Analytic Number Theory

Course Outline:

The following topics will be covered.

1. Primes in arithmetic progression (the theorem of Dirichlet): characters of finite abelian groups, analytic theory of Dirichlet series and $L$-functions, analytic continuation of the zeta function, Euler product formula, equi-distribution of primes in arithmetic progressions, a brief look at Chebotarev density theorem.

2. (Elliptic) Modular forms: the modular group and its action on Poincare upper half plane- the modular curve, modular functions and modular forms, examples (Eisenstein series, elliptic functions, theta functions), the algebra of modular forms and its structure. Hecke operators, Hecke $L$-functions, applications to Ramanujan's conjectures on $\tau (n).$ Applications of theta functions to representation of integers by quadratic forms.

3. An abelian trace formula (Poisson summation formula), Selberg trace formula in general, for Heisenberg groups, for $SL_2 (\mathbb{R})$, connections with spectral geometry and quantum mechanics, Weyl's law for hyperbolic Riemann surfaces, Riemann hypothesis for Selberg zeta function.

4. Seminar topics: Riemann-Weil explicit formula, Adele class space of Connes, noncommutative geometry and Connes' trace formula.

Student presentations:

1. Chebotarev density theorem (Sandeep Garch).

2. Prime number theorem (Travis Ens).

3. Selberg trace formula for Heisenberg group (Asghar Ghorbanpour).

4. Tate's thesis (Jason Haradyn).

5. Elliptic functions (Jiteng Jin).

6. Metaplectic representation and applications to Siegel modular forms (Ali Fathi).

7. Four squares theorem of Lagrange via theta functions ( Abdullah Abubakar).

8. Asymptotic evaluation of integrals (Steepest descent and stationary phase approximation methods, Airy function (Friday Michael).

9. Two dimensional Galois representations (Mitsuru Wilson).

10. Hecke algebras and Bost-Connes systems in class field theory (Sajad Sadeghi).

References:

1. A course in arithmetic; J. P. Serre (the last two chapters will be used).

2. Principles of Harmonic Analysis; A. Deitmar and S. Echterhoff (chapters 9, 10, 11 will be used).

3. Automorphic Forms; A. Deitmar (for self study).

4. Introduction to analytic number theory; T. M. Apostol (for self study).

5. Euler through time; V. S. Varadarajan (for self study; mainly of historical interest. highly recommended!).

6. Automorphic forms and the Langlands program (International press). Survey articles in this volume by Knapp, Gelbart, Gan, and Lapid are recommended.

7. The Riemann Hypothesis: Arithmetic and Geometry; Jeffrey C. Lagarias (in Surveys in Noncommutative Geometry)

8. Noncommutative Geometry and Number Theory; Paula Tretkoff (in Surveys in Noncommutative Geometry).