Research: Number Theory

Summer School 2018: Algorithms for Extensions of Large Degree

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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)

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))	Problem Session		Problem Session
17:00		Student Reports			Student Reports
	Old Town Draught House					Hillbilly Hideaway

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
Claus Fieker:
- Hecke, a software package for algebraic number theory maintained by Claus Fieker and Tommy Hofmann written
  in the Julia programming language
  and based on the
  computer algebra package
  Nemo.
Jordi Guàrdia:
- +Ideals, a package for ideal arithmetic in number fields for Magma

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).