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Shin Hattori

Publications and source records attributed to Shin Hattori.

15 recordsLinked to original sources
arXiv2026

On some constancy of Hecke eigensystems for Drinfeld cuspforms of finite slope

Let $p$ be a rational prime, let $q>1$ be a $p$-power integer, let $\mathbb{F}_q$ be the field of $q$ elements and let $A=\mathbb{F}_q[t]$ be the polynomial ring over $\mathbb{F}_q$. Let $\mathfrak{n}\in A$ be a nonzero element and let $\wp\in A$ be a monic irreducible polynomial of positive degree. Let $k\geq 2$ and $r\geq 1$ be integers. Let $S_k(Γ_1(\mathfrak{n}\wp^r))$ be the space of Drinfeld cuspforms of level $Γ_1(\mathfrak{n}\wp^r)$ and weight $k$. In this paper, we prove that the multiplicity of a Hecke eigensystem of finite $\wp$-slope in $S_k(Γ_1(\mathfrak{n}\wp^r))$ is equal to $q^{(r-1)\mathrm{deg}(\wp)}$ times that in $S_k(Γ_1(\mathfrak{n}\wp))$. In particular, this shows that a Hecke eigensystem of finite $\wp$-slope appears in $S_k(Γ_1(\mathfrak{n}\wp^r))$ if and only if it appears in $S_k(Γ_1(\mathfrak{n}\wp))$.

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arXiv2026

On autoduality of Drinfeld modules and Drinfeld modular forms

Let $\mathbb{F}_q$ be the field of $q$ elements and let $A=\mathbb{F}_q[t]$ be the polynomial ring over $\mathbb{F}_q$. Let $\mathfrak{n}\in A\setminus \mathbb{F}_q$ be a monic polynomial with a prime factor of degree prime to $q-1$. Let $Δ$ be a subgroup of $(A/(\mathfrak{n}))^\times$ such that the map $Δ\to (A/(\mathfrak{n}))^\times/\mathbb{F}_q^\times$ is bijective. Let $S$ be a scheme over $A[1/\mathfrak{n}]$ and let $R$ be an $A[1/\mathfrak{n}]$-algebra which is an excellent regular domain. In this paper, we show that any Drinfeld module $E$ of rank two over $S$ admitting a $Γ_1^Δ(\mathfrak{n})$-structure is isomorphic to its Taguchi dual $E^D$. As an application, for the Hodge bundle $\barω$ on the Drinfeld modular curve $X$ of level $Γ_1^Δ(\mathfrak{n})$ over $R$, we give a dual Kodaira--Spencer isomorphism of the form $\barω^{\otimes 2}\simeq Ω^1_{X/R}(2\mathrm{Cusps})$, in contrast with the usual one in the Drinfeld case in which $E^D$ is involved.

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arXiv2024

$\mathscr{D}$-elliptic sheaves and the Hasse principle

Let $p$ be a rational prime, $q>1$ a power of $p$ and $F=\mathbb{F}_q(t)$. For an integer $d\geq 2$, let $D$ be a central division algebra over $F$ of dimension $d^2$ which is split at $\infty$ and has invariant $\mathrm{inv}_x(D)=1/d$ at any place $x$ of $F$ at which $D$ ramifies. Let $X^D$ be the Drinfeld--Stuhler variety, the coarse moduli scheme of the algebraic stack over $F$ classifying $\mathscr{D}$-elliptic sheaves. In this paper, we establish various arithmetic properties of $\mathscr{D}$-elliptic sheaves to give an explicit criterion for the non-existence of rational points of $X^D$ over a finite extension of $F$ of degree $d$. As an application, for $d=2$, we present explicit infinite families of quadratic extensions of $F$ over which the curve $X^D$ violates the Hasse principle.

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arXiv2019

$\wp$-adic continuous families of Drinfeld eigenforms of finite slope

Let $p$ be a rational prime, $v_p$ the normalized $p$-adic valuation on $\mathbb{Z}$, $q>1$ a $p$-power and $A=\mathbb{F}_q[t]$. Let $\wp\in A$ be an irreducible polynomial and $\mathfrak{n}\in A$ a non-zero element which is prime to $\wp$. Let $k\geq 2$ and $r\geq 1$ be integers. We denote by $S_k(Γ_1(\mathfrak{n}\wp^r))$ the space of Drinfeld cuspforms of level $Γ_1(\mathfrak{n}\wp^r)$ and weight $k$ for $A$. Let $n\geq 1$ be an integer and $a\geq 0$ a rational number. Suppose that $\mathfrak{n}\wp$ has a prime factor of degree one and the generalized eigenspace in $S_k(Γ_1(\mathfrak{n}\wp^r))$ of slope $a$ is one-dimensional. In this paper, under an assumption that $a$ is sufficiently small, we construct a family $\{F_{k'}\mid v_p(k'-k)\geq \log_p(p^n+a)\}$ of Hecke eigenforms $F_{k'}\in S_{k'}(Γ_1(\mathfrak{n}\wp^r))$ of slope $a$ such that, for any $Q\in A$, the Hecke eigenvalues of $F_k$ and $F_{k'}$ at $Q$ are congruent modulo $\wp^κ$ with some $κ>p^{v_p(k'-k)}-p^n-a$.

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arXiv2018

Dimension variation of Gouvêa-Mazur type for Drinfeld cuspforms of level $Γ_1(t)$

Let $p$ be a rational prime and $q>1$ a $p$-power. Let $S_k(Γ_1(t))$ be the space of Drinfeld cuspforms of level $Γ_1(t)$ and weight $k$ for $\mathbb{F}_q[t]$. For any non-negative rational number $α$, we denote by $d(k,α)$ the dimension of the slope $α$ generalized eigenspace for the $U$-operator acting on $S_k(Γ_1(t))$. In this paper, we prove a function field analogue of the Gouvêa-Mazur conjecture for this setting. Namely, we show that for any $α\leq m$ and $k_1,k_2>α+1$, if $k_1\equiv k_2 \bmod p^m$, then $d(k_1,α)=d(k_2,α)$.

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arXiv2017

Duality of Drinfeld modules and $\wp$-adic properties of Drinfeld modular forms

Let $p$ be a rational prime and $q$ a power of $p$. Let $\wp$ be a monic irreducible polynomial of degree $d$ in $\mathbf{F}_q[t]$. In this paper, we define an analogue of the Hodge-Tate map which is suitable for the study of Drinfeld modules over $\mathbf{F}_q[t]$ and, using it, develop a geometric theory of $\wp$-adic Drinfeld modular forms similar to Katz's theory in the case of elliptic modular forms. In particular, we show that for Drinfeld modular forms with congruent Fourier coefficients at $\infty$ modulo $\wp^n$, their weights are also congruent modulo $(q^d-1)p^{\lceil \log_p(n)\rceil}$, and that Drinfeld modular forms of level $Γ_1(\mathfrak{n})\cap Γ_0(\wp)$, weight $k$ and type $m$ are $\wp$-adic Drinfeld modular forms for any tame level $\mathfrak{n}$ with a prime factor of degree prime to $q-1$.

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arXiv2017

Irreducible components of the eigencurve of finite degree are finite over the weight space

Let p be a rational prime and N a positive integer which is prime to p. Let W be the p-adic weight space for GL_{2,Q}. Let C_N be the p-adic Coleman-Mazur eigencurve of tame level N. In this paper, we prove that any irreducible component of C_N which is of finite degree over W is in fact finite over W. Combined with an argument of Chenevier and a conjecture of Coleman-Mazur-Buzzard-Kilford (which has been proven in special cases, and for general quaternionic eigencurves) this shows that the only finite degree components of the eigencurve are the ordinary components.

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arXiv2013

Ramification theory and perfectoid spaces

Let K and F be complete discrete valuation fields of residue characteristic p>0. Let m be a positive integer no more than their absolute ramification indices. Let s and t be their uniformizers. Let L/K and E/F be finite extensions such that the modulo s^m of the extension O_L/O_K and modulo t^m of O_E/O_F are isomorphic. Let j=<m be a positive rational number. In this paper, we prove that the ramification of L/K is bounded by j if and only if the ramification of E/F is bounded by j. As an application, we prove that the categories of finite separable extensions of K and F whose ramifications are bounded by j are equivalent to each other, which generalizes a theorem of Deligne to the case of imperfect residue fields. We also show the compatibility of Scholl's theory of higher fields of norms with the ramification theory of Abbes-Saito, and the integrality of small Artin and Swan conductors of abelian extensions of mixed characteristic.

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arXiv2012

Canonical subgroups via Breuil-Kisin modules

Let p>2 be a rational prime and K/Q_p be an extension of complete discrete valuation fields. Let G be a truncated Barsotti-Tate group of level n, height h and dimension d over O_K with 0<d<h. In this paper, we show that an upper ramification subgroup G^j+ is free of rank d over Z/p^nZ if the Hasse invariant of G is less than 1/(2p^(n-1)). We also prove the usual properties as the canonical subgroup.

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arXiv2012

On lower ramification subgroups and canonical subgroups

Let p be a rational prime, k be a perfect field of characteristic p and K be a finite totally ramified extension of the fraction field of the Witt ring of k. Let G be a finite flat commutative group scheme over O_K killed by some p-power. In this paper, we prove a description of ramification subgroups of G via the Breuil-Kisin classification, generalizing the author's previous result on the case where G is killed by p>2. As an application, we also prove that the higher canonical subgroup of a level n truncated Barsotti-Tate group G over O_K coincides with lower ramification subgroups of G if the Hodge height of G is less than (p-1)/p^n.

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arXiv2012

Canonical subgroups via Breuil-Kisin modules for p=2

Let p be a rational prime and K/Q_p be an extension of complete discrete valuation fields. Let G be a truncated Barsotti-Tate group of level n, height h and dimension d over O_K with 0<d<h. In this paper, we prove the existence of higher canonical subgroups with expected properties for G if the Hodge height of G is less than 1/(p^{n-2}(p+1)), including the case of p=2.

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arXiv2012

Ramification correspondence of finite flat group schemes over equal and mixed characteristic local fields

Let p>2 be a rational prime, k be a perfect field of characteristic p and K be a finite totally ramified extension of the fractional field of the Witt ring of k. Let G and H be finite flat commutative group schemes killed by p over O_K and k[[u]], respectively. In this paper, we show the upper and the lower ramification subgroups of G and H in the sense of Abbes-Saito are naturally isomorphic to each other when they are associated to the same Kisin module.

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arXiv2009

On a ramification bound of torsion semi-stable representations over a local field

For a rational prime p, let k be a perfect field of characteristic p, K be a finite totally ramified extension of Frac(W(k)) of degree e and r be a non-negative integer satisfying r u(K,r,n) acts trivially on the p^n-torsion semi-stable G_K-representations with the Hodge-Tate weights in {0,...,r}, where u(K,0,n)=0, u(K,1,n)=1+e(n+1/(p-1)) and u(K,r,n)=1-p^{-n}+e(n+r/(p-1)) for r>1.

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