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Topics in Number Theory: p-adic numbers and zeta function
| Credits | 8 credit points |
| Instructors | Jeu, R. de (Vrije Universiteit), Plazas Vargas, J. (Universiteit Utrecht) |
| jeu@few.vu.nl, J.A.PlazasVargas@uu.nl | |
| Aim | To provide a thorough introduction to both p-adic numbers and zeta-functions. |
| Description | This course deals with two topics. 1) For every prime number p the rational numbers can be provided with a metric that roughly states that, if the numerator of x is divisible by p^r, then |x|<= p^(-r). The completion of the rationals via this metric gives us the field of p-adic numbers. They were first described by Hensel in 1897 and they allow us to introduce tools from analysis in the theory of numbers. In this course we give an introduction to the field of p-adic numbers, its finite extensions, and some of its applications. 2) The Riemann zeta function zeta(s) is defined as the summation of 1/n^s over the integers n=1,2,3,... The variable s is a complex number and the series converges if the real part of s is bigger than 1. In his famous paper of 1857 Riemann discovered that zeta(s) can be continued analytically to the complex plane, that it satisfies a functional equation, and that the location of its zeros determines the behaviour of the distribution of the prime numbers. In this course we discuss these basic analytic properties together with a number of extensions to zeta-functions over number fields (= finite extensions of the rational numbers). We shall also touch upon the conjectured relation between the distribution of zeros of zeta(s) and eigenvalues of random matrices. |
| Organization | Each meeting will consist of 2x45 minutes of lectures followed by 45 minutes of problems class. |
| Examination | The final grade will be determined by biweekly homework assignments and an exam (oral or written; the precise structure may depend on the class size). The exam and the total of the homeworks will each count for 50% towards the grade. Written exam: June 8, 2010. 13:00h - 16:00h, Educatorium Alfazaal, Utrecht. Resit: June 22, 2010. 13:00h - 16:00h, BBL 083, Utrecht |
| Literature | The first part of the course will be mostly based on Gouvêa's book "p-adic Numbers: An Introduction", whereas the second part will be taught around notes on L-functions by Yiannis Petridis. For both parts other material will be taken from other sources as needed. |
| Prerequisites | Basic ring theory, group theory, and complex analysis. |
| Remarks | For information on the second part of the course see http://www.math.uu.nl/people/plazasva/LFunctions.html |