#### Results in Journal Journal of the Mathematical Society of Japan: 3,327

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Henrik Bachmann, Yoshihiro Takeyama, Koji Tasaka
Published: 22 October 2021
Journal of the Mathematical Society of Japan, Volume 73; https://doi.org/10.2969/jmsj/84348434

Motohiro Sobajima, Yuta Wakasugi
Published: 22 October 2021
Journal of the Mathematical Society of Japan, Volume 73; https://doi.org/10.2969/jmsj/83928392

Yohei Fujishima, Kazuhiro Ishige
Published: 22 October 2021
Journal of the Mathematical Society of Japan, Volume 73, pp 1-33; https://doi.org/10.2969/jmsj/84728472

Abstract:
Let $(u, v)$ be a solution to a semilinear parabolic system $$\mbox{(P)} \qquad \left\{ \begin{array}{ll} \partial_t u = D_1 \Delta u+v^p \quad \mbox{in} \quad \mathbf{R}^N \times (0,T),\\ \partial_t v = D_2 \Delta v+u^q \quad \mbox{in}\quad \mathbf{R}^N \times (0,T),\\ u,v \ge 0 \quad \mbox{in} \quad \mathbf{R}^N \times (0,T),\\ (u(\cdot,0),v(\cdot,0)) = (\mu,\nu) \quad \mbox{in} \quad \mathbf{R}^N, \end{array} \right.$$ where $N \ge 1$, $T > 0$, $D_1 > 0$, $D_2 > 0$, $0 < p \le q$ with $pq > 1$ and $(\mu, \nu)$ is a pair of Radon measures or nonnegative measurable functions in $\mathbf{R}^N$. In this paper we study qualitative properties of the initial trace of the solution $(u, v)$ and obtain necessary conditions on the initial data $(\mu, \nu)$ for the existence of solutions to problem (P).
Shinichiro Kobayashi
Published: 22 October 2021
Journal of the Mathematical Society of Japan, Volume 73, pp 1-11; https://doi.org/10.2969/jmsj/85088508

Abstract:
In this paper, we derive an upper bound for higher eigenvalues of the normalized Laplace operator associated with a symmetric finite graph in terms of lower eigenvalues.
Yasuaki Ogawa
Published: 22 October 2021
Journal of the Mathematical Society of Japan, Volume 73, pp 1-27; https://doi.org/10.2969/jmsj/84578457

Abstract:
The notion of extriangulated category was introduced by Nakaoka and Palu giving a simultaneous generalization of exact categories and triangulated categories. Our first aim is to provide an extension to extriangulated categories of Auslander's formula: for some extriangulated category $\mathcal{C}$, there exists a localization sequence $\operatorname{def}\mathcal{C} \to \mod\mathcal{C} \to \operatorname{lex}\mathcal{C}$, where $\operatorname{lex}\mathcal{C}$ denotes the full subcategory of finitely presented left exact functors and $\operatorname{def}\mathcal{C}$ the full subcategory of Auslander's defects. Moreover we provide a connection between the above localization sequence and the Gabriel–Quillen embedding theorem. As an application, we show that the general heart construction of a cotorsion pair $(\mathcal{U}, \mathcal{V})$ in a triangulated category, which was provided by Abe and Nakaoka, is the same as the construction of a localization sequence $\operatorname{def}\mathcal{U} \to \mod\mathcal{U} \to \operatorname{lex}\mathcal{U}$.
Takashi Aoki, Toshinori Takahashi, Mika Tanda
Published: 22 October 2021
Journal of the Mathematical Society of Japan, Volume 73, pp 1-44; https://doi.org/10.2969/jmsj/84528452

Abstract:
Relations between the hypergeometric function with a large parameter and Borel sums of WKB solutions of the hypergeometric differential equation with the large parameter are established. The confluent hypergeometric function is also investigated from the viewpoint of exact WKB analysis. As applications, asymptotic expansion formulas for those classical special functions with respect to parameters are obtained.
Published: 22 October 2021
Journal of the Mathematical Society of Japan, Volume 73, pp 1-34; https://doi.org/10.2969/jmsj/84618461

Abstract:
We present an algorithm that takes as input any element $B$ of the loop braid group and constructs a polynomial $f:\mathbb{R}^5 \to \mathbb{R}^2$ such that the intersection of the vanishing set of $f$ and the unit 4-sphere contains the closure of $B$. The polynomials can be used to create real analytic time-dependent vector fields with zero divergence and closed flow lines that move as prescribed by $B$. We also show how a family of surface braids in $\mathbb{C} \times S^1 \times S^1$ without branch points can be constructed as the vanishing set of a holomorphic polynomial $f:\mathbb{C}^3 \to \mathbb{C}$ on $\mathbb{C} \times S^1 \times S^1 \subset \mathbb{C}^3$. Both constructions allow us to give upper bounds on the degree of the polynomials.
Masao Jinzenji, Hayato Saito
Published: 22 October 2021
Journal of the Mathematical Society of Japan, Volume 73; https://doi.org/10.2969/jmsj/83148314

Yifei Chen
Published: 22 October 2021
Journal of the Mathematical Society of Japan, Volume 73, pp 1-14; https://doi.org/10.2969/jmsj/83898389

Abstract:
We answer a conjecture raised by Caucher Birkar of singularities of weighted blowups of $\mathbb{A}^{n}$ for $n \leq 3$.
Toshihiro Nakanishi
Published: 22 October 2021
Journal of the Mathematical Society of Japan, Volume 73, pp 1-32; https://doi.org/10.2969/jmsj/84998499

Abstract:
We introduce coordinate systems to the Teichmüller space of the twice-punctured torus and give matrix representations for the points of Teichmüller space. The coordinate systems allow representation of the mapping class group of the twice punctured torus as a group of rational transformations and provide several applications to the mapping class group and also to Kleinian groups.
Tai Melcher
Published: 22 October 2021
Journal of the Mathematical Society of Japan, Volume 73, pp 1-27; https://doi.org/10.2969/jmsj/84678467

Abstract:
We construct a class of iterated stochastic integrals with respect to Brownian motion on an abstract Wiener space which allows for the definition of Brownian motions on a general class of infinite-dimensional nilpotent Lie groups based on abstract Wiener spaces. We then prove that a Cameron–Martin type quasi-invariance result holds for the associated heat kernel measures in the non-degenerate case, and give estimates on the associated Radon–Nikodym derivative. We also prove that a log Sobolev estimate holds in this setting.
Toshiyuki Katsura, Natsuo Saito
Published: 22 October 2021
Journal of the Mathematical Society of Japan, Volume 73, pp 1-9; https://doi.org/10.2969/jmsj/85058505

Abstract:
We consider the multicanonical systems $|mK_{S}|$ of quasi-elliptic surfaces with Kodaira dimension 1 in characteristic 2. We show that for any $m \geq 6$ $|mK_{S}|$ gives the structure of quasi-elliptic fiber space, and 6 is the best possible number to give the structure for any such surfaces.
Toshiaki Hattori
Published: 27 July 2021
Journal of the Mathematical Society of Japan, Volume 73; https://doi.org/10.2969/jmsj/81358135

Anh T. Tran
Published: 27 July 2021
Journal of the Mathematical Society of Japan, Volume 73; https://doi.org/10.2969/jmsj/84058405

Soumen Sarkar, Vikraman Uma
Published: 27 July 2021
Journal of the Mathematical Society of Japan, Volume 73, pp 1-18; https://doi.org/10.2969/jmsj/83548354

Abstract:
Toric orbifolds are a topological generalization of projective toric varieties associated to simplicial fans. We introduce some sufficient conditions on the combinatorial data associated to a toric orbifold to ensure the existence of an invariant cell structure on it and call such a toric orbifold retractable. In this paper, our main goal is to study equivariant cohomology theories of retractable toric orbifolds. Our results extend the corresponding results on divisive weighted projective spaces.
Yoshiyuki Ohyama, Migiwa Sakurai
Published: 27 July 2021
Journal of the Mathematical Society of Japan, Volume 73, pp 1-12; https://doi.org/10.2969/jmsj/84478447

Abstract:
Satoh and Taniguchi introduced the $n$-writhe $J_{n}$ for each non-zero integer $n$, which is an integer invariant for virtual knots. The sequence of $n$-writhes $\{J_{n}\}_{n \neq 0}$ of a virtual knot $K$ satisfies $\sum_{n \neq 0} nJ_{n}(K) = 0$. They showed that for any sequence of integers $\{c_{n}\}_{n \neq 0}$ with $\sum_{n \neq 0} nc_{n} = 0$, there exists a virtual knot $K$ with $J_n(K) = c_{n}$ for any $n \neq 0$. It is obvious that the virtualization of a real crossing is an unknotting operation for virtual knots. The unknotting number by the virtualization is called the virtual unknotting number and is denoted by $u^{v}$. In this paper, we show that if $\{c_{n}\}_{n \neq 0}$ is a sequence of integers with $\sum_{n \neq 0} nc_{n} = 0$, then there exists a virtual knot $K$ such that $u^{v}(K) = 1$ and $J_{n}(K) = c_{n}$ for any $n \neq 0$.
Yoshinori Nishii, Hideaki Sunagawa, Hiroki Terashita
Published: 27 July 2021
Journal of the Mathematical Society of Japan, Volume 73; https://doi.org/10.2969/jmsj/84148414

Leo Murata
Published: 27 July 2021
Journal of the Mathematical Society of Japan, Volume 73, pp 1-10; https://doi.org/10.2969/jmsj/82968296

Abstract:
We consider a distribution property of the residual order (the multiplicative order) of the residue class $a \hspace{-.4em} \pmod{pq}$. It is known that the residual order fluctuates irregularly and increases quite rapidly. We are interested in how the residual orders $a \hspace{-.4em} \pmod{pq}$ distribute modulo 4 when we fix $a$ and let $p$ and $q$ vary. In this paper we consider the set $S(x) = \{(p, q); p, q \ \text{are distinct primes,} \ pq \leq x \}$, and calculate the natural density of the set $\{(p, q) \in S(x); \ \text{the residual order of} \ a \hspace{-.4em} \pmod{pq} \equiv l \hspace{-.4em} \pmod{4}\}$. We show that, under a simple assumption on $a$, these densities are $\{5/9,\, 1/18,\, 1/3,\, 1/18 \}$ for $l= \{0, 1, 2, 3 \}$, respectively. For $l = 1, 3$ we need Generalized Riemann Hypothesis.
Satoshi Nakamura
Published: 27 July 2021
Journal of the Mathematical Society of Japan, Volume 73, pp 1-15; https://doi.org/10.2969/jmsj/84408440

Abstract:
The notion of coupled Kähler–Einstein metrics was introduced recently by Hultgren–Witt Nyström. In this paper we discuss deformation of a coupled Kähler–Einstein metric on a Fano manifold. We obtain a necessary and sufficient condition for a coupled Kähler–Einstein metric to be deformed to another coupled Kähler–Einstein metric for a Fano manifold admitting non-trivial holomorphic vector fields. In addition we also discuss deformation for a coupled Käher–Einstein metric on a Fano manifold when the complex structure varies.
Hong Chuong Doan
Published: 27 July 2021
Journal of the Mathematical Society of Japan, Volume 73, pp 1-31; https://doi.org/10.2969/jmsj/83348334

Abstract:
Let $M$ be a non-doubling parabolic manifold with ends and $L$ a non-negative self-adjoint operator on $L^{2}(M)$ which satisfies a suitable heat kernel upper bound named the upper bound of Gaussian type. These operators include the Schrödinger operators $L = \Delta + V$ where $\Delta$ is the Laplace–Beltrami operator and $V$ is an arbitrary non-negative potential. This paper will investigate the behaviour of the Poisson semi-group kernels of $L$ together with its time derivatives and then apply them to obtain the weak type $(1, 1)$ estimate of the functional calculus of Laplace transform type of $\sqrt{L}$ which is defined by $\mathfrak{M}(\sqrt{L}) f(x) := \int_{0}^{\infty} \bigl[\sqrt{L} e^{-t \sqrt{L}} f(x)\bigr] m(t) dt$ where $m(t)$ is a bounded function on $[0, \infty)$. In the setting of our study, both doubling condition of the measure on $M$ and the smoothness of the operators' kernels are missing. The purely imaginary power $L^{is}$, $s \in \mathbb{R}$, is a special case of our result and an example of weak type $(1, 1)$ estimates of a singular integral with non-smooth kernels on non-doubling spaces.
Published: 27 July 2021
Journal of the Mathematical Society of Japan, Volume 73; https://doi.org/10.2969/jmsj/83918391

Taku Suzuki
Published: 27 July 2021
Journal of the Mathematical Society of Japan, Volume 73; https://doi.org/10.2969/jmsj/84428442

Published: 27 July 2021
Journal of the Mathematical Society of Japan, Volume 73, pp 1-34; https://doi.org/10.2969/jmsj/79987998

Abstract:
Let $q$ be a positive integer ($\geq 2$), $\chi$ be a Dirichlet character modulo $q$, $L(s, \chi)$ be the attached Dirichlet $L$-function, and let $L^{\prime} (s, \chi)$ denote its derivative with respect to the complex variable $s$. Let $t_{0}$ be any fixed real number. The main purpose of this paper is to give an asymptotic formula for the $2k$-th power mean value of $|(L^{\prime}/L)(1+it_0, \chi)|$ when $\chi$ runs over all Dirichlet characters modulo $q$ (except the principal character when $t_{0} = 0$).
David Leturcq
Published: 27 July 2021
Journal of the Mathematical Society of Japan, Volume 73, pp 1-46; https://doi.org/10.2969/jmsj/82908290

Abstract:
Bott, Cattaneo and Rossi defined invariants of long knots $\mathbb{R}^{n} \hookrightarrow \mathbb{R}^{n+2}$ as combinations of configuration space integrals for $n$ odd $\geq 3$. Here, we give a more flexible definition of these invariants. Our definition allows us to interpret these invariants as counts of diagrams. It extends to long knots inside more general $(n+2)$-manifolds, called asymptotic homology $\mathbb{R}^{n+2}$, and provides invariants of these knots.
Marcello Bernardara
Published: 27 July 2021
Journal of the Mathematical Society of Japan, Volume 73, pp 1-23; https://doi.org/10.2969/jmsj/82658265

Abstract:
Using a filtration on the Grothendieck ring of triangulated categories, we define the motivic categorical dimension of a birational map between smooth projective varieties. We show that birational transformations of bounded motivic categorical dimension form subgroups, which provide a nontrivial filtration of the Cremona group. We discuss some geometrical aspect and some explicit example. We can moreover define, in some cases, the genus of a birational transformation, and compare it to the one defined by Frumkin in the case of threefolds.
Hasse Carlsson
Published: 27 July 2021
Journal of the Mathematical Society of Japan, Volume 73, pp 1-21; https://doi.org/10.2969/jmsj/83298329

Abstract:
We prove sharp estimates for the renewal measure of a strongly nonlattice probability measure on the real line. In particular we consider the case where the measure has finite moments between 1 and 2. The proof uses Fourier analysis of tempered distributions.
Ismael Bailleul, Masato Hoshino
Published: 23 April 2021
Journal of the Mathematical Society of Japan, Volume 73; https://doi.org/10.2969/jmsj/81878187

Tohru Ozawa, Durvudkhan Suragan
Published: 23 April 2021
Journal of the Mathematical Society of Japan, Volume 73; https://doi.org/10.2969/jmsj/83738373

Tomoya Kato, Akihiko Miyachi, Naohito Tomita
Published: 23 April 2021
Journal of the Mathematical Society of Japan, Volume 73; https://doi.org/10.2969/jmsj/83468346

Published: 23 April 2021
Journal of the Mathematical Society of Japan, Volume 73; https://doi.org/10.2969/jmsj/83308330

Cheng Gong, Shinya Kitagawa, Jun Lu
Published: 23 April 2021
Journal of the Mathematical Society of Japan, Volume 73; https://doi.org/10.2969/jmsj/82438243

Akito Futaki, Yingying Zhang
Published: 23 April 2021
Journal of the Mathematical Society of Japan, Volume 73; https://doi.org/10.2969/jmsj/83558355

The Anh Bui, Xuan Thinh Duong
Published: 23 April 2021
Journal of the Mathematical Society of Japan, Volume 73, pp 597-631; https://doi.org/10.2969/jmsj/83938393

Abstract:
In this paper we first study the generalized weighted Hardy spaces $H^{p}_{L,w}(X)$ for $0 < p \le 1$ associated to nonnegative self-adjoint operators $L$ satisfying Gaussian upper bounds on the space of homogeneous type $X$ in both cases of finite and infinite measure. We show that the weighted Hardy spaces defined via maximal functions and atomic decompositions coincide. Then we prove weighted regularity estimates for the Green operators of the inhomogeneous Dirichlet and Neumann problems in suitable bounded or unbounded domains including bounded semiconvex domains, convex regions above a Lipschitz graph and upper half-spaces. Our estimates are in terms of weighted $L^{p}$ spaces for the range $1 < p <\infty$ and in terms of the new weighted Hardy spaces for the range $0 < p \le 1$. Our regularity estimates for the Green operators under the weak smoothness assumptions on the boundaries of the domains are new, especially the estimates on Hardy spaces for the full range $0 < p \le 1$ and the case of unbounded domains.
Stefan Schröer
Published: 23 April 2021
Journal of the Mathematical Society of Japan, Volume 73, pp 433-496; https://doi.org/10.2969/jmsj/83728372

Abstract:
We analyze the structure of simply-connected Enriques surfaces in characteristic two whose K3-like coverings are normal, building on the work of Ekedahl, Hyland and Shepherd-Barron. We develop general methods to construct such surfaces and the resulting twistor lines in the moduli stack of Enriques surfaces, including the case that the K3-like covering is a normal rational surface rather then a normal K3 surface. Among other things, we show that elliptic double points indeed do occur. In this case, there is only one singularity. The main idea is to apply flops to Frobenius pullbacks of rational elliptic surfaces, to get the desired K3-like covering. Our results hinge on Lang's classification of rational elliptic surfaces, the determination of their Mordell–Weil lattices by Shioda and Oguiso, and the behavior of unstable fibers under Frobenius pullback via Ogg's formula. Along the way, we develop a general theory of Zariski singularities in arbitrary dimension, which is tightly interwoven with the theory of height-one group schemes actions and restricted Lie algebras. Furthermore, we determine under what conditions tangent sheaves are locally free, and introduce a theory of canonical coverings for arbitrary proper algebraic schemes.
, Nathan Priddis
Published: 23 April 2021
Journal of the Mathematical Society of Japan, Volume 73; https://doi.org/10.2969/jmsj/79867986

Van Hoang Nguyen
Published: 23 April 2021
Journal of the Mathematical Society of Japan, Volume 73; https://doi.org/10.2969/jmsj/82258225

Atsushi Atsuji
Published: 23 April 2021
Journal of the Mathematical Society of Japan, Volume 73; https://doi.org/10.2969/jmsj/82398239

Mutsuo Oka
Published: 23 April 2021
Journal of the Mathematical Society of Japan, Volume 73; https://doi.org/10.2969/jmsj/83328332

Tomoyuki Arakawa, Hiromichi Yamada, Hiroshi Yamauchi
Published: 1 January 2021
Journal of the Mathematical Society of Japan, Volume 73; https://doi.org/10.2969/jmsj/83278327

Shigeyuki Kondō
Published: 1 January 2021
Journal of the Mathematical Society of Japan, Volume 73; https://doi.org/10.2969/jmsj/81778177

Zifei Shen, Bin Shang, Chenyin Qian
Published: 1 January 2021
Journal of the Mathematical Society of Japan, Volume 73; https://doi.org/10.2969/jmsj/83258325

Beata Osińska-Ulrych, Grzegorz Skalski, Stanisław SPODZIEJA
Published: 1 January 2021
Journal of the Mathematical Society of Japan, Volume 73; https://doi.org/10.2969/jmsj/83378337

Tsukasa Iwabuchi, Takayoshi Ogawa
Published: 1 January 2021
Journal of the Mathematical Society of Japan, Volume -1, pp 1-42; https://doi.org/10.2969/jmsj/81598159

Abstract:
We consider the compressible Navier–Stokes system in the critical Besov spaces. It is known that the system is (semi-)well-posed in the scaling semi-invariant spaces of the homogeneous Besov spaces $\dot{B}^{n/p}_{p,1} \times \dot{B}^{n/p-1}_{p,1}$ for all $1 \le p < 2n$. However, if the data is in a larger scaling invariant class such as $p > 2n$, then the system is not well-posed. In this paper, we demonstrate that for the critical case $p = 2n$ the system is ill-posed by showing that a sequence of initial data is constructed to show discontinuity of the solution map in the critical space. Our result indicates that the well-posedness results due to Danchin and Haspot are indeed sharp in the framework of the homogeneous Besov spaces.
Matsuyo Tomisaki, Toshihiro Uemura
Published: 1 January 2021
Journal of the Mathematical Society of Japan, Volume -1, pp 1-37; https://doi.org/10.2969/jmsj/85268526

Abstract:
We consider a homogenization problem for symmetric jump-diffusion processes by using the Mosco convergence and the two-scale convergence of the corresponding Dirichlet forms. Moreover, we show the weak convergence of the processes.
Jun-Muk Hwang
Published: 1 January 2021
Journal of the Mathematical Society of Japan, Volume -1, pp 1-20; https://doi.org/10.2969/jmsj/85868586

Abstract:
A nonsingular rational curve $C$ in a complex manifold $X$ whose normal bundle is isomorphic to $\mathcal{O}_{\mathbb{P}^1}(1)^{\oplus p} \oplus \mathcal{O}_{\mathbb{P}^1}^{\oplus q}$ for some nonnegative integers $p$ and $q$ is called an unbendable rational curve on $X$. Associated with it is the variety of minimal rational tangents (VMRT) at a point $x \in C$, which is the germ of submanifolds $\mathcal{C}^C_{x} \subset \mathbb{P} T_{x} X$ consisting of tangent directions of small deformations of $C$ fixing $x$. Assuming that there exists a distribution $D \subset TX$ such that all small deformations of $C$ are tangent to $D$, one asks what kind of submanifolds of projective space can be realized as the VMRT $\mathcal{C}^{C}_{x} \subset \mathbb{P} D_{x}$. When $D \subset TX$ is a contact distribution, a well-known necessary condition is that $\mathcal{C}_{x}^{C}$ should be Legendrian with respect to the induced contact structure on $\mathbb{P} D_{x}$. We prove that this is also a sufficient condition: we construct a complex manifold $X$ with a contact structure $D \subset TX$ and an unbendable rational curve $C \subset X$ such that all small deformations of $C$ are tangent to $D$ and the VMRT $\mathcal{C}^C_{x} \subset \mathbb{P} D_{x}$ at some point $x \in C$ is projectively isomorphic to an arbitrarily given Legendrian submanifold. Our construction uses the geometry of contact lines on the Heisenberg group and a technical ingredient is the symplectic geometry of distributions the study of which has originated from geometric control theory.
Jesús Álvarez López, Ramon Barral Lijo, Olga Lukina, Hiraku Nozawa
Published: 1 January 2021
Journal of the Mathematical Society of Japan, Volume -1, pp 1-26; https://doi.org/10.2969/jmsj/85748574

Abstract:
The discriminant group of a minimal equicontinuous action of a group $G$ on a Cantor set $X$ is the subgroup of the closure of the action in the group of homeomorphisms of $X$, consisting of homeomorphisms which fix a given point. The stabilizer and the centralizer groups associated to the action are obtained as direct limits of sequences of subgroups of the discriminant group with certain properties. Minimal equicontinuous group actions on Cantor sets admit a classification by the properties of the stabilizer and centralizer direct limit groups. In this paper, we construct new families of examples of minimal equicontinuous actions on Cantor sets, which illustrate certain aspects of this classification. These examples are constructed as actions on rooted trees. The acting groups are countable subgroups of the product or of the wreath product of groups. We discuss applications of our results to the study of attractors of dynamical systems and of minimal sets of foliations.
Hiro-Aki Narita
Published: 1 January 2021
Journal of the Mathematical Society of Japan, Volume 73; https://doi.org/10.2969/jmsj/81168116

Toshio Mikami
Published: 1 January 2021
Journal of the Mathematical Society of Japan, Volume 73; https://doi.org/10.2969/jmsj/81928192

Michael Reiter, Duong Ngoc Son
Published: 1 January 2021
Journal of the Mathematical Society of Japan, Volume 73; https://doi.org/10.2969/jmsj/82598259

Yuya Matsumoto
Published: 1 January 2021
Journal of the Mathematical Society of Japan, Volume -1, pp 1-24; https://doi.org/10.2969/jmsj/86318631

Abstract:
Let $\bar{Y}$ be a normal surface that is the canonical $\mu_2$- or $\alpha_2$-covering of a classical or supersingular Enriques surface in characteristic 2. We determine all possible configurations of singularities on $\bar{Y}$, and for each configuration we describe which type of Enriques surfaces (classical or supersingular) appear as quotients of $\bar{Y}$.
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