Courses

Courses will consist of lectures, exercise sessions and possible computer classes. 

Analytic number theory (10 classes)

Pieter Moree (Max Planck Institute for Mathematics), Francesco Pappalardi (Roma Tre University)

Basic notions of analytic number theory, including arithmetic functions, Riemann zeta function, Dirichlet series, Dirichlet characters and Dirichlet L-functions.

Algebraic number theory (11 classes)

Vandita Patel (University of Manchester), Peter Stevenhagen (Leiden University)

Introduction to classical algebraic number theory: Number fields and their subrings, the failure of unique factorisation in certain number fields and ideal factorisation in number rings with plenty of examples and exercises. Classical finiteness theorems for the class group and the unit group (Dirichlet's theorem), all with lots of examples and a supporting computer class, also including LMFDB. If time permits, we shall give some practical applications to the study of Diophantine equations. These equations require resolution over the integers, however, in some cases, one can succeed by passing to a field extension and working with powerful algebraic number theory knowledge! Additionally, we might also focus on the Galois theory.

Reference: Number Rings, Stevenhagen

Additional reference: The arithmetic of number rings, Stevenhagen

Algebraic curves (5 classes)

Laura Capuano (Roma Tre University)

Affine varieties, projective varieties, maps between varieties, Intersection multiplicity, Bézout Theorem.

Uniformizers, coordinate rings, local ring at a point and regular function, morphisms between affine curves. 

Divisors, derivations and differentials. Canonical divisors and the Riemann-Roch theorem. Genus of a curve. 

Main reference: Effective geometry and arithmetic of curves: an introduction, Ritzenthaler

Additional references: 

The Arithmetic of Elliptic Curves, Silverman 

Algebraic curves, Fulton

Commutative algebra (10 classes)

René Schoof (University of Rome Tor Vergata), Valerio Talamanca (Roma Tre University)

Commutative rings, basic notions, Stone's Theorem; modules: exactness, tensor products, flatness; localization, integrality, Nulstellensatz, Hilbert Basissatz.

Reference: Introduction to Commutative Algebra, Atiyah, Macdonald

Elliptic curves (10 classes)

Elisa Lorenzo Garcia (University of Neuchâtel), Steffen Müller (University of Groningen) 

Elliptic curves: introduction. Weierstrass equations, j-invariant; the group law and torsion points.

Isogenies and endomorphisms of elliptic curves.

Elliptic curves over finite fields.

Elliptic curves over the rationals

Besides regular exercise sessions, we will also see how to compute with elliptic curves using the computer algebra system SageMath.

Main reference: The Arithmetic of Elliptic Curves, Silverman 

Additional references: 

Rational Points on Elliptic Curves, Silverman-Tate

Elliptic Curves: Number Theory and Cryptography, Washington

p-adic numbers (5 classes)

Rachel Newton (King's College London) 

Valuations, Ostrowski's Theorem, definition of the p-adic numbers as a completion of Q, proof that Qp is a field, p-adic valuation, p-adic integers, p-adic units, power series representation of p-adic numbers, Hensel's lemma, structure of the p-adic units. If time permits, I will finish with a study of the quadratic extensions of Qp.

Transcendental number theory (5 classes)

Michel Waldschmidt (Sorbonne University) 

The Schneider Lang Criterion. Statement, corollaries: transcendence of e, π, Hermite-Lindemann, Gel'fond-Schneider. Proof,  introducing the following tools:  construction of an auxiliary function, the zero estimate, Schwarz Lemma, Liouville type estimate.

Diophantine approximation: Liouville (with proof), Thue--Siegel--Roth, Schmidt's Subspace Theorem, applications.

Linear independence of logarithms of algebraic numbers: Baker's results, effective (and explicit) lower bounds for linear forms. 

Algebraic independence. Lindemann-Weierstrass Theorem. Criteria for algebraic independence. Theorems of Gel'fond, Chudnovskii.

Schanuel's Conjecture; consequences.  

Transcendence for elliptic functions and modular functions.  Nesterenko's Theorem.

p-adic theory.