Variations around the Hilbert property

Lecture 1

Hilbert’s irreducibility theorem, Hilbertian fields, and the Hilbert property of varieties

(Mo) May 17, 2027 - 13:00-15:30 Lecturer: A. Fehm

Hilbert’s irreducibility theorem states that irreducible polynomials over the field of rational numbers have irreducible specializations, and has important applications in Galois theory and arithmetic geometry.

I will give an introduction to this topic and to two notions that derive from it: Hilbertian fields and the Hilbert property of varieties.

Keywords: Hilbert irreducibility; polynomal specialization; field and variety properties.

Lecture 2

Reducible fibers and their finiteness

(Tue) May 18, 2027 - 13:00-15:30 Lecturer: D. Neftin

For a polynomial map f over the rationals, Hilbert's irreducibility theorem asserts that the fiber f⁻¹(b) is irreducible over Q for all rational b outside a "thin" set of exceptions Red(f). In other words, f(x)-b is irreducible over Q for b outside Red(f).
The classical fiber reducibility problem (resp. Hilbert–Siegel problem) asks for which f the set Red(f) contains infinitely many rational (resp. integral) values outside obvious infinite families of exceptions, such as the value set f(Q).

We shall describe how new results on monodromy groups of polynomial maps can be used to attack these long-standing problems and discuss some of their applications to arithmetic dynamics and functional equations. Time permitting, we shall discuss extensions of these problems to rational functions on more general curves, and the uniformity of such results when b ranges over all algebraic numbers of fixed degree over Q.

Keywords:Hilbert-Siegel problen; thin set; monodromy groups; arithmetic dynamic.

Lecture 3

Integrally Hilbertian Rings

(Wed) May 19, 2027 - 13:00-15:30 Lecturer: A. Behajaina

The Hilbert Irreducibility Theorem asserts that, given an irreducible polynomial P(T,Y) in Q[T,Y], the specialized polynomial P(m,Y) remains irreducible in Q[Y] for infinitely many m in Q; it also holds over any number field. One can ask the same for a ring R: given an irreducible P(T,Y) in R[T,Y], does there exist m in R such that P(m,Y) remains irreducible in R[Y]?
Rings with this property are called integrally Hilbertian, first studied in 2022 by Bodin, Dèbes, König, and Najib. Examples include Z, k[X], and more generally Krull domains. This mini-course will give an overview of the topic, including recent work with Dèbes and König, and applications such as the polynomial version of Schinzel’s hypothesis.

Keywords: Specialization of polynomials; Irreducible polynomials; Integrally Hilbertian ring; Schinzel’s Hypothesis.

Lecture 4

Rational points on abelian varieties over number fields and function fields

(Thu) May 20, 2027 - 13:00-15:30 Lecturer: A. Javanpeykar

How many rational points does an abelian variety have? Over number fields, abelian varieties acquire a Zariski dense set of rational points after a finite extension. At the same time, one expects that ramified covers of an abelian variety have fewer rational points. This phenomenon can be made precise using a version of Hilbert irreducibility for abelian varieties.

In this lecture series, we explain such a Hilbert type statement and its consequences for the distribution of rational points on abelian varieties. If time permits, we will discuss the analogous picture over function fields of characteristic zero and indicate how these results relate to conjectures of Campana and Lang on rational points.

Keywords: Abelian varieties; ramified covers; Rational points; function fields; Campana and Lang conjectures.

Lecture 5

Thin Sets and Random Walks in Linear Groups

(Fr) May 21, 2027 - 13:00-15:30 Lecturer: L. Bary-Soroker

Let G be a linear algebraic group and Γ ⊆ G(ℚ) a finitely generated Zariski dense subgroup. The aim of this lecture is to understand what a “typical” element γ ∈ Γ looks like from an arithmetic and Galois-theoretic point of view. A guiding principle is that exceptional behavior is rare: for instance, one can show that for γ ∈ SL_n(ℤ), the characteristic polynomial is typically irreducible with Galois group S_n, while for γ ∈ Sp_{2n}(ℤ) the generic Galois group is the hyperoctahedral group.
We develop a general framework based on the notion of thin sets, and study their distribution inside finitely generated subgroups. The main results show that thin subsets are negligible when sampled along long random walks on Cayley graphs, providing a quantitative form of Hilbert’s irreducibility theorem in this context.
The methods combine tools from several areas. For semisimple groups, they rely on expansion properties and strong approximation, leading to exponential decay estimates. For unipotent groups, one uses polynomial approximation and counting techniques, while in situations where expansion is not available (such as tori like G_m^2), one turns to analytic methods.

Overall, the lecture presents a unified perspective on generic behavior in arithmetic groups, connecting random walks, Galois theory, and the geometry of algebraic groups.

Keywords: Quantitative HIT; thin set; random walk; algebraic groups.

Lectures - Anabelian reconstruction principles

May 17-June 21, 2027 RIMS

Lecturers: B. Collas (RIMS, JP), C. Merten (Utrecht, NL), K. Sawada (RIMS, JP), S. Tsujimura (RIMS, JP)

Conference - Homotopy, rationality, and geometry

May 24 - 28, 2027 RIMS

Scientific committee: B. Collas (RIMS, JP), P. Dèbes (Lille, FR), C. Demarche (IMJ-PRG, FR), A. Fehm (TU Dresden, DE) & S. Mochizuki (RIMS, JP).

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Workshop - Arithmetic invariants from homotopy Galois theory

May 31- June 1, 2027 RIMS

Org.: S. Philip (Stockholm, SE), N. Yamaguchi (Tokyo Denki, JP)

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