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Seidai Yasuda

Publications and source records attributed to Seidai Yasuda.

15 recordsLinked to original sources
arXiv2025

Semi-stable representations as limits of crystalline representations

We construct an explicit sequence $V_{k_n,a_n}$ of crystalline representations of exceptional weights converging to a given irreducible two-dimensional semi-stable representation $V_{k,{\mathcal{L}}}$ of $\mathrm{Gal}({\overline{\mathbb{Q}}}_p/{\mathbb{Q}}_p)$. The convergence takes place in the blow-up space of two-dimensional trianguline representations studied by Colmez and Chenevier. The process of blow-up is described in detail in the rigid analytic setting and may be of independent interest. Also, we recover a formula of Stevens expressing the ${\mathcal{L}}$-invariant as a logarithmic derivative. Our result can be used to compute the reduction of $V_{k,{\mathcal{L}}}$ in terms of the reductions of the $V_{k_n,a_n}$. For instance, using the zig-zag conjecture we recover (resp. extend) the work of Breuil-Mézard and Guerberoff-Park computing the reductions of the $V_{k,{\mathcal{L}}}$ for weights at most $p-1$ (resp. $p+1$), at least on the inertia subgroup. In the cases where zig-zag is known, we are further able to obtain some new information about the reductions for small odd weights. Finally, we explain some apparent violations to local constancy in the weight of the reductions of crystalline representations of small weight.

math.NT↗
arXiv2022

Local newforms for the general linear groups over a non-archimedean local field

In [12], Jacquet--Piatetskii-Shapiro--Shalika defined a family of compact open subgroups of $p$-adic general linear groups indexed by non-negative integers, and established the theory of local newforms for irreducible generic representations. In this paper, we extend their results to all irreducible representations. To do this, we define a new family of compact open subgroups indexed by certain tuples of non-negative integers. For the proof, we introduce the Rankin--Selberg integrals for Speh representations.

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arXiv2021

Distributions and Euler systems for the general linear group

The main aim is to give a rigorous statement and proof of the slogan "the d-fold tensor product of distributions is an Euler system for GL_d". Of the few known examples of Euler systems, we look at those of cyclotomic units and of Beilinson-Kato elements. The cyclotomic units satisfy distribution property, and this is the key to the proof of the norm relation property for GL_1. For the Beilinson-Kato elements, the Siegel units satisfy distribution property, and the 2-fold tensor product, giving rise to elements in the K_2 of modular curves, satisfies the norm relation for GL_2. We make this common property clear, generalizing everything to GL_d. As an application (our main arithmetic result), we construct elements in the motivic cohomology of Drinfeld modular schemes with integral coefficient and show that the norm relation common to Euler systems (i.e., the norm of one element is described using the local L-factor times another element) hold. We use the language of Y-sites which was introduced in [Kon-Ya3] in order to simplify the computation common to the theory of automorphic forms. Instead of double cosets, we work more systematically with torsion modules and Q-morphisms between them. The idea is that torsion modules are level structures, and Q-morphisms induce morphisms between some moduli spaces. Chapters 1 and 3 serve as a sequel: further generalities on Y-sites, more examples of Y-sites, and proofs of some statements in loc.cit are given. An application is also given: we provide a group theoretic formulation of a conjecture of Tamagawa on affine curves over an algebraic closure of a finite field.

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arXiv2021

Arithmetic quotients of the Bruhat-Tits building for projective general linear group in positive characteristic

Let $d \ge 1$. We study a subspace of the space of automorphic forms of $\mathrm{GL}_d$ over a global field of positive characteristic (or, a function field of a curve over a finite field). We fix a place $\infty$ of $F$, and we consider the subspace $\mathcal{A}_{\mathrm{St}}$ consisting of automorphic forms such that the local component at $\infty$ of the associated automorphic representation is the Steinberg representation (to be made precise in the text). We have two results. One theorem (Theorem 16) describes the constituents of $\mathcal{A}_{\mathrm{St}}$ as automorphic representation and gives a multiplicity one type statement. For the other theorem (Theorem 12), we construct, using the geometry of the Bruhat-Tits building, an analogue of modular symbols in $\mathcal{A}_{\mathrm{St}}$ integrally (that is, in the space of $\mathbb{Z}$-valued automorphic forms). We show that the quotient is finite and give a bound on the exponent of this quotient.

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arXiv2020

Category of mixed plectic Hodge structures

The purpose of this article is to investigate the properties of the category of mixed plectic Hodge structures defined by Nekovář and Scholl. We give an equivalent description of mixed plectic Hodge structures in terms of the weight and partial Hodge filtrations. We also construct an explicit complex calculating the extension groups in this category.

math.AG↗
arXiv2019

Regularity of quotients of Drinfeld modular schemes

Let $A$ be the coordinate ring of a projective smooth curve over a finite field minus a closed point. For a nontrivial ideal $I \subset A$, Drinfeld defined the notion of structure of level $I$ on a Drinfeld module. We extend this to that of level $N$, where $N$ is a finitely generated torsion $A$-module. The case where $N=(I^{-1}/A)^d$, where $d$ is the rank of the Drinfeld module,coincides with the structure of level $I$. The moduli functor is representable by a regular affine scheme. The automorphism group $\mathrm{Aut}_{A}(N)$ acts on the moduli space. Our theorem gives a class of subgroups for which the quotient of the moduli scheme is regular. Examples include generalizations of $Γ_0$ and of $Γ_1$. We also show that parabolic subgroups appearing in the definition of Hecke correspondences are such subgroups.

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arXiv2018

Belyi's theoerm in characteristic two

We prove an analogue of Belyi's theorem in characteristic two. Our proof consists of the following three steps. We first introduce a new notion called "pseudo-tame" for morphisms between curves over an algebraically closed field of characteristic two. Secondly, we prove the existence of a "pseudo-tame" rational function by proving vanishing of an obstruction class. Finally we will construct a tamely ramified rational function from the "pseudo-tame" rational function.

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arXiv2017

First and second $K$-groups of an elliptic curve over a global field of positive characteristic

In this paper, we show that the maximal divisible subgroup of groups $K_1$ and $K_2$ of an elliptic curve $E$ over a function field is uniquely divisible. Further those $K$-groups modulo this uniquely divisible subgroup are explicitly computed. We also calculate the motivic cohomology groups of the minimal regular model of $E$, which is an elliptic surface over a finite field.

math.KT↗
arXiv2017

Sites whose topoi are the smooth representations of locally profinite groups

We define a class of sites such that the associated topos is equivalent to the category of smooth sets (representations) of some locally prodiscrete monoids (to be defined). Examples of locally prodiscrete monoids include profinite groups and finite adele valued points of algebraic groups. This is a generalization of the fact that the topos associated with the étale site of a scheme is equivalent to the category of sets with continuous action by the étale fundamental group. We then define a subclass of sites such that the topos is equivalent to the category of discrete sets with a continuous action of a locally profinite group.

math.NT↗
arXiv2014

What makes a multi-complex exact?

In this paper, we give a sufficient condition which makes the total complex of a cube exact. This can be regarded as a variant of the Buchsbaum-Eisenbud theorem which gives a characterization of what makes a complex of finitely generated free modules exact in terms of the grade of the Fitting ideals of boundary maps of the complex.

math.AC↗
arXiv2014

The Borel-Moore homology of an arithmetic quotient of the Bruhat-Tits building of PGL of a non-archimedean local field in positive characteristic and modular symbols

We study the homology and the Borel-Moore homology with coefficients in $\mathbb{Q}$ of a quotient (called arithmetic quotient) of the Bruhat-Tits building of $\mathrm{PGL}$ of a nonarchimedean local field of positive characteristic by an arithmetic subgroup (a special case of the general definition in Harder's article (Invent.\ Math.\ 42, 135-175 (1977)). We define an analogue of modular symbols in this context and show that the image of the canonical map from homology to Borel-Moore homology is contained in the sub $\mathbb{Q}$-vector space generated by the modular symbols. By definition, the limit of the Borel-Moore homology as the arithmetic group becomes small is isomorphic to the space of $\mathbb{Q}$-valued automorphic forms that satisfy certain conditions at a distinguished (fixed) place (namely that it is fixed by the Iwahori subgroup and the center at the place). We show that the limit of the homology with $\mathbb{C}$-coefficients is identified with the subspace consisting of cusp forms. We also describe an irreducible subquotient of the limit of Borel-Moore homology as an induced representation in a precise manner and give a multiplicity one type result.

math.NT↗
arXiv2007

On Haagerup's list of potential principal graphs of subfactors

We show that any graph, in the sequence given by Haagerup in 1991 as that of candidates of principal graphs of subfactors, is not realized as a principal graph except for the smallest two. This settles the remaining case of a previous work of the first author.

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