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Takuya Yamauchi

Publications and source records attributed to Takuya Yamauchi.

At least 19 recordsLinked to original sources
arXiv2026

An Explicit Belyi Map for the Wiman Sextic and Cusp Forms for a Noncongruence Subgroup

In this paper, we explicitly determine an algebraic Belyi function on a unique smooth projective model $\widetilde{W}$ of the Wiman sextic curve $W$ and describe its complex uniformization in terms of modular functions associated with a certain noncongruence subgroup $\Gamma_{\widetilde{W}} \subset {\rm SL}_2(\mathbb{Z})$. As an application, we give a direct proof of the unbounded denominators conjecture in weight $2$ for $\Gamma_{\widetilde{W}}$. The conjecture is now known in full generality by the work of Calegari, Dimitrov, and Tang, following earlier progress including work of Dong, Lin, and Ng. Our proof, however, uses a degeneration of $\widetilde{W}$ over $\mathbb{F}_{5}$ together with explicit Puiseux series expansions and is substantially different from the methods employed in their work.

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arXiv2026

Isogeny graphs associated to Moret-Bailly families of supersingular abelian surfaces

For an odd prime $p$ and any prime $\ell\neq p$, we study finite directed graphs arising from the set of all equivalence classes of Moret-Bailly families of abelian surfaces in characteristic $p$ together with relative $(\ell,\ell)$-isogenies. We relate these graphs to the space of algebraic modular forms on an inner form of $GSp_4/\mathbb{Q}$ that is compact modulo its center and, using the Jacquet-Langlands correspondence, estimate the eigenvalues of their adjacency matrices. We further investigate a Sarnak-Xue type theorem in this setting, providing a first step toward the study of cut-off phenomena for these graphs.

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arXiv2026

On explicit Fourier expansions of theta lifts to ${\rm SO}(3,n+1)$ arising from elliptic newforms of level one

Using degenerate Whittaker functions and explicit computations of Eisenstein series, we obtain explicit formulas for the Fourier expansions of theta lifts to the special orthogonal group $G={\rm SO}(3,n+1)$ over $\mathbb{Q}$, where $n\ge 3$ and $G$ splits at all finite places. The theta lifts in question are Hecke eigen, non-cuspidal, square-integrable automorphic forms of weight $l$ ($l\ge n+2$, even), arising from elliptic newforms for $\SL_2(\Z)$ of weight $l-\frac{n-2}{2}$ when $n$ is even and $2l-n+1$ when $n$ is odd.

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arXiv2025

Regulators on some abelian coverings of $\mathbb{P}^1$ minus $n+2$ points

In this paper, we construct certain rational or integral elements in the motivic cohomology of superelliptic curves which are quotient curves of abelian coverings of $\mathbb{P}^1$ minus $n+2$ points, and prove that these elements are non-trivial by expressing their regulators in terms of Appell-Lauricella hypergeometric functions. We also check that such elements are integral under a mild assumption. We also give various numerical examples for the Beilinson conjecture on special values of $L$-functions of the superelliptic curves by using hypergeometric expressions.

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arXiv2025

The $L$-function of the surface parametrizing cuboids

In this note, we compute the $L$-function of the projective smooth surface $S$ over $\mathbb{Q}$ that parametrizes cuboids whose geometric properties are studied in detail by Stoll and Testa. As a byproduct, we completely determine the structure of ${\rm Pic}(S_{\overline{\mathbb{Q}}})$ as a ${\rm Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$-module.

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arXiv2025

The non-existence of some Galois representations of moderate dimension in small characteristic

Refining arguments of Hyunsuk Moon, under the assumption of the Generalized Riemann Hypothesis, we prove the non-existence of irreducible mod 2 Galois representations unramified outside 2 of dimensions $\leq 4$, and of totally real such representations of dimensions $\leq 8$. We also prove the non-existence of irreducible totally real mod 3 representations unramified outside 3 of dimensions $\leq 4$. We show unconditionally that the image of an irreducible mod 2 symplectic 4-dimensional Galois representation that is unramified outside 2 must be large. Under GRH, we then deduce the non-existence of such representations.

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arXiv2025

On the Fourier expansion of Gan-Gurevich lifts on the exceptional group of type $G_2$

By using the degenerate Whittaker functions, we study the Fourier expansion of the Gan-Gurevich lifts which are Hecke eigen quaternionic cusp forms of weight $k$ ($k\geq 2$, even) on the split exceptional group $G_2$ over $\mathbb{Q}$ which come from elliptic newforms of weight $2k$ without supercuspidal local components. In particular, our results give a partial answer to Gross' conjecture.

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arXiv2025

The Kodaira dimension of even-dimensional ball quotients

We prove that, up to scaling, there exist only finitely many isometry classes of Hermitian lattices over $O_E$ of signature $(1,n)$ that admit ball quotients of non-general type, where $n>12$ is even and $E=\mathbb{Q}(\sqrt{-D})$ for an odd discriminant $-D<-3$. Furthermore, we show that even-dimensional ball quotients, associated with arithmetic subgroups of $\mathrm{U}(1,n)$ defined over $E$, are always of general type if $n > 207$, or $n>12$ and $D>2557$. To establish these results, we construct a nontrivial full-level cusp form of weight $n$ on the $n$-dimensional complex ball. A key ingredient in our proof is the use of Arthur's multiplicity formula from the theory of automorphic representations.

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arXiv2025

A new theta cycle for $GSp_4$ and an Edixhoven type theorem

In this paper, we investigate a new theta cycle for $GSp_4/\mathbb{Q}$ by using author's theta operators defined in the previous work. In the course of the construction, we also modify the theta operators so that they work in any characteristic, including $p = 2$, and for any weight. As an application, we discuss an Edixhoven type theorem for $GSp_4/\mathbb{Q}$.

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arXiv2025

Restriction of modular forms on $E_{7,3}$ to $Sp_6$

In this paper, we study the restriction of modular forms such as Ikeda type lifts and the Eisenstein series on the exceptional group of type $E_{7,3}$ to the symplectic group $Sp_6$ (rank 3). As an application, we explicitly write down the restriction when modular forms have small weight. The restriction may contain Miyawaki lifts of type I,II (CAP forms) and genuine forms whose description is compatible with Arthur's classification.

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arXiv2025

The residual monodromy for the Dwork family in even characteristic and its applications to Galois representations

We study the residual monodromy representations associated to the Dwork family in characteristic two. Various applications involving 2-adic and mod 2 Galois representations are discussed. Combining the author's previous work with Tsuzuki and recent results of Boxer, Calegari, Gee, and Pilloni, we also prove the automorphy of certain rank 4 symplectic motives over a totally real field, arising from the Dwork quintic family, under suitable conditions.

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arXiv2025

Computing algebraic Belyi functions on Bring's curve

In this paper, we explicitly compute two kinds of algebraic Belyi functions on Bring's curve. One is related to a congruence subgroup of ${\rm SL}_2(\mathbb{Z})$ and the other is related to a congruence subgroup of the triangle group $Δ(2,4,5)\subset \SL_2(\R)$. To carry out the computation, we use elliptic cusp forms of weight 2 for the former case and the automorphism group of Bring's curve for the latter case. We also discuss a suitable base field (a number field) for describing isomorphisms between Hulek-Craig's curve, Bring's curve, and another algebraic model obtained as a modular curve.

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arXiv2025

Central limit theorem for Hecke eigenvalues

In this paper, we obtain the central limit theorem of Hecke eigenvalues in very general setting of split simple algebraic groups over $\mathbb{Q}$, using irreducible characters of compact Lie groups.

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arXiv2024

On finiteness theorems for automorphic forms

In this paper, for any Shimura datum $(G,\mathcal{D})$ satisfying reasonable conditions so that many interesting cases satisfy, we prove some finiteness theorems for any graded vector space consisting of automorphic forms on $\mathcal{D}$ of some weights over the graded ring of automorphic forms on $X$ with positive parallel weights. We also discuss the integral base ring which we can work on. To realize automorphic forms as global sections on some coherent sheaves on the minimal compactification, we use the notion of reflexive sheaves and higher Koecher principle due to Kai-Wen Lan. Further, we give a more finer version of finiteness results for Siegel modular forms by using only the results of Chai-Faltings.

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arXiv2023

Automorphy of mod 2 Galois representations associated to certain genus 2 curves over totally real fields

Let $C$ be a genus two hyperelliptic curve over a totally real field $F$. We show that the mod 2 Galois representation $\barρ_{C,2}\colon\mathrm{Gal}(\bar{F}/F)\to \mathrm{GSp}_4(\mathbb{F}_2)$ attached to $C$ is automorphic when the image of $\barρ_{C,2}$ is isomorphic to $S_5$ and it is also a transitive subgroup under a fixed isomorphism $\mathrm{GSp}_2(\mathbb{F}_2)\cong S_6$. To be more precise, there exists a Hilbert--Siegel Hecke eigen cusp form on $\mathrm{GSp}_4(\mathbb{A}_F)$ of parallel weight two whose mod 2 Galois representation is isomorphic to $\barρ_{C,2}$.

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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.

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arXiv2023

Transfers of some Hecke elements for possibly ramified base change in GL_n

In this paper we prove an explicit matching theorem for some Hecke elements in the case of (possibly ramified) cyclic base change for general linear groups over local fields of characteristic zero with odd residue characteristic under a mild assumption. A key observation, based on the works of Waldspurger and Ganapathy-Varma, is to regard the base change lifts with twisted endoscopic lifts and replace the condition for the matching orbital integrals with one for semi-simple descend in the twisted space according to Waldspurger's fundamental work "L'endoscopie tordue n'est pas si tordue".

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