## UNCG Summer School in Computational Number Theory 2017

### Modular Forms

From May 22 to May 26, 2017, the University of North Carolina at
Greensboro will host the UNCG Summer School in Computational Number
Theory: **Modular Forms**.

#### Speakers

- Matt Greenberg (University of Calgary)
- Paul Gunnells (UMass Amherst)
- Mark McConnell (Princeton University)
- David Roe (University of Pittsburgh)
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Modular forms play an increasingly important role in number theory and arithmetic geometry. The main focus of the summer school is the study of computational aspects of modular forms and related objects. Topics may include classical modular forms and modular symbols, group cohomology and Galois representations, and lattice enumeration and isometry testing techniques to compute with spaces of modular forms for compact forms of classical groups.

On a typical day, external and local experts give talks in the morning, and in the afternoon students solve problems related to this material. The talks early in the week introduce the students to the subject. Talks later in the week cover related areas of current research and unsolved problems. The problems given to the students might be of a theoretical nature but could also involve programming problems and computer experiments. All problems are aimed at increasing the studentsâ€™ understanding of the material by working with it.

#### Schedule

All talks will take place in Room 213 in the Petty Science Building (campus maps and directions, Google map).

- Monday 5/22
**9:15-9:30**Organizers: Welcome**9:30-11:00**Participants: Introductions**11:15-12:15**Paul Gunnells: Modular forms and modular symbols I (notes (PDF))- Lunch
**2:00-3:00**David Roe: Congruences between and Interpolation of Modular Forms**3:30-5:00**Problem Session- Gunnells: Exercises 1
(PDF)

GP script sample to play with Eisenstein series - Roe: Exercises 1 (link)

CoCalc formerly known as SageMathCloud: Create a free account, and contact David to get access to his worksheet.

- Gunnells: Exercises 1
(PDF)
- Tuesday 5/23
**9:30-10:30**Paul Gunnells: Modular forms and modular symbols II**11:15-12:15**Matt Greenberg: Lattices -- structure and symmetry groups- Lunch
**2:00-3:00**David Roe: Computing with Modular Symbols in Sage**3:30-5:00**Problem Session- Gunnells: Exercises 2
(PDF)

GP script sample to play with Eisenstein series - Greenberg: Exercises 1 (PDF)
- Roe: Exercises 2 (link)

CoCalc formerly known as SageMathCloud: Create a free account, and contact David to get access to his worksheet.

- Gunnells: Exercises 2
(PDF)
- Wednesday 5/24
**9:30-10:30**Paul Gunnells: Modular forms and modular symbols III**11:15-12:15**Matt Greenberg: Automorphic forms and lattices- Lunch
**3:30**David Roe: An introduction to Sage- Gunnells: Exercises 3
(PDF)

GP script sample to play with modular symbols - Greenberg: Exercises 2 (PDF)
- Roe: link

- Gunnells: Exercises 3
(PDF)
- Thursday 5/25
**9:30-10:30**Matt Greenberg: Computing automorphic forms in the totally definite case**11:15-12:15**Mark McConnell: The Well-Rounded Retract and the Voronoi Polyhedron- Lunch
**2:00-3:00**David Roe: Overconvergent Modular Symbols and \(p\)-adic L-functions**3:30-5:00**Problem Session- Greenberg: Exercises 3 (PDF)
- McConnell: Exercises 1
(PDF)

- Roe: link

- Friday 5/26
**9:30-10:30**Mark McConnell: An algorithm for Hecke Operators for SL in higher rank**11:15-12:15**Problem Session- End
- Angie Babei (Dartmouth College)
- Matthew Bates (UMass Amherst)
- Ben Breen (Dartmouth College)
- Benjamin Carrillo (Arizona State University)
- Sara Chari (Dartmouth College)
- Mariagiulia De Maria (University of Luxembourg and UniversitĂ© de Lille 1)
- Marcus Elia (University of Vermont)
- Ricky Farr (UNCG)
- Nathan Fontes (UNCG)
- Yu Fu (UMass Amherst)
- Hugh Geller (Clemson University)
- Matt Greenberg (University of Calgary)
- Paul Gunnells (UMass Amherst)
- Seoyoung Kim (Brown University)
- Andrew Kobin (University of Virginia)
- Michael Leshowitz (UNCG)
- Huixi Li (Clemson University)
- Mark McConnell (Princeton University)
- Jonathan Milstead (UNCG)
- Sebastian Pauli (UNCG)
- Michael Reed (UNCG)
- David Roe (University of Pittsburgh)
- Manami Roy (University of Oklahoma)
- James Rudzinski (UNCG)
- Sandi Rudzinski (UNCG)
- Filip Saidak (UNCG)
- Brian Sinclair (NSA)
- Chris Steinhart (Universitaet des Saarlandes)
- Makoto Suwama (University of Georgia)
- Brett Tangedal (UNCG)
- Debbie White (UNCG)
- Luciena Xiao (California Institute of Technology)
- Yuan Yan (UMass Amherst)
- Dan Yasaki (UNCG)

#### Participants

#### Acknowledgements

The summer school in computational number theory is supported by UNCG and the NSA (H98230-16-1-0027) and the NSF (DMS-1602025).