65 research outputs found
Arithmetical problems in number fields, abelian varieties and modular forms
La teoria de nombres, una àrea de la matemàtica fascinant i de les més antigues, ha experimentat un progrés espectacular durant els darrers anys. El desenvolupament d'una base teòrica profunda i la implementació d'algoritmes han conduït a noves interrelacions matemàtiques interessants que han fet palesos teoremes importants en aquesta àrea. Aquest informe resumeix les contribucions a la teoria de nombres dutes a terme per les persones del Seminari de Teoria de Nombres (UB-UAB-UPC) de Barcelona. Els seus resultats són citats en connexió amb l'estat actual d'alguns problemes aritmètics, de manera que aquesta monografia cerca proporcionar al públic lector una ullada sobre algunes línies específiques de la recerca matemàtica actual.Number theory, a fascinating area in mathematics and one of the oldest, has experienced spectacular progress in recent years. The development of a deep theoretical background and the implementation of algorithms have led to new and interesting interrelations with mathematics in general which have paved the way for the emergence of major theorems in the area. This report summarizes the contribution to number theory made by the members of the Seminari de Teoria de Nombres (UB-UAB-UPC) in Barcelona. These results are presented in connection with the state of certain arithmetical problems, and so this monograph seeks to provide readers with a glimpse of some specific lines of current mathematical research
A refinement of Stark's conjecture over complex cubic number fields
AbstractWe study the first-order zero case of Stark's conjecture over a complex cubic number field F. In that case, the conjecture predicts the absolute value of a complex unit in an abelian extension of F. We present a refinement of Stark's conjecture by proposing a formula (up to a root of unity) for the unit itself instead of its absolute value
Artin's primitive root conjecture -a survey -
This is an expanded version of a write-up of a talk given in the fall of 2000
in Oberwolfach. A large part of it is intended to be understandable by
non-number theorists with a mathematical background. The talk covered some of
the history, results and ideas connected with Artin's celebrated primitive root
conjecture dating from 1927. In the update several new results established
after 2000 are also discussed.Comment: 87 pages, 512 references, to appear in Integer
On Hilbert modular threefolds of discriminant 49
Let K be the totally real cubic field of discriminant 49, let O be its ring
of integers, and let p be the prime over 7. Let Gamma (p)\subset Gamma =
SL_2(O) be the principal congruence subgroup of level p. This paper
investigates the geometry of the Hilbert modular threefold attached to Gamma
(p) and some related varieties. In particular, we discover an octic in P^3 with
84 isolated singular points of type A_2
Theory of the Riemann Zeta and Allied Functions
[no abstract available
Explicit Methods in Number Theory
These notes contain extended abstracts on the topic of explicit methods in number theory. The range of topics includes the Sato-Tate conjecure, Langlands programme, function fields, L-functions and many other topics
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