Given a field $k$ of characteristic zero and an indeterminate $T$, the main topic of the paper is the construction of specializations of any given finite extension of $k(T)$ of degree $n$ that are degree $n$ field extensions of $k$ with specified local behavior at any given finite set of primes of $k$. First, we give a full non-Galois analog of a result with a ramified type conclusion from a preceding paper and next we prove a unifying statement which combines our results and previous work devoted to the unramified part of the problem in the case $k$ is a number field.
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.
Given a finite group $G$ and a number field $K$, we investigate the following question: Does there exist a Galois extension $E/K(t)$ with group $G$ whose set of specializations yields solutions to all Grunwald problems for the group $G$, outside a finite set of primes? Following previous work, such a Galois extension would be said to have the "Hilbert-Grunwald property". In this paper we reach a complete classification of groups $G$ which admit an extension with the Hilbert-Grunwald property over fields such as $K=\mathbb{Q}$. We thereby also complete the determination of the ``local dimension…
The purpose of this paper is to investigate the number of classical weight one specializations of a non-CM ordinary Hida family of parallel weight Hilbert cusp forms. It is known that a specialization of a primitive ordinary Hida family at any arithmetic points of weight at least two is a classical holomorphic Hilbert cusp form. However, this is not always the case for weight one specializations. Balasubramanyam, Ghate and Vatsal proved that such a Hida family admits infinitely many classical weight one specializations if and only if it is of CM type. We give an explicit upper bound on the num…
Integral Specializations of Polynomials: Hilbert Irreducibility, Coprimality, and Density
Institute of Science Tokyo（东京科学大学）；Univ. Lille, CNRS, UMR 8524, Laboratoire Paul Painlevé（里尔大学，法国国家科学研究中心，联合研究实验室8524，保罗·潘勒瓦实验室）；TU Graz, Institute of Analysis and Number Theory（格拉茨工业大学，分析与数论研究所）
We establish a version "over the ring" of the celebrated Hilbert Irreducibility Theorem. Given finitely many polynomials in $k+n$ variables, with coefficients in $\mathbb Z$, of positive degree in the last $n$ variables, we show that if they are irreducible over $\mathbb Z$ and satisfy a necessary "Schinzel condition", then the first $k$ variables can be specialized in a Zariski-dense subset of ${\mathbb Z}^k$ in such a way that irreducibility over ${\mathbb Z}$ is preserved for the polynomials in the remaining $n$ variables. The Schinzel condition, which comes from the Schinzel Hypothesis, is…
Given a field $k$ of characteristic zero and an indeterminate $T$ over $k$, we investigate the local behaviour at primes of $k$ of finite Galois extensions of $k$ arising as specializations of finite Galois extensions $E/k(T)$ (with $E/k$ regular) at points $t_0 \in \mathbb{P}^1(k)$. We provide a general result about decomposition groups at primes of $k$ in specializations, extending a fundamental result of Beckmann concerning inertia groups. We then apply our result to study crossed products, the Hilbert--Grunwald property, and finite parametric sets.
Given a hilbertian field $k$ of characteristic zero and a finite Galois extension $E/k(T)$ with group $G$ such that $E/k$ is regular, we produce some specializations of $E/k(T)$ at points $t_0 \in \mathbb{P}^1(k)$ which have the same Galois group but also specified inertia groups at finitely many given primes. This result has two main applications. Firstly we conjoin it with previous works to obtain Galois extensions of $\mathbb{Q}$ of various finite groups with specified local behavior - ramified or unramified - at finitely many given primes. Secondly, in the case $k$ is a number field, we pr…
The central topic is this question: is a given $k$-étale algebra $\prod_lE_l/k$ the specialization of a given $k$-cover $f:X\rightarrow B$ at some point $t_0\in B(k)$? Our main tool is a {\it twisting lemma} that reduces the problem to finding $k$-rational points on a certain $k$-variety. Previous forms of this twisting lemma are generalized and unified. New applications are given: a Grunwald form of Hilbert's irreducibility theorem over number fields, a non-Galois variant of the Tchebotarev theorem for function fields over finite fields, some general specialization properties of covers over P…
We prove Tchebotarev type theorems for function field extensions over various base fields: number fields, finite fields, p-adic fields, PAC fields, etc. The Tchebotarev conclusion - existence of appropriate cyclic residue extensions - also compares to the Hilbert specialization property. It is more local but holds in more situations and extends to infinite extensions. For a function field extension satisfying the Tchebotarev conclusion, the exponent of the Galois group is bounded by the l.c.m. of the local specialization degrees. Further local-global questions arise for which we provide answer…
We investigate specializations of infinite families of regular Galois extensions over number fields. The problem to what extent the local behaviour of specializations of one single regular Galois extension can be prescribed has been investigated by Dèbes and Ghazi in the unramified case, and by Legrand, Neftin and the author in general. Here, we generalize these results and give a partial solution to Grunwald problems using Galois extensions arising as specializations of a family of regular Galois extensions.
Let f(t,X) be an irreducible polynomial over the field of rational functions k(t), where k is a number field. Let O be the ring of integers of k. Hilbert's irreducibility theorem gives infinitely many integral specializations of t to values a in O such that f(a,X) is still irreducible. In this paper we study the set Red(O) of those a in O with f(a,X) reducible. We show that Red(O) is a finite set under rather weak assumptions. In particular, several results of K. Langmann, obtained by Diophantine approximation techniques, appear as special cases of some of our results. Our method is completely…
Let $t$ and $x$ be indeterminates, let $ϕ(x)=x^2+t\in\mathbb Q(t)[x]$, and for every positive integer $n$ let $Φ_n(t,x)$ denote the $n^{\text{th}}$ dynatomic polynomial of $ϕ$. Let $G_n$ be the Galois group of $Φ_n$ over the function field $\mathbb Q(t)$, and for $c\in\mathbb Q$ let $G_{n,c}$ be the Galois group of the specialized polynomial $Φ_n(c,x)$. It follows from Hilbert's irreducibility theorem that for fixed $n$ we have $G_n\cong G_{n,c}$ for every $c$ outside a thin set $E_n\subset\mathbb Q$. By earlier work of Morton (for $n=3$) and the present author (for $n=4$), it is known that $E…
The paper is concerned with the following version of Hilbert's irreducibility theorem: if $π: X \to Y$ is a Galois $G$-covering of varieties over a number field $k$ and $H \subset G$ is a subgroup, then for all sufficiently large and sufficiently divisible $n$ there exist a degree $n$ closed point $y \in |Y|$ and $x \in π^{-1}(y)$ for which $k(x)/k(y)$ is a Galois $H$-extension, and $k(y)/k$ is an $S_n$-extension. The result has interesting corollaries when applied to moduli spaces of various kinds. For instance, for every finite group $G$ there is a constant $N$ such that for all $n>N$ there …
The paper has three main applications. The first one is this Hilbert-Grunwald statement. If $f:X\rightarrow \Pp^1$ is a degree $n$ $\Qq$-cover with monodromy group $S_n$ over $\bar\Qq$, and finitely many suitably big primes $p$ are given with partitions $\{d_{p,1},..., d_{p,s_p}\}$ of $n$, there exist infinitely many specializations of $f$ at points $t_0\in \Qq$ that are degree $n$ field extensions with residue degrees $d_{p,1},..., d_{p,s_p}$ at each prescribed prime $p$. The second one provides a description of the se-pa-ra-ble closure of a PAC field $k$ of characteristic $p\not=2$: it is ge…
We prove the indecomposability of Galois representation restricted to the p-decomposition group attached to a non CM nearly p-ordinary weight two Hilbert modular form under mild conditions.
We prove some new cases of local--global compatibility for the Galois representations associated to Hilbert modular forms of low weight (that is, partial weight one).
Effective Hilbert's Irreducibility Theorem for global fields
Instituto Argentino de Matemáticas Alberto P. Calderón-CONICET（阿尔贝托·P·卡尔德龙阿根廷数学研究所-CONICET）；Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires（布宜诺斯艾利斯大学精确与自然科学学院数学系）
The field-of-norms functor is applied to deduce explicit reciprocity formulae for the Hilbert symbol in the mixed characteristic case from the explicit formula for the Witt symbol in characteristic p > 2 in the context of higher local fields. Is is shown that a "very special case" of this construction gives Vostokov's explicit formula
The rank one Gross conjecture for Deligne-Ribet $p$-adic $L$-functions was solved in works of Darmon-Dasgupta-Pollack and Ventullo by the Eisenstein congruence among Hilbert modular forms. The purpose of this paper is to prove an analogue of the Gross conjecture for the Katz $p$-adic $L$-functions attached to imaginary quadratic fields via the congruences between CM forms and non-CM forms. The new ingredient is to apply the $p$-adic Rankin-Selberg method to construct a non-CM Hida family which is congruent to a Hida family of CM forms at the $1+\varepsilon$ specialization.
The aim of this article is to establish the specialization method on characteristic ideals for finitely generated torsion modules over a complete local normal domain R that is module-finite over $O[[x_1, ..., x_d]]$, where $O$ is the ring of integers of a finite extension of the field of p-adic integers $Q_p$. The specialization method is a technique that recovers the information on the characteristic ideal $char_R(M)$ from $char_{R/I}(M/IM)$, where I varies in a certain family of nonzero principal ideals of R. As applications, we prove Euler system bound over Cohen-Macaulay normal domains by …
André and Maulik--Poonen proved that for any smooth proper family $X\to B$ of varieties over an algebraically closed field of characteristic $0$, there is a closed fiber whose Néron-Severi group has the same rank as that of the Néron-Severi group of the geometric generic fiber. We prove the analogous statement over algebraically closed fields of characteristic $p>0$ which are not isomorphic to $\overline{\mathbb{F}}_p$. Furthermore, we prove that for any algebraically closed field $k$ of characteristic $p>0$ and smooth proper family $X\to B$ of $k$-varieties, there exists a dense open subvarie…
Usually the boundary map in K-theory localization only gives the tame symbol at $K_{2}$. It sees the tamely ramified part of the Hilbert symbol, but no wild ramification. Gillet has shown how to prove Weil reciprocity using such boundary maps. This implies Hilbert reciprocity for curves over finite fields. However, phrasing Hilbert reciprocity for number fields in a similar way fails because it crucially hinges on wild ramification effects. We resolve this issue, except at p=2. Our idea is to pinch singularities near the ramification locus. This fattens up K-theory and makes the wild symbol vi…
This is a review of the vast area of explicit formulas for the (wild) Hilbert symbol (not only in the one-dimensional case but in the higher dimensional case as well). An extensive bilbiography is included.
