Connecticut Summer School 16

All talks will be at the Pharmacy/Biology Building (PBB 129 and 131), and the coffee breaks will be outside of PBB 129. A campus map pointing to PBB can be found here (Google labels the building as the “School of Pharmacy”).

Goals of the Summer School

The organizers of the summer school hope that the students attending this event will learn fundamental ideas in contemporary number theory and have a sense of some directions of current research. For undergraduates, the summer school will expose them to topics not available in a typical college curriculum and we will encourage applications from students at institutions where advanced topics in number theory are not ordinarily taught. The school will provide a chance for participants to meet fellow students, as well as faculty, interested in number theory.

Expected Background of Students

  • Undergraduate Students: a semester each of elementary number theory, abstract algebra, and complex analysis.
  • Graduate Students: a year of abstract algebra. Suggested, but not required: a semester of algebraic number theory and familiarity with p-adic numbers.

Structure of the Summer School

The summer school will take place at the Storrs campus of the University of Connecticut. Activities will be designed at two levels, targeting advanced undergraduate and beginning graduate students. Lectures will be scheduled so that a student could attend almost all lectures if desired, choosing according to their background and interests. The daily schedule in the summer school will be as shown in the following table.

  • On the first day only, we will all meet at 8:50 at PBB 131 for some welcoming remarks.
Time PBB 129 PBB 131
8:15 – 9 Breakfast
9 – 9:50 Lecture A1 Lecture B1
9:50 – 10:10 Coffee Break
10:10 – 11 Exercises Lecture B2
11:10 – 12 Plenary Lecture (PBB 131)
12 – 2 Lunch
2 – 2:50 Magma/Sage Tutorials (PBB 131)
3 – 3:50 Exercises Lecture A2
3:50 – 4:10 Coffee Break
4:10 – 5 Exercises Lecture B3
5 – 7 Dinner
After 7 Projects/Latex/Discussion (PBB 131)

 

On the last day of the summer school, students will present the progress on their projects in the morning, prior to the beginning of the research conference in the afternoon.

Lecture series

Each day’s events at the summer school is as follows. The videos for the lectures can be found at the UConn Math YouTube Channel.

  • Plenary Lectures: Each day will have a plenary talk, where a number theorist will give an overview (accessible to advanced undergraduates and beginning graduate students) of a current trend in number theory. Titles of the lectures and speakers:
    • “Gauss and the Arithmetic-Geometric Mean” by David Cox (Amherst C.) – Video Slides
    • “Mock and Quantum Modular Forms” by Amanda Folsom (Amherst C.) – Video Slides
    • “Sphere Packings and Modular Forms” by Stephen D. Miller (Rutgers) – Video
    • (Cancelled due to illness) “The Geometric Approach to Modular Forms”  by Ila Varma (Harvard)
      • Replaced by “Why is the Riemann Hypothesis important?” by Keith Conrad (UConn) – Slides
  • Lecture A1: “Introduction to Modular Forms,” by Keith Conrad. Topics will include Eisenstein series and q-expansions, applications to sums of squares and zeta-values, Hecke operators, eigenforms, and the L-function of a modular form.
  • Lecture A2: “Introduction to Elliptic Curves,” by Álvaro Lozano-Robledo. This will be an overview of the theory of elliptic curves, discussing the Mordell-Weil theorem, how to compute the torsion subgroup of an elliptic curve, the 2-descent algorithm, and what is currently known about rank and torsion subgroups of elliptic curves.
  • Lecture B1: “Computational methods for modular and Shimura curves,” by John Voight. The classical method of modular symbols on modular curves is introduced to compute the action of the Hecke algebra and corresponding spaces of modular forms. Generalizations to Shimura curves will then be discussed using Dirichlet modular symbols.
  • Lecture B2: “Introduction to the local-global principle,” by Liang Xiao. The plan is starting with an introduction to Qp, then introducing Hilbert symbols and the local-global principle for quaternion algebras and central simple algebras, and ending with examples of the failure of the local-global principle.
  • Lecture B3: “Gauss sums and the Weil Conjectures,” by Bin Zhao. The topics include will Gauss sums, Jacobi sums, and Weil’s original argument for diagonal hypersurfaces when he raised his conjectures. Further developments towards the Langlands program and the modularity theorem will be mentioned at the end.
  • Exercise sessions: Each day will have a period set aside for students to work on exercises together, led by senior graduate students.
  • Magma and Sage tutorials, by Harris Daniels. Both Magma and Sage are extremely useful computer algebra packages for doing research in number theory. The goal for these sessions is to give an introduction to both packages so that students can solve proposed computational exercises from the lecture courses and projects of the summer school.
  • Evening Latex tutorial: (LaTeX intro files) The plan is to cover the following topics:
    • Basic use and setup of LaTeX.
    • Commutative diagrams.
    • Preparing talks with the Beamer package.
    • An introduction to graphics packages.
  • Evening project and discussion session: The students will be grouped into different projects to discuss and work collaboratively. Instructors and graduate student mentors are available to assist the students. The projects will consist of open-ended problems or more involved exercises, with computational aspects to them, that are related to the given lecture series and possibly leading to some current research topics.

Lecture Series: day by day

  • Lecture A1: “Introduction to Modular Forms,” by Keith Conrad. Videos
    • Lecture 1: Definition of modular forms, Eisenstein series, and q-expansions.
    • Lecture 2: The q-expansion of Eisenstein series, fundamental domains, and modular forms for a subgroup.
    • Lecture 3:  Modular forms and sums of four squares, computing dimensions of spaces of modular forms.
    • Lecture 4: Computing dimensions of spaces of modular forms (continued), application to zeta-values, Hecke operators and the L-function for the discriminant modular form of weight 12.
  • Lecture A2: “Introduction to Elliptic Curves,” by Álvaro Lozano-Robledo. Videos
    • Lecture 1: What is an elliptic curve? Curves by degree and genus. The addition law. Curves over finite fields.
    • Lecture 2: Torsion points on elliptic curves, the rank, and Z-linear independence of points.
    • Lecture 3: Calculating the Mordell-Weil, Selmer, and Sha groups.
    • Lecture 4: The modular form and L-function of an elliptic curve, Taniyama-Shimura-Weil, and BSD.
  • Lecture B1: “Computational methods for modular and Shimura curves,” by John Voight. Videos
    • Lecture 1: Modular curves, fundamental domains, Farey symbols.
    • Lecture 2: Homology of modular curves via modular symbols.
    • Lecture 3: Examples, applications, and higher weight.
    • Lecture 4: Shimura curves, Dirichlet symbols, and examples.
  • Lecture B2: “Introduction to the local-global principle,” by Liang Xiao. Videos
    • Lecture 1: Quaternion algebras and Qp.
    • Lecture 2:  Hilbert symbols, basic properties, and their relation to quaternion algebras.
    • Lecture 3: Product formula for Hilbert symbols, local-global principles for quaternion algebras.
    • Lecture 4: Central simple algebras, Brauer groups, and failure of local-global principles.
  • Lecture B3: “Gauss sums and the Weil Conjectures,” by Bin Zhao. Videos
    • Lecture 1: Gauss sum, counting numbers of solutions of equations in finite fields, zeta functions of curves over finite fields, the Riemann hypothesis/Weil conjecture for curves over finite fields;
    • Lecture 2: Divisors, Riemann-Roch Theorem for smooth projective curves, rationality of zeta functions of curves over finite fields;
    • Lecture 3: Intersection theory on surfaces, Weil’s proof of Riemann hypothesis;
    • Lecture 4: Generalization of Weil conjecture, modularity and Langlands programme.
  • Magma and Sage Tutorials, by Harris Daniels.
    • Lecture 1: The basic syntax and working with elliptic curves in Sage.
    • Lecture 2: Congruence subgroups, modular forms and Farey symbols in Sage.
    • Lecture 3:  An overview of Magma.
    • Lecture 4:  A tour of the LMFDB database.