermat quartic surface

Let $Z$ be the quotient of the Siegel modular threefold $\mathcal{A}^{\rm sa}(2,4,8)$ which has been studied by van Geemen and Nygaard. They gave an implication that some 6-tuple $F_Z$ of theta constants which is in turn known to be a Klingen type Eisenstein series of weight 3 should be related to a holomorphic differential $(2,0)$-form on $Z$. The variety $Z$ is birationally equivalent to the tangent cone of Fermat quartic surface in the title. In this paper we first compute the L-function of two smooth resolutions of $Z$. One of these, denoted by $W$, is a kind of Igusa compactification such that the boundary $\partial W$ is a strictly normal crossing divisor. The main part of the L-function is described by some elliptic newform $g$ of weight 3. Then we construct an automorphic representation $Π$ of ${\rm GSp}_2(\A)$ related to $g$ and an explicit vector $E_Z$ sits inside $Π$ which creates a vector valued (non-cuspidal) Siegel modular form of weight $(3,1)$ so that $F_Z$ coincides with $E_Z$ in $H^{2,0}(\partial W)$ under the Poincaré residue map and various identifications of cohomologies.

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arXiv2014

On the class numbers of the fields of the p^n-torsion points of certain elliptic curves over Q

Let E be an elliptic curve over Q with prime conductor p. For each non-negative integer n we put K_n:=Q(E[p^n]). The aim of this paper is to estimate the order of the p-Sylow group of the ideal class group of K_n. We give a lower bounds in terms of the Mordell-Weil rank of $E(\Q)$. As an application of our result, we give an example such that p^{2n} divides the class number of the field $K_n$ in the case of $p=5077$ for each positive integer n.

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arXiv2013

An explicit computation of p-stabilized vectors

In this paper, we give a method to compute p-stabilized vectors in the space of parahori-fixed vectors for connected reductive groups over p-adic fields. The application to the global setting is also discussed. In particular, we give an explicit p-stabilized form of a Saito-Kurokawa lift.

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arXiv2012

Endoscopic lifts to the Siegel modular threefold related to Klein's cubic threefold

Let $A^{lev}_{11}$ be the moduli space of (1,11)-polarized abelian surfaces with level structure of canonical type. Let $χ$ be a finite character of order 5 with conductor 11. In this paper we construct five endoscopic lifts $Π_i,0\le i\le 4$ from two elliptic modular forms $f\otimesχ^i$ of weight 2 and $g\otimesχ^i$ of weight 4 with complex multiplication by $Q(\sqrt{-11})$ such that ${Π_i}_\infty$ gives a non-holomorphic differential form on $A^{lev}_{11}$ for each $i$. Then the spinor L-function is of form $L(f\otimesχ^i,s-1)L(g\otimesχ^i,s)$ such that $L(g\otimesχ^i,s)$ does not appear in the L-function of $A^{lev}_{11}$ for any $i$. The existence of such lifts is motivated by the computation of the L-function of Klein's cubic hypersurface which is a birational smooth model of $A^{lev}_{11}$.

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arXiv2012

Local newforms and formal exterior square L-functions

Let F be a non-archimedean local field of characteristic zero. Jacquet and Shalika attached a family of zeta integrals to unitary irreducible generic representations $π$ of GL_n(F). In this paper, we show that Jacquet-Shalika integral attains a certain L-function, so called the formal exterior square L-function, when the Whittaker function is associated to a newform for $π$. By consideration on the Galois side, formal exterior square L-functions are equal to exterior square L-functions for some principal series representations.

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