arXiv subjects

Miyu Suzuki

Publications and source records attributed to Miyu Suzuki.

12 recordsLinked to original sources

arXiv2026

Root lattices over totally real fields

A root lattice is a finite rank $\mathbb{Z}$-lattice generated by elements $x$ satisfying $x\cdot x=2$. It is well-known that the root lattices have an $ADE$ classification and they play a prominent role in the study of even unimodular lattices. The notion of root lattices can be naturally generalized to lattices over the ring of integers $\mathcal{O}$ of a totally real field $K$. In the case where $K$ is a real quadratic field, such lattices were classified by Mimura in 1979, and this classification has been used by several researchers in the study of even unimodular $\mathcal{O}$-lattices. In this paper, we extend this classification to arbitrary totally real fields. The irreducible root lattices of rank greater than $2$ are indexed by finite Coxeter systems. All the rank $2$ root lattices are realized as orders in quadratic extensions of $K$ and their classification requires some technique from algebraic number theory.

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arXiv2026

Discrete series representations of quaternionic ${\rm GL}_n(D)$ with symplectic periods

For a non-Archimedean locally compact field $F$ of odd residue characteristic and characteristic $0$, we prove a conjecture of D. Prasad predicting that, for an integer $n \geq 1$ and a non-split quaternionic $F$-algebra $D$, a discrete series representation of ${\rm GL}_n(D)$ has a symplectic period if and only if it is cuspidal and its Jacquet--Langlands transfer to ${\rm GL}_{2n}(F)$ is non-cuspidal.

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arXiv2026

Quaternionic symplectic model for discrete series representations

Let $D$ be the quatenion division algebra over a non-Archimedean local field $F$ of characteristic zero and odd residual characterisitc. We show that an irreducible discrete series representation of $\mathrm{GL}_n(D)$ is $\mathrm{Sp}_n(D)$-distinguished only if it is supercuspidal. Here, $\mathrm{Sp}_n(D)$ is the quaternionic symplectic group. Combined with the recent study on $\mathrm{Sp}_n(D)$-distinguished supercuspidal representations by Sécherre and Stevens, this completes the classification of $\mathrm{Sp}_n(D)$-distinguished discrete series representations, as predicted by Dipendra Prasad.

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arXiv2025

Denominator identity for the affine Lie superalgebra $\widehat{\mathfrak{spo}}(2m,2m+1)$ and indefinite theta functions

In 1994, Kac and Wakimoto found the denominator identity for classical affine Lie superalgebras, generalizing that for affine Lie algebras. As an application, they obtained power series identities for some powers of $\triangle(q)$, where $\triangle(q)$ is the generating function of triangular numbers. In this article, we give a different proof of one of their identities. The main step is to prove that a certain indefinite theta function involving spherical polynomials is a modular form. We use the technique recently developed by Roehrig and Zwegers.

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arXiv2025

Indefinite theta functions arising from affine Lie superalgebras and sums of triangular numbers

We extend the recently developed theory of Roehrig and Zwegers on indefinite theta functions to prove certain power series are modular forms. As a consequence, we obtain several power series identities for powers of the generating function of triangular numbers. We also show that these identities arise as specializations of denominator identities of affine Lie superalgebras.

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arXiv2025

A reformulation of the conjecture of Prasad and Takloo-Bighash

Prasad and Takloo-Bighash proposed a conjecture which predicts a necessary condition in terms of epsilon factors for representations of $\mathrm{GL}_n(F)$ and its inner forms to have linear periods. In this rather expository article, we reformulate their conjecture in the following form: The distinguished members in each generic $L$-packet $Π_ϕ$ are determined by the characters of the component group $S_ϕ$ and local epsilon factors. We follow Aubert et al.\,for the definitions of the $L$-packets and the component groups. We observe that under some hypotheses, the reformulated conjecture follows from the conjectural multiplicity formula recently proposed by Chen Wan for general spherical varieties and the conjectural integral formula for epsilon factors which we propose in this article.

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arXiv2024

Zeta functions and nonvanishing theorems for toric periods on $\mathrm{GL}_2$

Let $F$ be a number field and $D$ a quaternion algebra over $F$. Take a cuspidal automorphic representation $π$ of $D_\mathbb{A}^\times$ with trivial central cahracter. We study the zeta functions with period integrals on $π$ for the perhomogeneous vector space $(D^\times\times D^\times\times\mathrm{GL}_2, D\oplus D)$. We show their meromorphic continuation and functional equation, determine the location and orders of possible poles and compute the residue. Arguing along the theory of Saito and computing unramified local factors, the explicit formula of the zeta functions is obtained. Counting the order of possible poles of this explicit formula, we show that if $L(1/2, π)\neq0$, there are infinitely many quadratic extension $E$ of $F$ which embeds in $D$, such that $π$ has nonvanishing toric period with respect to $E$.

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arXiv2024

Explicit mean value theorems for toric periods and automorphic $L$-functions

Let $F$ be a number field and $D$ a quaternion algebra over $F$. Take a cuspidal automorphic representation $π$ of $D_{\mathbb{A}}^\times$ with trivial central character and a cusp form $ϕ$ in $π$. Using the prehomogeneous zeta function, we find an explicit mean value of the toric periods of $ϕ$ with respect to quadratic algebras over $F$. The result can also be written as a mean value formula for the central values of automorphic $L$-functions twisted by quadratic characters.

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arXiv2023

Epsilon dichotomy for linear models: the Archimedean case

Let $G=\mathrm{GL}_{2n}(\mathbb{R})$ or $G=\mathrm{GL}_n(\mathbb{H})$ and $H=\mathrm{GL}_n(\mathbb{C})$ regarded as a subgroup of $G$. Here, $\mathbb{H}$ is the quaternion division algebra over $\mathbb{R}$. For a character $χ$ on $\mathbb{C}^\times$, we say that an irreducible smooth admissible moderate growth representation $π$ of $G$ is $χ_H$-distinguished if $\mathrm{Hom}_H(π, χ\circ\det_H)\neq0$. We compute the root number of a $χ_H$-distinguished representation $π$ twisted by the representation induced from $χ$. This proves an Archimedean analogue of the conjecture by Prasad and Takloo-Bighash (J. Reine Angew. Math., 2011). The proof is based on the analysis of the contribution of $H$-orbits in a flag manifold of $G$ to the Schwartz homology of principal series representations. A large part of the argument is developed for general real reductive groups of inner type. In particular, we prove that the Schwartz homology $H_\ast(H, π\otimesχ)$ is finite-dimensional and hence it is Hausdorff for a reductive symmetric pair $(G, H)$ and a finite-dimensional representation $χ$ of $H$.

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arXiv2022

Distribution of toric periods of modular forms on definite quaternion algebras

Let $D$ be a definite quaternion algebra over $\mathbb{Q}$ and $\mathcal{O}$ an Eichler order in $D$ of square-free level. We study distribution of the toric periods of algebraic modular forms of level $\mathcal{O}$. We focus on two problems: non-vanishing and sign changes. Firstly, under certain conditions on $\mathcal{O}$, we prove the non-vanishing of the toric periods for positive proportion of imaginary quadratic fields. This improves the known lower bounds toward Goldfeld's conjecture in some cases and provides evidence for similar non-vanishing conjectures for central values of twisted automorphic $L$-functions. Secondly, we show that the sequence of toric periods has infinitely many sign changes. This proves the sign changes of the Fourier coefficients $\{a(n)\}_n$ of weight 3/2 modular forms, where $n$ ranges over fundamental discriminants. In the final section, we present numerical experiments in some cases and formulate several conjectures based on them.

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arXiv2020

Classification of standard modules with linear periods

Suppose that $F$ is a non-Archimedean local field and $D$ is a central division algebra over $F$. Let $n$ be a positive integer. We show a classification modulo essentially square-integrable representations of standard modules of $\mathrm{GL}_n(D)$ which have non-zero linear periods. By this classification, the conjecture of Prasad and Takloo-Bighash is reduced to the case of essentially square integrable representations.

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arXiv2019

Quaternion distinguished generic representations of $\mathrm{GL}_{2n}$

Let $E/F$ be a quadratic extension of non-Archimedean local fields of characteristic 0. Let $D$ be the unique quaternion division algebra over $F$ and fix an embedding of $E$ to $D$. Then, $\mathrm{GL}_m(D)$ can be regarded as a subgroup of $\mathrm{GL}_{2m}(E)$. Using the method of Matringe, we classify irreducible generic $\mathrm{GL}_m(D)$-distinguished representations of $\mathrm{GL}_{2m}(E)$ in terms of Zelevinsky classification. Rewriting the classification in terms of corresponding representations of the Weil-Deligne group of $E$, we prove a sufficient condition for a generic representation in the image of the unstable base change lift from the unitary group $\mathrm{U}_{2m}$ to be $\mathrm{GL}_m(D)$-distinguished.

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