Math 2540: Number Theory (Preliminary Syllabus)
Instructor:
Junehyuk Jung (Assistant professor)
Class meeting:
TTh 2:30-3:50 PM Kassar 105
Office Hour:
By appointments only. Zoom preferred.
Preferred contact:
Website:
https://sites.google.com/brown.edu/junehyuk
Textbooks:
No textbooks for this course. I will let you know recommended readings during the class.
Prerequisites:
I intend the course to be self-contained, but a little knowledge about Fourier analysis and complex analysis will be helpful.
Topics:
This course is an introduction to automorphic forms on $PSL_2$ over rationals and imaginary quadratic number fields. These include holomorphic modular forms and Maass forms. Topics that I plan to cover are:
-Spectral theory of Laplace-Beltrami operator on $PSL(2,\mathbb{Z}) \backslash \mathbb{H}$. (Maass forms)
-Analytic continuation of Eisenstein series on a hyperbolic surface.
-Hecke Theory.
-Selberg's trace formula.
-Kuznetsov trace formula.
-Hyperbolic geometry and dynamics.
-Various estimates concerning Maass forms.
-Prime geodesic theorem.
-Arithmetic hyperbolic 3 manifolds
-Bianchi subgroups
-Binary Hermitian quadratic forms and immersed totally geodesics surfaces.
-etc.
However, what I will actually cover depends on students' background and interest.
Goals:
The main purpose of this course is to understand various analytic behavior of automorphic forms and corresponding $L$-functions. At the beginning of the semester, if necessary, I will review basic materials that will be used throughout the course. For the first half of the semester, I will introduce basic tools for studying Maass forms and their $L$-functions. The tools include Selberg's trace formula, and I will spend a few weeks deriving the formula by carefully going over the analytic continuation of Eisenstein series. For the second half of the semester, I will introduce various arithmetic problems concerning arithmetic hyperbolic manifolds and Maass forms defined on them.
Grading:
The grade will be based on 8 homework assignments (80%) and attendance (20%).
A total score of 90% or more guarantees an A, a score of 80% or more a B, 70% or more a C, 60% or more a D.
Collaboration policy:
You may discuss with anyone in the department, and you may use any resource as long as you state/cite clearly in your work.
Credit hours and estimate of work load.
Expect 12 weekly hours over 12 weeks of class + reading + assignments + self study. 144 hours in total.
Accommodations for students with disabilities.
Brown University is committed to full inclusion of all students. Please inform me early in the term if you may require accommodations or modification of any of course procedures. You may speak with me after class, during office hours, or by appointment. If you need accommodations around online learning or in classroom accommodations, please be sure to reach out to Student Accessibility Services (SAS) for their assistance (seas@brown.edu, 401-863-9588). Students in need of short-term academic advice or support can contact one of the deans in The College.
Other resources.
Especially during this period it is important to care for one's well-being. The university has resources available, to all students, in particular CAPS, the college, the graduate school, and in our department the directors of undergraduate and graduate studies.