The general aim of this course is to provide a modern approach to number theory by emphasizing harmonic analysis on topological groups. The more particular aim to cover John Tate's thesis that treated Dirichlet L-functions from the adelic point of view. This approach is now used to treat more general L-functions coming from modular forms as well as automorphic forms on higher rank groups and time permitting, we will talk a little about them. As a rule, we will try to spend more time on the interesting questions; many of the background technical facts used will be either left as exercises or pointed to appropriate references.
Prerequisites
This course will assume some knowledge of algebra, basic algebraic number theory and basic point set topology. More precisely, you should be comfortable with the following: groups, rings, fields, Galois theory, Q_p, Z_p, metric spaces, topological spaces, open/closed/compact/locally compact/dense/discrete sets, basic measure theory, basic complex analysis and at least somewhat familiar with the following: number fields and their rings of integers, prime decomposition in extensions of number fields, ideal class group of number fields, local fields, discrete valuation rings, extensions of local fields.
Some familiarity with L-functions (coming from Dirichlet characters, modular forms, etc.) and their expected properties will be helpful for motivation but not required.
Reading
J.P. Serre: Local fields (for background on local fields)
Dinakar Ramakrishnan: Fourier analysis on number fields (a good reference for topological groups and fields, Fourier analysis, adeles, and Tate's thesis)
Jürgen Neukirch: Algebraic number theory (for background on algebraic number theory, though this book has much more than we will need)
Cassels and Frohlich: Algebraic number theory (another excellent reference for algebraic number theory and class field theory)
Lecture notes, homeworks, etc.
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Abhishek Saha HG G 68.1 abhishek.saha@math.ethz.ch
