M2R 2026/27: Number Theory
Differential Galois theory (Marina Poulet)
This course provides an introduction to the differential Galois theory and a presentation of its applications to the study of the solutions of differential equations.
After introducing differential Galois groups and motivating their study, we will highlight the similarities but also the differences between differential and classical Galois theories. We will see that the differential Galois group is not necessarily a finite group but it is an algebraic group. We will discuss analogues of classical results, such as the Galois correspondence and radical solvability. Then, we will detail some applications of this theory, in particular the study of algebraic relations between the solutions of differential equations and their derivatives.
We will then focus more particularly on differential equations which are called regular singular equations. We will talk about Schlesinger's density theorem which gives a Zariski-dense subgroup of the Galois group, an algebraic object, thanks to the monodromy, which comes from analytical continuation.
If we have enough time, we will discuss other Galois theories which concern for example the so-called Mahler equations, which are functional equations linked in particular to automatic sequences and to the complexity of the developments of real numbers.
Prerequisites :
- Basic notions in algebra, topology and complex analysis, corresponding roughly to undergraduate studies in mathematics.
- Basics notions on algebraic groups and classical Galois theory (to be covered in the beginning of the course of Samuel Le Fourn).
Tentative program :
I. 1) Differential algebra (differential rings, fields and ideals ; differential equations, systems and modules).
I. 2) Picard-Vessiot rings.
I. 3) Differential Galois groups (comparison with the classical Galois theory, Galois correspondence, Liouvillian solutions, etc).
I. 4) Application : study of the algebraic relations between the solutions of differential equations and their derivatives.
II. 1) Regular singular differential equations.
II. 2) Monodromy.
II. 3) Schlesinger's density theorem.
(II. 4) extension of the Schlesinger's density theorem for irregular differential equations).
(III. Galois theory of Mahler equations and some applications.)
References :
- Teresa Crespo, Zbigniew Hajto, Algebraic groups and differential Galois theory, GSM 122, Americal Mathematical Society, 2011.
- Kumiko Nishioka. Mahler Functions and Transcendence, volume 1631 of Lecture Notes in Math. Springer-Verlag Berlin Heidelberg, 1996
- Marius van der Put and Michael F. Singer. Galois theory of linear differential equations, volume 328 of Grundlehren Math. Wiss. Springer-Verlag, Berlin, 2003.
Algebraic number theory (Samuel Le Fourn and Vanessa Vitse) :
This course provides an introduction to classical notions of algebraic number theory, aiming to understand how the properties of integers generalise (or not) to algebraic integers. Algebraic number theory is also fundamental to understand the properties of diophantine equations, i.e. equations involving integers themselves, such as a description of the solutions of the Pell equation $x^2 - n y^2 = 1$.
The modern theory relies on Dedekind rings (and rings of integers in number fields are such rings). Those rings are often not principal ideal domains, but behave similarly when one replaces elements of the ring by ideals: indeed, ideals of a Dedekind ring factor uniquely as products of prime ideals as will be shown in this course.
We will use this especially for some specific families of number fields and Galois extensions of Q, and provide applications of these facts to diophantine equations or other famous problems involving integers.
Prerequisites :
- Field extensions, linear algebra, some group theory and arithmetics.
- Basic notions of commutative algebra (ideals, quotient rings).
Tentative program :
- Algebraic extensions and Galois theory, with application to finite fields
- Number fields and their rings of (algebraic) integers, trace and norm
- Discrete valuation rings and Dedekind rings
- Decomposition of prime ideals and applications to factoring of large integers (number field sieve)
- Class groups and unit group of a ring of integers
- Places of a number field and Weil height
References :
[1] Da. Marcus, Number Fields
[2] K. Ireland & M. Rosen - A classical introduction to modern number theory
Elliptic curves and abelian varieties (Samuel Le Fourn and Vanessa Vitse) :
Elliptic curves are algebraic curves over a field having the very specific property of being endowed with a (commutative) group structure, defined by rational fractions. Abelian varieties are their generalisation to algebraic varieties of any dimension.
At first objets of curiosity (related initially to the computation of so-called "elliptic integrals", dating back to Euler), they have become ubiquitous in modern number theory, from their key role to the proof of Fermat's last theorem by Wiles in 1995 to their daily use for asymmetric cryptographic protocols in online transactions.
This course aims to provide the basic notions about elliptic curves and how to manipulate and compute with them, with an opening to their generalisation to abelian varieties, in particular jacobian varieties, constructing for any smooth algebraic curve (not of genus 0) an abelian variety containing it.
Prerequisites :
- Good knowledge of rings of polynomials
- Some general topology, group theory and notions of differential geometry
- Complex analysis
Tentative program :
- Algebraic geometry over a field: algebraic varieties, Zariski topology, Nullstellensatz, projective algebraic varieties.
- Algebraic curves and function fields, divisors, Riemann-Roch theorem
- Elliptic curves : equations, change of variables, group law
- Complex elliptic curves and isomorphism with complex tori
- Isogenies and endomorphism rings of elliptic curves
- Elliptic curves over finite fields, Hasse bound, and applications to cryptography
- Computation of torsion points (and their Weil pairing) of elliptic curves
- Abelian varieties : general notions of algebraic groups and some fundamental theorems, polarisation
- Construction of the jacobian variety, and computations of hyperelliptic curves of genus 2 and their jacobians
References :
[1] J. Silverman - The Arithmetic of Elliptic Curves
[2] G. Kempf - Algebraic varieties
Introduction to p-adic analysis and p-adic differential equations (Andrea Pulita)
We will provide the basic foundations of p-adic analysis over the affine line and the theory of differential modules on some specific p-adic domains of the affine line. Then, in connection with the other courses of this Master 2 program, we will explore certain themes from the book by Dwork–Gerotto–Sullivan, oriented toward a proof of Chudnovsky's theorem using p-adic differential equations.
Tentative program :
1. Introduction to p-adic analysis on the affine line ; Differential modules over a p-adic field ; Radii of convergence of the solutions
2. G-functions ; Chudnowsky's theorem
3. Nilpotence ; Radii of convergence ; André-Bombieri's theorem
References :
[1] B.Dwork, G.Gerotto, F.Sullivan "An introduction to G-functions".
[2] Kyran S. Kedlaya "p-adic differential equations"
[3] G.Christol P.Robba "Equations différentielles p-diques"
[4] G.Lepetit https://arxiv.org/pdf/2109.10239
Diophantine Approximation and Transcendence Theory (S. Checcoli and T. Rivoal)
This course will be devoted to the Diophantine properties of real or complex numbers defined by analytic or arithmetic means. A central question is whether such a number is irrational or not, and whether it is algebraic or transcendental. We will explore classical theorems as well as more recent results.
The first part of the course (Sara Checcoli) will be an introduction to classical Diophantine approximation and some aspects of transcendence theory. We will begin with continued fractions, Liouville’s and Roth’s theorems, the Subspace Theorem, and basic results about the approximation of algebraic numbers. We will then study Diophantine properties of certain continued fractions arising from combinatorial constructions (e.g., automatic sequences), introducing along the way notions such as automatic sequences and word complexity. If time permits, we will also present some elements of the theory of p-adic continued fractions. These parts may include recent research results and is intended to provide both classical tools and modern perspectives.
The second part of the course (Tanguy Rivoal) will focus on the transcendence of values taken by certain special functions at algebraic points. We will start with $\exp(z)$ and $\log(z)$, and show how certain “explicit” constructions based on Hermite-type approximants lead to the proof of the transcendence of $\exp(\alpha)$ for every non-zero algebraic $\alpha$, and of $\log(\alpha)$ for every algebraic $\alpha \neq 0,1$, culminating in the Gel'fond–Schneider Theorem. We will then study $E$-functions (a class of functions including $\exp(z)$), which are power series satisfying a linear differential equations and whose coefficient enjoy certain arithmetic conditions. We will discuss, in particular, the Siegel–Shidlovskii Theorem about the algebraic independence of the values of E-functions at algebraic points. Time permitting, we will also briefly discuss $G$-functions (a class of function including $log(1+z)$) and the Chudnovsky Theorem, which offers an analogue of the Siegel–Shidlovskii result in this context.
Prerequisites : A general background in number theory will be useful. Attendance of the introductory number theory course offered in the first semester is recommended. It will also be useful to attend the Galois differential Theory course in the first semester.
Tentative program:
* Classical theory of continued fractions
* Liouville and Roth theorems
* The Subspace Theorem
* Automatic sequences and word complexity
* Diophantine properties of continued fractions arising from automatic sequences
* (If time allows) Some results on p-adic continued fractions
* Transcendence at algebraic points of $\exp(z)$ and $\log(z)$
* Gel'fond–Schneider Theorem
* Siegel–Shidlovskii Theorem for $E$-functions
* (If time allows) Chudnovsky-type results for $G$-functions
References
* J.-P. Allouche and J. Shallit. Automatic Sequences: Theory, Applications, Generalizations. Cambridge University Press, 2003.
* A. Baker, Transcendental Number Theory, Cambridge University Press, 1990
* K. Nishioka, Mahler Functions and Transcendence, Lecture Notes in Mathematics 1631, Springer, 1996
* W. M. Schmidt, Diophantine approximation, vol. 785, Lecture Notes in Mathematics. Berlin-Heidelberg-New York: Springer-Verlag, 1980, new ed. 2001.
* A. Shidlovskii, Transcendental Numbers, de Gruyter, 1987