STUDENT SEMINAR, NUMBER THEORY COURSE
Date and time: Thursday, May 5th, from 14.30
Room: Aula 4
1) Daniele Casazza
Title: Analytic and geometric tecniques for Dirichlet series.
Abstract: First we find analytical properties of Dirichlet series as region of convergence and residue in a simple pole, then we define zeta function for algebraic number fields and for ray classes of a modulus. Using geometric tools as lattices and the geometric view of the action of automorphism group we estimate the number of integral ideals of given norm in a ray class.
2) Fabio Malanchini
Title: Residue formula for zeta functions on number fields.
Abstract: Calculating the volume of a particular solid using lattice point counting we find the residue of a zeta function of a ray class and an explicit formula relating class number, regulator and discriminant of an algebraic number fields to the residue at s=1 of its zeta function. Using these informations we can obtain a slightly larger region of convergence for the zeta function of a number field.
3) Gao Ziyang (Jerry).
Title: An Introduction to Arakelov Geometry: Slope Theory (PDF note of the seminar)
Abstract:
In this lecture, I aim to introduce the slope theory, which is an important tool in Arakelov Geometry. I will start from the basic definitions of Arakelov Geometry and ends in the statement and proof of the slope inequality. At last, a proof of Siegel’s Lemma, which is one of the fundamental theorems of Arithmetic Algebraic Geometry, in flavor of the slope theory will be presented as an application.