# Research: Number Theory

## Summer School 2018: Algorithms for Extensions of Large Degree

2020 2019 2018 2017 2016 2015 2014 2013 2012

### UNCG Summer School in Computational Number Theory 2018

From May 28 to June 1, 2018, the University of North Carolina at

Greensboro will host the UNCG Summer School in Computational Number

Theory: **Algorithms for Extensions of Large Degree**.

Algorithms for extensions of global and local fields are the backbone of computational number theory.

Improvements in computing power have made it feasible to conduct computations in larger and larger degree.

As the complexity of algorithms for field extensions depends on the degree of the extensions,

this has increased the interest in asymptotically fast algorithms.

Among others we will consider algorithms for integral bases computation and ideal arithmetic.

#### Speakers

- Peter Bruin (Universiteit Leiden)
- Claus Fieker (Technische Universität Kaiserslautern)
- Jordi Guàrdia (Universitat Politècnica de Catalunya)

#### Application

We expect to be able to provide support for approximately ten graduate students (travel, dorm accomodation, food).

Interested students should complete the online

application

and have a letter of reference sent via email to batanged@uncg.edu by

**March 22, 2018**. We will respond with funding decisions by April 13, 2018.

#### Schedule

All talks will take place in room 213 in the Petty Building (campus

maps and directions). The problem sessions will take place in Petty 213 and the computer lab in Petty 211.

Time | Sunday 5/27 | Monday 5/28 | Tuesday 5/29 | Wednesday 5/30 | Thursday 5/31 | Friday 6/1 | |
---|---|---|---|---|---|---|---|

9:00 | Arrival | Welcome | Coffee | ||||

9:30 | Participants: Introductions slides (pdf) |
Peter Bruin: Curves and function fields Notes: Curves, Function Fields, Picard Groups I, II, and III (pdf) |
Jordi Guàrdia: MacLane valuations vs Okutsu frames | Claus Fieker: Some hope | Peter Bruin: Algorithms for curves and Picard groups Cocalc Worksheet |
||

10:45 | Coffee | ||||||

11:15 | Claus Fieker: The curse of the large degree Classgroup Computations in Large Degree (pdf) |
Claus Fieker: The power of asymptotics | Peter Bruin: Differentials, \(L(X,D)\), Riemann-Roch | Jordi Guàrdia: Montes algorithm and applications | Jordi Guàrdia: Montes algorithm examples | ||

12:30 | Lunch | ||||||

14:00 | Jordi Guàrdia: The origins: Kummer, Dedekind, Ore | Problem Session | Excursion | Problem Session | |||

15:30 | Problem Session I (Claus (pdf), Jordi (pdf)) |
||||||

17:00 | Student Reports | Student Reports | |||||

Old Town Draught House | Hillbilly Hideaway | Grashoppers |

#### Materials

For notes and problems please see the schedule. The following are software package, background reading, and references ordered by speaker.

- Peter Bruin:
- W. Fulton’s book “Algebraic Curves: An Introduction to Algebraic

Geometry” can be downloaded (legally) at

http://www.math.lsa.umich.edu/~wfulton/CurveBook.pdf. - modgalrep, a package for computing in Jacobians of projective curves over finite fields, with

applications to modular Galois representations, based on

PARI/GP

- W. Fulton’s book “Algebraic Curves: An Introduction to Algebraic
- Claus Fieker:
- Jordi Guàrdia:

#### Participants

- Vishal Arul (MIT)
- Chad Awtrey (Elon University)
- Alex Best (Boston University)
- Peter Bruin (Universiteit Leiden)
- Benjamin Carrillo (Arizona State University)
- Lance Everhart (UNCG)
- Endrit Fejzullahu (University of Florida)
- Claus Fieker (Technische Universität Kaiserslautern)
- Nathan Fontes (UNCG)
- Richard Gottesman (University of California, Santa Cruz)
- Jordi Guàrdia (Universitat Politècnica de Catalunya)
- Cole Love (UNCG)
- Jonathan Milstead (UNCG)
- Michael Musty (Dartmouth College)
- Sebastian Pauli (UNCG)
- James Rudzinski (UNCG)
- Sandi Rudzinski (UNCG)
- Filip Saidak (UNCG)
- Bernd Sing (University of the West Indies)
- Carlo Sircana (Technische Universität Kaiserslautern)
- Brett Tangedal (UNCG)
- Shuai Wei (Clemson University)
- Dan Yasaki (UNCG)
- William Youmans (University of South Florida)
- Anthoula Zervou (University of Paderborn)
- Dena Zhu (Duke University)

#### Organizers

- Sebastian Pauli (UNCG)
- Filip Saidak (UNCG)
- Brett Tangedal (UNCG)
- Dan Yasaki (UNCG)

#### Acknowledgements

The summer school in computational number theory is supported by UNCG

and the NSF (DMS-1602025).