Caput Algebra and Geometry 19/20
Faculteit  Science and Engineering 
Jaar  2019/20 
Vakcode  WMMA1920C 
Vaknaam  Caput Algebra and Geometry 19/20 
Niveau(s)  master 
Voertaal  Engels 
Periode  semester I b 
ECTS  5 
Rooster  rooster.rug.nl 
Uitgebreide vaknaam  Caput Algebra and Geometry 19/20  
Leerdoelen  The student is able to 1. apply the bachelor’s topics treated in abstract algebra in more advanced examples relevant to the course in question. 2. apply the theory presented in the course in concrete examples (the specific theory varies each year). 3. explain the basics of class field theory. 4. construct Hilbert class fields over imaginary quadratic fields. 

Omschrijving  The subject of the course varies with the national Mastermath program. This course intends to bridge the gap between the algebra taught in our bachelor program (Group Theory, Rings, (Finite) Fields, Modules), and various courses in Algebra and Geometry offered regularly in the national Mastermath program. 19/20: Complex multiplication in dimension one: This course focuses on the theory of complex multiplication (CM) for lattices of rank 2. Our main aim will be understanding the complex multiplication theory to construct class fields over imaginary quadratic fields. If the time permits we will discuss the computational aspect of the theory; or the connection of the CM theory with elliptic curves. Topics include: 1. Elliptic functions 2. Modular functions, in particular the jfunction 3. Number fields (ramification theory, class group) 4. Class field theory (brief summary of the main theorems) 5. Main theorem of complex multiplication 6. Constructing Hilbert class fields. if the time permits: 7. Computations or 7. Elliptic curves with CM. In previous years, the following topics were discussed: 18/19: Computational algebraic number theory 17/18: The arithmetic of hyperelliptic curves 16/17: Galois Theory and applications 15/16: JP. Serre's book "A Course in Arithmetic". 2014/15 Representation Theory of Finite Groups 2013/14 Galois and Differential Galois Theory 2012/13 Algebraic structures of differential calculus in (non)commutative geometry. 2011/12 Group methods in the geometry of differential equations 2010/11 Galois correspondence 2009/10 Commutative Algebra 2008/09 Arithmetic of elliptic curves 2007/08 Algebraic geometry: cubic surfaces 2006/07 Representation Theory of Finite Groups 2005/06 Galois and Differential Galois Theory 

Uren per week  
Onderwijsvorm 
Hoorcollege (LC), Opdracht (ASM), Practisch werk (PRC)
(Depending on the number of participants, the course is given as a series of lectures or as a seminar in which the participants each prepare and present a chapter from a chosen textbook.) 

Toetsvorm 
Mondeling tentamen (OR), Schriftelijk tentamen (WE)
(Final grade is based on the answers of the homeworks (50%) and the oral examination (50%)) 

Vaksoort  master  
Coördinator  P. Kilicer, PhD.  
Docent(en)  P. Kilicer, PhD.  
Entreevoorwaarden  Linear Algebra, Complex Analysis, Group Theory, Algebraic Structures and Advanced Algebraic Structures are required.  
Opmerkingen  The course unit prepares students for various mastermath courses in which the learning objectives attained are required as prior knowledge.  
Opgenomen in 
