Distribution of primes (9 hours)
Lecturer: Francesco Pappalardi and Cihan Pehlivan
We shall first cover the basics of the Theory of Distribution of prime numbers.
This will include the proof to the Chebicev Theorem, the Mertens Theorems and the Dirichlet Theorem for primes in arithmetic progressions.
Next we concentrate on the Riemann zeta function discussing its fundamental properties
following the original paper by Riemann (Functional equation, Hadamard product expansion, Vertical distribution of the zeros and zero
Time will force us to omit several proofs but we will emphasise the link between the position of the zeros of zeta and the distribution of prime
Basic properties of primitive roots. History of the Artin Conjecture. Kummerian Extension.
Explicit Chebotarev for Kummerian Extensions. Consequences of the GRH. The proof of Hooley's Theorem.
Sketch of the proof of the quasi resolution of the Artin Conjecture by Gupta, Murty and Heath--Brown.
|L-functions and ζ-functions (14 hours)
Lecturers: Paul Voutier, Michel Waldschmidt and Fernando Rodriguez Villegas
Definition of the Riemann zeta function and Dirichlet L-functions. Elementary analytic properties.
Values at even integers, Euler product. Functional equation.
Vertical distribution of the zeros. Hadamard product expansion. The Riemann hypothesis.
Other examples of L-functions and ζ-functions. L-functions coming from geometry.
The Selberg class of L-functions.
Zeta functions of varieties over finite fields. Interpretation in terms of the Frobenius automorphism.
Heuristics on motives based on
concrete examples (including hypergeometric motives). Global L-function associated to a motive: Euler factor at infinity,
at good and bad primes, conductor, functional equation, etc.
Chebotarev Density Theorem (12 hours)
Lecturers: Ekim Ozman and Adriana Salerno
Review of algebraic number thoery: Number fields and their ring of integers. Extension of number fields:
Decomposition and ramification, the ramification index and the residue class degree of a prime ideal: their definitions and relations.
Galois extensions, spilt and inert primes. The discriminant of an extension. Characterization of ramified prime ideals as those dividing the discriminant.
Explicit theory for quadratic and cubic number fields.
Definition of the inertia group and the decomposition group and the Frobenius symbol. The Chebotarev density theorem.
|Elliptic curve analogues (9 hours)
Lecturers: Peter Stevenhagen and Valerio Talamanca
Elliptic curves: Addition law on a plane cubic in Weierstrass form. Abstract elliptic curves and their planar model. Rational points on an elliptic curve.
Elliptic curves over finite fields: the Weil bound for the number of points.
Reduction of elliptic curves modulo a prime: first properties and the standard short exact sequence. Definition of good and bad reduction
of an ellptic curve. Definition of the L-function of an elliptic curve defined over ther rationals. Primitive points modulo a prime. The Lang-Trotter conjecture on
the density of primitive points. Reduction modulo a prime