MATH-GA.2210-001 Number Theory, Tue 3:20pm

Instructor: Yuri Tschinkel

Office hours:

• Tue 2:30-3:20pm or by appointment

Syllabus:

• Elementary number theory, quadratic reciprocity
• Diophantine equations
• Local-global principles
• p-adic numbers
• Riemann zeta function
• Prime Number Theorem
• L-functions, primes in arithmetic progressions
• Basic algebraic number theory: quadratic and cubic fields
• p-adic measures, Kummer congruences
• Transcendence

Grading:

• Homework: 20%
• Midterm March 9: 30%
• Final May 4: 50%

Homework:

1. Introduction / Homework (due Febr. 16)
2. Diophantine equations / Homework (due Febr. 23)
3. Rational points, valuations, p-adic numbers / Homework (due March 2)
4. p-adic interpolation, number-theoretic functions / Homework (due March 9)
5. Arithmetic functions, Fourier methods
6. Riemann zeta function: functional equation, special values, Dirichlet L-functions / Homework (due March 30)
7. p-adic measures, Kummer congruences, p-adic L-functions / Homework (due April 6)
8. Prime Number Theorem / Homework (due April 13)
9. Quadratic number fields / Homework (due April 20)
10. Algebraic number theory / Homework (due April 27)

Recommended books:

• A. Chambert-Loir, A Field guide to Algebra, Springer
• J.-P. Serre, A course in arithmetic, Springer