The CTNT 2018 Summer School will take place May 28 – June 1 (followed by a conference June 1-3). All talks during the summer school 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, and a semester of algebraic number theory.

## 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 8:45 at PBB 131 for some welcoming remarks.

Schedule MONDAY MAY 28:

Time |
PBB 129 |
PBB 131 |

8:15 – 9 | Breakfast | |

9 – 9:50 | Mini-course A | Mini-course B |

9:50 – 10:10 | Coffee Break | |

10:10 – 11 | Mini-course C | |

11:10 – 12 | Guest Lecture | |

12 – 2 | Lunch | |

2 – 2:50 | Mini-course D | |

3 – 3:50 | Mini-course E | |

3:50 – 4:10 | Coffee Break | |

4:10 – 5 | Mini-course F | |

5 – 7 | Dinner | |

After 7 | Evening sessions |

Schedule TUESDAY MAY 29 – THURSDAY MAY 31:

Time |
PBB 129 |
PBB 131 |

8:15 – 9 | Breakfast | |

9 – 9:50 | Mini-course E | |

9:50 – 10:10 | Coffee Break | |

10:10 – 11 | Mini-course C | |

11:10 – 12 | Guest Lecture | |

12 – 2 | Lunch | |

2 – 2:50 | Mini-course D | |

3 – 3:50 | Mini-course A | Mini-course B |

3:50 – 4:10 | Coffee Break | |

4:10 – 5 | Mini-course F | |

5 – 7 | Dinner | |

After 7 | Evening sessions |

## 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.

**Guest 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:- May 28: Farshid Hajir (UMass Amherst)
**Title**: The Tsfasman-Vladut Generalization of the Brauer-Siegel Theorem.

**Abstract:**The Brauer-Siegel theorem is a classic result about the growth of the basic invariants of families of number fields whose root discriminants go to infinity. The Tsfasman-Vladut Generalization gives less precise information for more general families. In this talk, I will assume very little familiarity with algebraic number theory and build up the definitions from scratch, then try to explain the significance of these theorems.

M. A. Tsfasman and S. G. Vladut, Infinite Global Fields and the Generalized Brauer–Siegel Theorem.

- May 29: Keith Conrad (UConn)
**Title**: The Biggest Known Prime Number.

**Abstract**: There are infinitely many primes, but at any moment there is a biggest*known*prime. Earlier this year it was announced that a new prime number was discovered larger than all previously known primes, with over 23 million digits.

This prime, like most other large known primes, is a Mersenne prime, for which there is a tailor-made primality test (not applicable to general numbers). In this talk, we will discuss some background about Mersenne primes and how their primality is checked.- Slides.

- May 30: Jeremy Teitelbaum (UConn)
**Title**: Factoring with elliptic curves.

**Abstract**: Lenstra’s elliptic curve algorithm ([1]) for factoring is a standard piece of the toolkit for computational number theory. I will give a brief introduction to this algorithm.

[1] H. Lenstra, Factoring integers with elliptic curves, Annals of Math, 126(3): 649-673.

- May 31: Amanda Folsom (Amherst College)
**Title**: Quantum modular and quantum Jacobi forms

**Abstract**: Quantum modular forms were defined by Zagier in 2010; they are similar to ordinary modular forms in some ways, but are not quite modular on their domain. We will discuss the developing theory and some questions of interest surrounding quantum modular forms, including relationships to mock modular forms, and quantum Jacobi forms, which were recently defined by Bringmann and the speaker.

- May 28: Farshid Hajir (UMass Amherst)
**Mini-course A**: “Basic Algebraic Number Theory” by Liang Xiao (UConn). The goal of this mini-course is to give an overview of basic facts of algebraic number theory such as prime ideal factorization, ideal class group, Dirichlet’s unit theorem, and inertia and decomposition groups. We will focus on explaining these concepts through explicit examples. The examples will be revisited in Mini-course D.**Mini-course B**: “Arithmetic Statistics” by Álvaro Lozano-Robledo (UConn). The area of Arithmetic Statistics is concerned with the average or asymptotic behavior of number theoretic objects in families (such as prime numbers, ideal class groups, zeta functions, or elliptic curves). In each lecture of this course, we will introduce different instances where heuristics and statistics have inspired theorems and famous conjectures. This course will assume a background in algebraic number theory.**Mini-course C**: “Function Field Arithmetic” by Christelle Vincent (University of Vermont). This will be an introduction to function fields over finite fields. We will focus on the simplest example: the polynomial ring**F**_{q}[u] and its fraction field**F**_{q}(u). In particular, we will compare and contrast them to**Z**and**Q**. In the last lecture we will introduce Drinfeld modules of rank 1 and 2 and give an overview of their analogies with the usual exponential function and elliptic curves.**Mini-course D**: “Computational Number Theory” by Harris Daniels (Amherst College). Both Magma and Sage/CoCalc 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 computational questions that appear in number theory.

**Mini-course E**: “Elliptic Curves over Finite Fields” by Erik Wallace (UConn). The course will introduce basic concepts of elliptic curves and associated Galois representations on torsion points, with an emphasis on the case of elliptic curves over finite fields. If time permits, Hasse’s theorem and/or elliptic curve cryptography will be discussed.**Mini-course F**: “L-functions and the Riemann Hypothesis” by Keith Conrad (UConn). Basic properties of L-functions of Dirichlet characters will be described, including special values, analytic continuation, and functional equation. Then we will describe the Riemann Hypothesis for these L-functions: what it says, how it can be tested numerically, and some applications.- Notes.

- Notes.
**Other sessions**: Participants will have time scheduled outside of the lectures to discuss exercises or review lecture notes from the courses. Instructors and graduate assistants will be available to answer questions. We will also offer the following presentations:- Beamer tutorial: we will cover basic guidelines for creating slide talks using Beamer.
- Graduate school preparation panel: we will give advice and answer questions about the process of applying to graduate school and choosing graduate programs.
- Graduate school advising panel: we will give advice and answer questions about the process of selecting a research area and picking a thesis advisor.