Lattices, heights and diophantine approximation






The school aims at introducing the students to a few lines of research in number theory revolving around the concepts of lattices, heights and diophantine approximation. Part of the school is devoted to presenting the recent proof by Maryna Viazovska that the densest sphere packing in dimension 8 is the one given by the E_8 lattice. In 2022 she was awarded the Fields medal. To this end we introduce the students to lattices, sphere packings and modular forms. We will prove the Cohn-Elkies bound and finally we will study Viazovska's construction of the function which optimizes such bound and proves that the E_8 lattice is the densest packing in dimension 8. In the other part of the program we will introduce the machinery of heights, which are standard tools of Diophantine geometry used to measure arithmetic complexity of objects. We will then demonstrate the use of height functions in Diophantine approximation and Diophantine geometry discussing results and conjectures such as the Mordell-Weil theorem, Faltings' theorem, Siegel's lemma, Cassels' theorem, Lehmer's conjecture, and many others. On the Diophantine approximation side, we will discuss the central themes of equidistribution and approximation of reals by algebraic numbers. We expect the participants to gain familiarity with these central and vibrant areas of mathematics that have been at the forefront of mathematical research for over a hundred years. Our program is specifically designed in a way that assumes a rather modest background, but with a concentrated and focused approach takes the audience to some of the modern-day research directions.
Le programme scientifique est disponible sur le site local de l'Ecole :

Langue officielle de l'Ecole : Anglais

Coordinateurs administratifs et scientifiques

Zafar IBRAGIMOV (Urgench State University,
, )
Elisa LORENZO GARCIA (Université de Neuchâtel & Université de Rennes 1,
, )

Programme scientifique

Cours 1: "introductory course - Basic properties of Modular forms. Dimension of the space of modular form of a given weight. Construction of Eisenstein series in particular of E_2, E_4, E_6. Fourier Expansions of Eisenstein Series. Generators for the graded algebra of all modular forms. Properties of the discriminant function, in particular its non-vanishing. The rings {\bf C}[E_4,E_6] \subset {\bf C}[E_2,E_4,E_6]. Construction of the Thetanullwerte. Jacobi modular forms and their identities. Quasi-modular forms and weakly modular forms, and their Fourier transform. References: Serre, A course in arithmetic, Chapter VII, Graduate Texts in Mathematics, Springer. Zagier, Elliptic Modular Forms and Their Applications in “The 1-2-3 of Modular Forms “ Universitext, Springer.", Francesco PAPPALARDI (Universita' Roma Tre, Italy)

Cours 2: "introductory course - Harmonic analysis for $\mathbf{R}^n,\mathbf{Z}^n$ and $\mathbf{T}^n$, Fourier series and Fourier transform and their main properties. Poisson summation formula. Laplace transform and its properties. Bessel functions and their elementary properties. Radial functions, Fourier transform of radial functions in terms of Bessel functions. Admissible functions, properties of admissible function: convergence of Fourier Series, Poisson summation formula. Cohn-Elkies boud for sphere packing. References G. Andrews, R. Askey, and R. Roy, Special Functions, Cambridge University Press, Cambridge, 1999. H. Cohn and N. Elkies, New upper bounds on sphere packings I, Ann. of Math. (2) 157 (2003), no. 2, 689–714. E. M. Stein and G. L. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Math. Series 32, Princeton Univ. Press, Princeton, NJ, 1971. ", Valerio TALAMANCA (Universita' Roma Tre, Italy)

Cours 3: "introductory course - This course will be an introduction to the concept of equidistribution (also called uniform distribution) along with many examples. Equidistribution is a natural property of a sequence which is fundamental to many deep open questions in math- ematics today. We will first introduce the notion of equidistribution along with examples and non-examples and discuss connections between equidistribution and density, as well as rearrangements of sequences. The idea of equidistribution under- lies many classical theorems about prime numbers, for example Dirichlet’s theorem for primes in arithmetic progressions, which we will explore in one lecture. We will prove Weyl’s criterion, a fundamental tool in equidistribution theory. Weyl’s criterion naturally connects questions of equidistribution with the analytical prop- erties of L-series. In order to study this connection, we will introduce L-series and characters. We then discuss Tauberian theorems, which provide information about the analytic properties of these L-series. Finally, we conclude the course with some examples of equidistribution in compact Lie groups other than the circle, where we often order elements by height. • Lecture 1: An introduction to equidistribution. • Lecture 2: Weyl’s criterion. • Lecture 3: Characters and L-series. • Lecture 4: Tauberian Theorems. • Lecture 5: Equidistribution in compact Lie groups other than the circle. For lack of slots in the on-line form the training session for this course is included here: Lecture 6: Training session, 1 hour – a selection of exercises on equidistribution will be discussed with featured student presentations. ", Kate PETERSEN (University of Minnesota Duluth. , United States)

Cours 4: "introductory course - A set of algebraic points of bounded degree and height over a fixed number field is finite by a theorem of Northcott. Therefore establishing explicit bounds on height of solutions of polynomial equations over a fixed number field potentially allows for a search algorithm for such solutions, which is closely related to the Hilbert’s 10th problem. The goal of this lecture course is to introduce the subject of search bounds for solutions of Diophantine equations with respect to height over different fields and rings with a special focus on the linear and quadratic cases. Here is the plan of lectures. • Lecture 1: Hilbert’s 10th problem and search bounds. (1) Hilbert’s 10th problem. (2) Search bound approach to the problem. (3) Overview of some known results: Siegel’s lemma and the inhomogeneous linear case, Cassels’ theorem and the inhomogeneous quadratic case, cubic forms and higher degree polynomials. • Lecture 2: Linear and quadratic cases over Q in more detail. (1) The classical Thue-Siegel lemma with proof. (2) The classical theorem of Cassels on small zeros of quadratic forms with proof. • Lecture 3: Height machinery. (1) Homogeneous and inhomogeneous heights over number fields and $\overline{\mathbb Q}$. (2) Arakelov-Schmidt height on subspaces. (3) Heightinequalities (comparing heights, height of a linear combination). (4) Northcott’s theorem. (5) Search bounds with respect to height. Lecture 4: Siegel’s lemma over algebraic numbers. (1) Bombieri-Vaaler version of Siegel’s lemma over number fields. (2) Roy-Thunder “absolute” version of Siegel’s lemma over $\overline{\mathbb Q}$. (3) A new “absolute” monotone basis version of Siegel’s lemma. Lecture 5: Search bounds for higher degree polynomials. (1) Cassels’ and Masser’s theorems for a single quadratic polynomial over number fields. (2) Multilinear polynomials over a number field. (3) Higher degree polynomials over $\overline{\mathbb Q}$. (4) A system of quadratic polynomials over $\overline{\mathbb Q}$. For lack of slots in the on-line form the training session for this course is included here: • Lecture 6: Training session, 1 hour – a selection of exercises on Siegel’s lemma, Cassels’ theorem and search bounds for higher degree polynomials will be discussed with featured student presentations.", Lenny FUKSHANSKY (Claremont McKenna College, United States)

Cours 5: "introductory course - This course is an introduction to the notion of height with a view to applications to Diophantine approximation and transcendence. The proofs of Diophantine approximation share the characteristic that at some point, one uses the fact that a nonzero rational integer has absolute value at least 1. One important problem is to extend this result to nonzero algebraic numbers: the lower bound will not be 1 but will involve the height of the nonzero algebraic number. There are several notions of height on the set of algebraic numbers, each of them with the so–called Northcott property that given D and H, the set of algebraic numbers with degree at most D and height at most H is finite. Among these different heights, one of them has several advantages with respect to the others, coming from the fact that it may be defined in three equivalent ways, one involving an integral involving the minimal polynomial, another a product involving the complex conjugates, a third one involving the different absolute values of the given number. This is the Weil absolute logarithmic height. We give these definitions and prove that they produce the same value, and we discuss some estimates comparing different heights. Let us come back to the main issue occurring in Diophantine approximation: since the set of algebraic numbers with degree at most D and height at most H is finite, a nonzero complex algebraic number in this set has a modulus bounded from below by a function of D and H. The basic estimate in this direction is Liouville’s inequality, and there are many different versions of it. We will review some of them, give some applications and introduce open problems. • Lecture 2: Usual Height and Size (1) Definitions: usual height, house, size (2) Inequalities among the different notions of height • Lecture 3: Liouville’s Inequalities (1) Introduction (2) The Main Lower Bound (3) Further Lower Bounds (4) Estimates for Determinants • Lecture 4: Lower Bound for the Height (1) Kronecker’s Theorem (2) Lehmer’s Problem (3) Dobrowolski’s Theorem (4) Fermat’s Little Theorem (5) Dobrowolski’s Proof (6) Proof of Dobrowolski’s Theorem Following Cantor-Straus and Rausch (7) Further Related Questions and Results • Lecture 5: Recent results and Open Problems (1) Recent results (to be updated in 2024) (2) Lehmer’s problem (1933) (3) Conjecture of Schinzel and Zassenhaus (1965) (4) Problem of A. Dubickas (5) Problem of D. Boyd (1980) Reference. Chap. 3 of Michel Waldschmidt, Diophantine Approximation on Linear Algebraic Groups — Transcendence Properties of the Exponential Function in Several Variables, Grundlehren der Mathematischen Wissenschaften 326. Springer-Verlag, Berlin-Heidelberg, 2000. MR Zbl This book is available on the website", Michel WALDSCHMIDT (Sorbonne Université , France)

Cours 6: "introductory course - Rational numbers are algebraic numbers of degree 1. For the past two or three centuries an extensive study of how a number can be approximated by rationals has been developed. A very natural generalisation of this approach is to extend the set of admissible approximants to the set of algebraic numbers of bounded degree. The course is aimed at describing the main problems arising in this direction and tools that can be applied to solve them. We shall start with elementary classical theo- ems and gradually come to some recent achievements in this area of Diophantine approximation. • Lecture 1: Approximation by rationals (1) Minkowski’s convex body theorem (2) Geometry of Dirichlet’s approximation theorem (3) Geometry of continued fractions • Lecture 2: Approximation by quadratic irrationals (1) A polynomial, its derivative, and distance to its roots (2) Deduction of Davenport and Schmidt’s theorem from a theorem on two linear forms (3) Best approximation vectors for a linear form • Lecture 3: Davenport and Schmidt’s theorem (1) Application of geometry of continued fractions for the analysis of best approximation vectors (2) Proof of the theorem on two linear forms by Davenport and Schmidt • Lecture 4: Approximation by algebraic integers (1) Approximation by algebraic integers of degree 2 (2) Minkowski’s and Mahler’s theorems on successive minima (3) Davenport and Schmidt’s lemma (4) Roy’s extremal numbers • Lecture 5: Mahler’s and Koksma’s classifications (1) Mahler’s measure and the height of a polynomial (2) Application of generalised Liouville’s theorem and Schmidt’s subspace theorem (3) Wirsing’s theorem and Wirsing’s conjecture For lack of slots in the on-line form the training session for this course is included here: Lecture 6: Training session, 1 hour – a selection of exercises on approximation by algebraic numbers will be discussed with featured student pre- sentations. ", Oleg GERMAN (Moscow state university, Russian Federation)

Cours 7: "introductory course - In this course we recall the basic definitions concerning lattices and sphere packings: the fundamental domain and covolume of a lattice. The notions of shortest vectors and the kissing number. We define the dual of a lattice, integral and even lattices. We discuss lattice sphere packings, periodic packings, arbitrary packings as well as the density of a sphere packing. On the way, we introduce the Gamma function and formulas for the volume of the spheres. We describe elementary bounds for the density of an $n$-dimensional packing. We construct low dimensional examples, in particular in ${\bf R}^2$ and ${\bf R}^3$. We study the lattice $E_8$ in some detail. For lack of slots in the on-line form the training session for this course is included here: Student will be asked to present their solution to exercises assigned during the course", René SCHOOF (Università di Roma Tor Vergata, Italy)

Cours 8: "advanced course - The course is devoted to the proof of Viazovska’s theorem, stating that the E_8 lattice packing is the densest packing in dimension 8. The proof consists of the construction of an optimal function $f\colon \R^8\to \R$ satisfying the Cohn-Elkies criterion, i.e. a radial rapidly decreasing function such that $f(x)\le 0$, for $\Vert x\vert >\sqrt2$, $\widehat f(x)\ge 0$, for all $x\in\R^8$ and $f(0)=\widehat f(0)>0$. The function $f$ is a linear combination of two radial Schwartz functions $a$ and $b$ defined as integrals of rational functions of modular and quasi-modular forms on suitable paths in the complex plane. The functions $a$ and $b$ are $\pm 1$ eigenfunctions of the Fourier transform, respectively. To prove that the resulting function $f$ has the required properties, different integral representations of $a$ and $b$ are used, as well as estimates of the Fourier coefficients of the modular and quasi-modular forms involved.", Laura GEATTI (Università di Roma Tor Vergata, Italy)

Cours 9: "other interactive sessions - This course will be a short introduction to Diophantine Geometry. The main object of study is heights: we will study their properties, their constructions and their applications. We will start by introducing absolute values and valuations to define heights on projective spaces and later on on varieties, more precisely on abelian varieties via the Weil heights machinery. We revisit the Mordell-Weil theorem on the group of rational points on abelian varieties and Faltings’ theorem on the finiteness of rational points on curves of genus greater or equal than 2. We finish the course by discussing some open problems of Diophantine Geometry, such as the abc conjecture. This course is already available at the Youtube CIMPA channel in here. The main references are Hindry - Silverman book “Diophantine Geometry” and Bombieri - Gubler book “Heights in diophantine geometry”. The original online course is a 9h course. The students will be asked to watch the videos in advance. So the 6 lectures in person during the CIMPA school will be approximately structured as a 15min presentation of the mais results and ideas by the instructor, 15min of questions by the students and 20min of exercises • Lecture 1: Heights in projective spaces. The height, the logarithmique height and the absolut height of a point in a projective space. Finitude of points of bounded height. Kronecker’s theorem. • Lecture 2: Curves and abelian varieties. Algebraic curves and varieties. Genus. Divisors. Riemann-Roch Theo- rem. Riemann-Hurwitz Theorem. Abelian varieties. The Jacobian variety. • Lecture 3: The N ́eron-Tate height on abelian varieties. Height of a point in a variety. The Weil’s height machine. The N ́eron- Tate height. • Lecture 4: The (weak) Mordell-Weil Theorem. The (weak) Mordell-Weil Theorem. The Descent lemma. Hensel’s lemma. A Mazur’s Theorem. Lutz-Nagell Theorem. • Lecture 5: Faltings’ Theorem and the proof strategy. Falting’s Theorem. Vojta’s inequality. Ideas of the proofs. • Lecture 6: Height bounds and height conjectures. The abc-Conjecture. The abc-Conjecture implies Fermat’s Last Theo- rem and Falting’s Theorem. Other Diophantine Conjectures: Szpiro, Frey, Lang. Roth’s Theorem.", Elisa LORENZO GARCIA (Université de Neuchâtel & Université de Rennes 1, Switzerland)

Cours 10: "exercise session - Training session, 1 hour – a selection of exercises on the topics treated in the course Heights of algebraic numbers will be discussed with featured student presentation", Michel WALDSCHMIDT (Sorbonne Université , France)

Cours 11: "exercise session - During this hour the student will present their work on the verification of some details of the proof which will be left as exercises.", Laura GEATTI (Università di Roma Tor Vergata, Italy)

Cours 12: "other interactive sessions - In this introductory class the lecturer will interactively review the material recorded in Nesin Mathematical Village in November 2022 which the student will be asked to study before the school: (1) Absolute values (a) Absolute values on a field, ultrametric, archimedian, trivial (b) p-adic valuation and p-adic Absolute Values over Q (c) Number Fields (d) Archimedean Absolute Values over a Number Field (e) Ultrametric Absolute Values over a Number Field (f) The Product Formula (2) The Absolute Logarithmic Height (Weil) (a) Definition (b) First properties (c) Upper bound for the Weil height of values of polynomials (3) Mahler’s Measure (a) An integral equal to a product (b) Connection with Weil’s height", Michel WALDSCHMIDT (Sorbonne Université , France)

Cours 13: "other interactive sessions - This will be a review of basic facts of complex analysis up to Cauchy integral theorem and the Residue theorem. The student will be asked to watch the first 7 classes of the course of Jorge Mozo "Complex Analysis" avaialble at The review will be done by the students who will be asked to present their solution to exercises assigned before the school. ", Francesco PAPPALARDI (Universita' Roma Tre, Italy)

Cours 14: "exercise session - In this session the students will present their solution of exercises assigned during the course.", Francesco PAPPALARDI (Universita' Roma Tre, Italy)

Cours 15: "exercise session - In this session the students will present their solution of exercises assigned during the course.", Valerio TALAMANCA (Universita' Roma Tre, Italy)

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Date limite d'inscription : mars 15, 2024