Hilbert's Tenth Problem for function fields of varieties over C

希尔伯特第十问题在有限生成环上的情况

Hilbert's tenth problem for finitely generated rings

本文探讨了有限生成环上希尔伯特第十问题的最新进展，通过椭圆曲线理论中的问题连接整数上的负解结果，并概述了最近的解决方法。

Towards Hilbert's Tenth Problem for rings of integers through Iwasawa theory and Heegner points

For a positive proportion of primes $p$ and $q$, we prove that $\mathbb{Z}$ is Diophantine in the ring of integers of $\mathbb{Q}(\sqrt[3]{p},\sqrt{-q})$. This provides a new and explicit infinite family of number fields $K$ such that Hilbert's tenth problem for $O_K$ is unsolvable. Our methods use Iwasawa theory and congruences of Heegner points in order to obtain suitable rank stability properties for elliptic curves.

Hilbert's Tenth Problem for function fields of characteristic zero

In this article we outline the methods that are used to prove undecidability of Hilbert's Tenth Problem for function fields of characteristic zero. Following Denef we show how rank one elliptic curves can be used to prove undecidability for rational function fields over formally real fields. We also sketch the undecidability proofs for function fields of varieties over the complex numbers of dimension at least 2.

整数上的希尔伯特第十问题的十个未知数

Ten unknowns for Hilbert's tenth problem over the integers

本文改进了希尔伯特第十问题的相关结论，证明不存在算法可判定含十个未知数的整数系数多项式方程是否有整数解。

Hilbert's Tenth Problem and Mazur's Conjecture for large subrings of Q

We give the first examples of infinite sets of primes S such that Hilbert's Tenth Problem over Z[S^{-1}] has a negative answer. In fact, we can take S to be a density 1 set of primes. We show also that for some such S there is a punctured elliptic curve E' over Z[S^{-1}] such that the topological closure of E'(Z[S^{-1}]) in E'(R) has infinitely many connected components.

Hilbert's Tenth Problem for function fields over valued fields in characteristic zero

Let K be a field with a valuation satisfying the following conditions: both K and the residue field k have characteristic zero; the value group is not 2-divisible; there exists a maximal subfield F in the valuation ring such that Gal(\bar{F}/F) and Gal(\bar{k}/k) have the same 2-cohomological dimension and this dimension is finite. Then Hilbert's Tenth Problem has a negative answer for any function field of a variety over K. In particular, this result proves undecidability for varieties over C((T)).

通过加性组合学解决希尔伯特第十问题

Hilbert's tenth problem via additive combinatorics

本文通过构造无秩增长的椭圆曲线，结合加性组合学与 $2$-descent 方法，证明了对于所有无限环 $R$，若 $R$ 在 $\mathbb{Z}$ 上有限生成，则希尔伯特第十问题无解。

有理数域上希尔伯特第十问题的86个未知量

86 unknowns for Hilbert's tenth problem over the rationals

OpenAI已否定性解决有理数域上希尔伯特第十问题，本文进一步证明不存在算法判定含86个未知量的整数系数多项式在有理数域上是否有解。

Baire category theory and Hilbert's Tenth Problem inside $\mathbb{Q}$

For a ring R, Hilbert's Tenth Problem HTP(R) is the set of polynomial equations over R, in several variables, with solutions in R. We consider computability of this set for subrings R of the rationals. Applying Baire category theory to these subrings, which naturally form a topological space, relates their sets HTP(R) to the set HTP($\mathbb{Q}$), whose decidability remains an open question. The main result is that, for an arbitrary set C, HTP($\mathbb{Q}$) computes C if and only if the subrings R for which HTP(R) computes C form a nonmeager class. Similar results hold for 1-reducibility, for …

Hilbert's Tenth Problem for rational function fields over p-adic fields

Let K be a p-adic field (a finite extension of some Q_p) and let K(t) be the field of rational functions over K. We define a kind of quadratic reciprocity symbol for polynomials over K and apply it to prove isotropy for a certain class of quadratic forms over K(t). Using this result, we give an existential definition for the predicate "v_t(x) >= 0" in K(t). This implies undecidability of diophantine equations over K(t).

有界阶全纯函数环的希尔伯特第十问题

Hilbert's tenth problem for rings of holomorphic functions of bounded order

研究单变量复整函数环中希尔伯特第十问题类似问题，针对增长阶至多为\(\rho\)的情况给出否定解，为希尔伯特第十问题扩展研究提供了相关成果。

Hilbert's Tenth Problem over Function Fields of Positive Characteristic Not Containing the Algebraic Closure of a Finite Field

We prove that the existential theory of any function field $K$ of characteristic $p> 0$ is undecidable in the language of rings provided that the constant field does not contain the algebraic closure of a finite field. We also extend the undecidability proof for function fields of higher transcendence degree to characteristic 2 and show that the first-order theory of {\bf any} function field of positive characteristic is undecidable in the language of rings without parameters.

Hilbert's Tenth Problem and Mazur's Conjectures in Complementary Subrings of Number Fields

We show that Hilbert's Tenth Problem is undecidable for complementary subrings of number fields and that the p-adic and archimedean ring versions of Mazur's conjectures do not hold in these rings. More specifically, given a number field K, a positive integer t>1, and t nonnegative computable real numbers delta_1,..., delta_t whose sum is one, we prove that the nonarchimedean primes of K can be partitioned into t disjoint recursive subsets S_1,..., S_t of densities delta_1,..., delta_t, respectively such that Hilbert's Tenth Problem is undecidable for each corresponding ring O_{K,S_i}. We also …

虚二次域的 $\mathbb{Z}_p$-扩张族中的希尔伯特第十问题

Hilbert's tenth problem for families of $ \mathbb{Z}_p $-extensions of imaginary quadratic fields

本文利用Iwasawa理论，在虚二次域的$\mathbb{Z}_p$-扩张族中研究希尔伯特第十问题，确定了使所有有限层中该问题有否定答案的直线条件，并证明了$p=3$时正比例域满足条件，其他素数情形则依赖于Tate--Shafarevich群的消失。

关于黎曼猜想与希尔伯特第十问题

On the Riemann Hypothesis and Hilbert's Tenth Problem

本文概述了证明黎曼猜想为$Π_1^0$语句的既有方法，并尝试将Elliott-Halberstam猜想也归入此类。

As Easy as $\mathbb Q$: Hilbert's Tenth Problem for Subrings of the Rationals and Number Fields

Hilbert's Tenth Problem over the field $\mathbb Q$ of rational numbers is one of the biggest open problems in the area of undecidability in number theory. In this paper we construct new, computably presentable subrings $R$ of $\mathbb Q$ having the property that Hilbert's Tenth Problem for $R$, denoted $HTP(R)$, is Turing equivalent to $HTP(\mathbb Q)$. We are able to put several additional constraints on the rings $R$ that we construct. Given any computable nonnegative real number $r \leq 1$ we construct such a ring $R = Z[\frac1p : p \in S]$ with $S$ a set of primes of lower density $r$. We …

二次扩张中的秩稳定性及数域整数环的希尔伯特第十问题

Rank stability in quadratic extensions and Hilbert's tenth problem for the ring of integers of a number field

本文证明了任意数域二次扩张中存在基变换下秩不增长的正秩 Abel 簇，由此推出任意数域整数环上的希尔伯特第十问题具有否定解答。

某些非交换环上的希尔伯特第十问题

Hilbert's Tenth Problem for some Noncommutative Rings

本文研究两类非交换环上的希尔伯特第十问题，证明扭多项式环及微分多项式环等情形具有否定答案，并建立相关丢番图集与归约关系。

丢番图映射

Diophantine Maps

本文提出丢番图映射与丢番图等价映射概念，比较丢番图与递归情形，并证明有效丢番图映射可在环之间传递 Hilbert 第十问题的肯定或否定答案。

Numerical simulations of a quantum algorithm for Hilbert's tenth problem

We employ quantum mechanical principles in the computability exploration of the class of classically noncomputable Hilbert's tenth problem which is equivalent to the Turing halting problem in Computer Science. The Quantum Adiabatic Theorem enables us to establish a connection between the solution for this class of problems and the asymptotic behaviour of solutions of a particular type of time-dependent Schrödinger equations. We then present some preliminary numerical simulation results for the quantum adiabatic processes corresponding to various Diophantine equations.

数域反分圆塔中的希尔伯特第十问题

Hilbert's tenth problem in Anticyclotomic towers of number fields

本研究在虚二次域上满足p-挠模同构的一对椭圆曲线的显式条件下，证明其反分圆Zp-扩张是整丢番图的，从而得到该扩张内所有数域上希尔伯特第十问题不可解的新情形，并给出p=3、K=Q(√-5)的实例。

A Note on Hilbert's "Geometric" Tenth Problem

This paper explores undecidability in theories of positive characteristic function fields in the "geometric" language of rings $\mathcal{L}_F = \{0, 1, +, \cdot, F\}$, with a unary predicate $F$ for nonconstant elements. In particular we are motivated by a question of Fehm on the decidability of $\mbox{Th}_{\exists}(\mathbb{F}_p(t); \mathcal{L}_F)$; equivalently, that of $\mbox{Th}_{\exists}(\mathbb{F}_p(t); \mathcal{L}_r)$ without parameters. We indicate how to generalise existing machinery to prove the undecidability of $\mbox{Th}_{\forall^1\exists}(K; \mathcal{L}_F)$ without parameters, whe…

对角二次型方程组的希尔伯特第十问题与 Büchi 问题

Hilbert's tenth problem for systems of diagonal quadratic forms, and Büchi's problem

该研究通过证明五个平方数在二阶差分恒为2时必连续的断言，补全 Büchi 证明，确立对角二次型方程组整数可解性的不可判定性。

The Hilbert's-Tenth-Problem Operator

For a ring $R$, Hilbert's Tenth Problem $HTP(R)$ is the set of polynomial equations over $R$, in several variables, with solutions in $R$. We view $HTP$ as an operator, mapping each set $W$ of prime numbers to $HTP(\mathbb Z[W^{-1}])$, which is naturally viewed as a set of polynomials in $\mathbb Z[X_1,X_2,\ldots]$. For $W=\emptyset$, it is a famous result of Matiyasevich, Davis, Putnam, and Robinson that the jump $\emptyset~\!'$ is Turing-equivalent to $HTP(\mathbb Z)$. More generally, $HTP(\mathbb Z[W^{-1}])$ is always Turing-reducible to $W'$, but not necessarily equivalent. We show here th…

Quantum Algorithm for Hilbert's Tenth Problem

We explore in the framework of Quantum Computation the notion of {\em Computability}, which holds a central position in Mathematics and Theoretical Computer Science. A quantum algorithm for Hilbert's tenth problem, which is equivalent to the Turing halting problem and is known to be mathematically noncomputable, is proposed where quantum continuous variables and quantum adiabatic evolution are employed. If this algorithm could be physically implemented, as much as it is valid in principle--that is, if certain hamiltonian and its ground state can be physically constructed according to the propo…

希尔伯特第十问题的局部-整体方法综述

A survey of local-global methods for Hilbert's Tenth Problem

本文综述数域（尤其 $\mathbb{Q}$）上 H10 的研究，梳理利用二次型与中心单代数局部-整体原理的近期进展，并展望模型论与算术几何两个方向。

Further results on Hilbert's Tenth Problem

Hilbert's Tenth Problem (HTP) asks for an algorithm to test whether an arbitrary polynomial Diophantine equation with integer coefficients has solutions over the ring $\mathbb Z$ of the integers. This was finally solved by Matiyasevich negatively in 1970. In this paper we obtain some further results on HTP over $\mathbb Z$. We prove that there is no algorithm to determine for any $P(z_1,\ldots,z_9)\in\mathbb Z[z_1,\ldots,z_9]$ whether the equation $P(z_1,\ldots,z_9)=0$ has integral solutions with $z_9\ge0$. Consequently, there is no algorithm to test whether an arbitrary polynomial Diophantine…

Extensions in the cohomology of Hilbert modular varieties

We describe extension classes arising in the $\ell$-adic and Hodge cohomology of Hilbert modular varieties, generalising results of Caspar to arbitrary dimensions. We show that this description is consistent with the "plectic conjectures" of Nekovář and the second author.

Plectic Lie代数在Hilbert模簇上的作用

Plectic Lie Algebra Action on Hilbert modular varieties

本文构造了Hilbert模簇局部解析完备上同调上plectic Galois作用的一部分，证明了中维度以下完备上同调消失，并使用了部分Sen算子。

Galois representations and torsion in the coherent cohomology of Hilbert modular varieties

Let $F$ be a totally real number field and let $p$ be a prime unramified in $F$. We prove the existence of Galois pseudo-representations attached to mod $p^m$ Hecke eigenclasses of paritious weight occurring in the coherent cohomology of Hilbert modular varieties for $F$ of level prime to $p$.

Hilbert specialization of parametrized varieties

Hilbert specialization is an important tool in Field Arithmetic and Arithmetic Geometry, which has usually been intended for polynomials, hence hypersurfaces, and at scalar values. In this article, first, we extend this tool to prime ideals, hence affine varieties, and offer an application to the study of the irreducibility of the intersection of varieties. Then, encouraged by recent results, we consider the more general situation in which the specialization is done at polynomial values, instead of scalar values.

Cohomology vanishing for Ekedahl--Oort strata on Hilbert--Blumenthal Shimura varieties

We prove vanishing results for the $0$th cohomology of automorphic line bundles on the Ekedahl--Oort strata of the special fiber of Hilbert--Blumenthal Shimura varieties. This work vastly extends to all Ekedahl--Oort strata previous results of Diamond--Kassaei and Goldring--Koskivirta. Furthermore, our results also cover the case of unitary Shimura varieties of rank $\leq 2$ attached to an arbitrary CM extension. We also prove a conjecture of Goldring--Koskivirta on the set of possible weights of automorphic forms.

希尔伯特模簇中低度完备上同调

Completed cohomology of Hilbert modular varieties below middle degree

研究希尔伯特模簇低度完备上同调的局部解析向量，证明通用包络代数作用需通过商分解，并给出维数界与稠密性结果。

Iwahori级Hilbert模簇的紧化

Compactifications of Iwahori-level Hilbert modular varieties

本文研究Iwahori级Hilbert模簇p-整模型的极小与环面紧化，将理论推广到更精细的级结构，并应用紧化结果分析Hilbert模形式的q-展开及Hecke算子作用。

Hyperbolic distribution problems on Siegel 3-folds and Hilbert modular varieties

We generalize to Hilbert modular varieties of arbitrary dimension the work of W. Duke (Inventiones 1988) on the equidistribution of Heegner points and of primitive positively oriented closed geodesics in the Poincare upper half plane, subject to certain subconvexity results. We also prove vanishing results for limits of cuspidal Weyl sums associated with analogous problems for the Siegel upper half space of degree 2. In particular, these Weyl sums are associated with families of Humbert surfaces in Siegel 3-folds and of modular curves in these Humbert surfaces.

Motivic action on coherent cohomology of Hilbert modular varieties

We propose an action of a certain motivic cohomology group on the coherent cohomology of Hilbert modular varieties, extending conjectures of Venkatesh, Prasanna, and Harris. The action is described in two ways: on cohomology modulo $p$ and over $\mathbb C$, and we conjecture that they both lift to an action on cohomology with integral coefficients. The conjecture is supported by theoretical evidence based on Stark's conjecture on special values of Artin $L$-functions, and by numerical evidence in base change cases.

论带挠系数的希尔伯特模簇的平展上同调

On the étale cohomology of Hilbert modular varieties with torsion coefficients

本文研究带挠系数的希尔伯特模簇平展上同调，借鉴酉志村簇方法得到一般情形下的中间次数集中零化定理与非一般情形的次数界，借助几何Jacquet--Langlands函子性证明并应用于p进局部朗兰兹对应。

关于与 Hilbert 模形式相关的 Abel 簇的自同态代数

On the Endomorphism Algebra of Abelian Varieties Associated with Hilbert Modular Forms

本文在基域相对于 Q 的次数为奇数的条件下，将 Quer 关于权 2 模形式对应 Abel 簇自同态代数 Brauer 类的显式公式推广至平行权 2 的 Hilbert 模形式。

Finite descent obstruction for Hilbert modular varieties

Let $S$ be a finite set of primes. We prove that a form of finite Galois descent obstruction is the only obstruction to the existence of $\mathbb{Z}_{S}$-points on integral models of Hilbert modular varieties, extending a result of D.Helm and F.Voloch about modular curves. Let $L$ be a totally real field. Under (a special case of) the absolute Hodge conjecture and a weak Serre's conjecture for mod $\ell$ representations of the absolute Galois group of $L$, we prove that the same holds also for the $\mathcal{O}_{L,S}$-points.

Overconvergent Hilbert modular forms via perfectoid modular varieties

We give a new construction of $p$-adic overconvergent Hilbert modular forms by using Scholze's perfectoid Shimura varieties at infinite level and the Hodge--Tate period map. The definition is analytic, closely resembling that of complex Hilbert modular forms as holomorphic functions satisfying a transformation property under congruence subgroups. As a special case, we first revisit the case of elliptic modular forms, extending recent work of Chojecki, Hansen and Johansson. We then construct sheaves of geometric Hilbert modular forms, as well as subsheaves of integral modular forms, and vary ou…

Abelian Varieties and Galois Extensions of Hilbertian Fields

In a recent paper, Moshe Jarden proposed a conjecture, later named the Kuykian conjecture, which states that if A is an abelian variety defined over a Hilbertian field K, then every intermediate field of K(A_{tor})/K is Hilbertian. We prove that the conjecture holds for Galois extensions of K in K(A_{tor}).

Hilbert 模簇的控制定理

Control Theorems for Hilbert Modular Varieties

本文在Hida理论意义下，利用Loeffler-Rockwood-Zerbes技术与Caraiani-Tamiozzo的消失定理，证明了Hilbert模簇中间阶上同调普通部分的精确控制定理，为后续研究Asai-Flach Euler系统在Hida族中的变化奠定基础。

Partial Hasse invariants on splitting models of Hilbert modular varieties

Let $F$ be a totally real field of degree $g$, and let $p$ be a prime number. We construct $g$ partial Hasse invariants on the characteristic $p$ fiber of the Pappas-Rapoport splitting model of the Hilbert modular variety for $F$ with level prime to $p$, extending the usual partial Hasse invariants defined over the Rapoport locus. In particular, when $p$ ramifies in $F$, we solve the problem of lack of partial Hasse invariants. Using the stratification induced by these generalized partial Hasse invariants on the splitting model, we prove in complete generality the existence of Galois pseudo-re…

On certain cohomology groups attached to $\mathfrak{p}^{\infty}$-towers of quaternionic Hilbert modular varieties

For a totally real number field $F$ and a nonarchimedean prime $\mathfrak{p}$ of $F$ lying above a prime number $p$ we introduce certain sheaf cohomology groups that intertwine the $\mathfrak{p}^{\infty}$-tower of a quaternionic Hilbert modular variety associated to a quaternion algebra $D$ over $F$ that is split at $\mathfrak{p}$ and a $p$-adically admissible representation of $\mbox{PGL}_2(F_{\mathfrak{p}})$. Applied to infinitesimal $p$-adic deformations of the local factor at $\mathfrak{p}$ of a cuspidal automorphic representation $π$ of $D^*(\mathbb{A})$ this yields a natural construction…

On Ihara's lemma for Hilbert Modular Varieties

Let ρbe a modulo p representation of the absolute Galois group of a totally real number field. Under the assumptions that ρhas large image and admits a low weight crystalline modular deformation we show that any low weight crystalline deformation of ρunramified outside a finite set of primes will be modular. We follow the approach of Wiles as generalized by Fujiwara. The main new ingredient is an Ihara type lemma for the local component at ρof the middle degree cohomology of a Hilbert modular variety. As an application we relate the algebraic p-part of the value at 1 of the adjoint L-function …

模 p Jacquet-Langlands 关系与通过 Hilbert 模簇几何的 Serre 滤过：拼接与切分

A mod p Jacquet-Langlands relation and Serre filtration via the geometry of Hilbert modular varieties: Splicing and dicing

本文通过构造特征 p 下 Hilbert 模簇的几何 Jacquet-Langlands 关系，建立模 p Hilbert 模形式与四元数模形式之间的联系，推广 Serre 的经典结果，并证明上同调消失定理以关联 Galois 表示。

特征零函数域上阿贝尔簇的希尔伯特不可约性

Hilbert irreducibility for abelian varieties over function fields of characteristic zero

该研究证明了特征零函数域上阿贝尔簇的希尔伯特不可约定理，完善了相关数论与代数几何理论。

Integral points on Hilbert moduli schemes

We use the method of Faltings (Arakelov, Paršin, Szpiro) in order to explicitly study integral points on a class of varieties over $\mathbb Z$ called Hilbert moduli schemes. For instance, integral models of Hilbert modular varieties are classical examples of Hilbert moduli schemes. Our main result gives explicit upper bounds for the height and the number of integral points on Hilbert moduli schemes.

具有nef切丛的簇的Hilbert不可约性

Hilbert irreducibility for varieties with a nef tangent bundle

本文证明了具有Hilbert型性质的簇的纤维化性质，并将其应用于研究具有nef切丛的簇上的有理点。

Canonical Subgroups over Hilbert Modular Varieties

We obtain new results on the geometry of Hilbert modular varieties in positive characteristic and morphisms between them. Using these results and methods of rigid geometry, we develop a theory of canonical subgroups for abelian varieties with real multiplication.

Hilbert property for double conic bundles and del Pezzo varieties

In this paper we prove that, over a Hilbertian ground field, surfaces with two conic fibrations whose fibres have non-zero intersection product have the Hilbert property. We then give an application of this result, namely the verification of the Hilbert property for certain del Pezzo varieties.

Tautological rings of Hilbert modular varieties

We compute the tautological ring for a Hilbert modular variety at an unramified prime. The method generalises that of van der Geer from the Siegel case.

Lifts of Hilbert modular forms and application to modularity of abelian varieties

In this paper, we prove the existence of certain lifts of Hilbert cusp forms to general odd spin groups. We then use those lifts to provide evidence for a conjecture of Gross on the modularity of abelian varieties not of ${\rm GL}_2$-type.

有理点存在性的有效性是不可判定的

Effectivity for existence of rational points is undecidable

该研究针对数论中不可判定性领域的开放问题，证明了带有有限个高度比较条件的有理数域上的希尔伯特第十问题是不可判定的，推进了有理点存在性相关的有效性问题研究。