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Course, academic year 2018/2019
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Modular forms and L-functions I - NMAG462
Title in English: Modulární formy a L-funkce I
Guaranteed by: Department of Algebra (32-KA)
Faculty: Faculty of Mathematics and Physics
Actual: from 2018
Semester: winter
E-Credits: 3
Hours per week, examination: winter s.:2/0 Ex [hours/week]
Capacity: unlimited
Min. number of students: unlimited
State of the course: not taught
Language: English, Czech
Teaching methods: full-time
Additional information: https://sites.google.com/site/vitakala/teaching/17mf
Guarantor: Mgr. Vítězslav Kala, Ph.D.
Class: M Mgr. MSTR
M Mgr. MSTR > Povinně volitelné
Classification: Mathematics > Algebra
Annotation -
Last update: doc. Mgr. et Mgr. Jan Žemlička, Ph.D. (10.05.2017)

Modular forms and L-functions are central objects in modern number theory, which played an important role in the proof of Fermat's Last Theorem. They are certain complex functions encoding information of number-theoretic interest, e.g., about the distribution of prime numbers, or numbers of solutions of diophantine equations. Combining analytic and algebraic methods, the course will cover their basic properties and some applications. Specific choice of topics will depend on the interests of participants.
Literature -
Last update: doc. Mgr. et Mgr. Jan Žemlička, Ph.D. (10.05.2017)

J. S. Milne: Modular Functions and Modular Forms,

S. Lang: Algebraic Number Theory, Second Edition, GTM, Springer 1994

F. Diamond, J. Shurman: A First Course in Modular Forms, GTM, Springer 2005

D. Bump: Automorphic Forms and Representations, Cambridge Studies in Advanced Mathematics 55 (1998)

Requirements to the exam - Czech
Last update: doc. Mgr. et Mgr. Jan Žemlička, Ph.D. (10.05.2017)

Zkouška bude ústní s 30-60 minutami na přípravu jedné nebo dvou otázek, odpovídajících probrané látce na přednáškách.

Syllabus -
Last update: doc. Mgr. et Mgr. Jan Žemlička, Ph.D. (10.05.2017)

Riemann surfaces

Upper half plane and SL(2, R)

Elliptic functions

Modular forms

Eisenstein's series, Ramanujan's tau function

Hecke operators

Zeta function and Dirichlet L-functions

Analytic continuation and functional equation

Theta functions

L-functions of modular forms and elliptic curves

FLT and modularity theorem

 
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