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Notes from the International Autumn School on Computational Number Theory

Ilker Inam and Engin Büyükasik, eds.
Publisher: 
Birkhäuser
Publication Date: 
2019
Number of Pages: 
376
Format: 
Paperback
Series: 
Tutorials, Schools, and Workshops in the Mathematical Sciences
Price: 
84.99
ISBN: 
978-3-030-12557-8
Category: 
Collection
[Reviewed by
Nathan Ryan
, on
03/15/2020
]
In October 2017, some Turkish mathematicians organized an "International Autumn School on Computational Number Theory" at the Izmir Institute of Technology. The aims of the conference were to:
 
  • Introduce Master and PhD students to computational number theory.
  • Teach some topics in modern number theory related to current research.
  • Introduce the practical use of computer algebra systems for studying research questions in number theory
The notes from this summer school are in two parts: the first part is made up of four chapters written by an all-star cast: Henri Cohen writes about modular forms and L-functions, Gabor Wiese writes about the modular symbol algorithm and Florian Luca writes about exponential Diophantine equations. The second (much shorter) part consists of four research papers on topics related to the topics covered in the first part; beyond being nominally about the same material, there is little obvious connection between the two parts (one deals with modular forms, one deals with cyclic codes, one deals with Diophantine equations and one deals with the Nullstellensatz). This review focuses on the first part, the course notes from Cohen, Wiese and Luca.
 
Cohen has influenced a generation of computational number theorists both with the textbooks he has written and for the creation of PARI/GP. His perspective always seems to be to move from the explicit and concrete to the general, and his two chapters follow that pattern. Additionally, the well-chosen exercises in both chapters often require explicit computer calculation to solve. They’re instances of experimental number theory at its best.  An important thing to note, especially for the chapter on L-functions, about which he has written less, is that we are being shown an expert’s deep bag of tricks for computing things. To have all of these written down in one place is particularly convenient.
 
Wiese is responsible for a lot of MAGMA packages related to modular forms and Hecke algebras. He has a long track record of doing and presenting on algebraic approaches to computing modular forms. These notes are a self-contained introduction to the modular symbols algorithm, including its proof and all the necessary ingredients from group cohomology. Along the way, there are both theoretical and computational exercises which, by the end, will allow the reader to implement the modular symbols algorithm over an arbitrary ring.
 
Luca is a renowned problem-poser and solver, and has published extensively on solving Diophantine equations and linear recurrences. While his chapter is rather different from the other three (and less in my area of expertise), I found it to be as gentle an introduction to the area of exponential Diophantine equations as he promises in the introduction of the chapter. Two highlights for me are the collection of 60 or so problems in various states of solution that he poses and an application of the LLL algorithm that was new to me.
 
These four chapters written by three illustrious number theorists should have accomplished the aims of the workshop. I didn’t attend the workshop but, if these chapters are any indication, it would have been a great experience.

 

Nathan Ryan is a Professor of Mathematics at Bucknell University and an affiliated faculty member in Latin American Studies.  Whenever he can, he tries to get to South America with his family; most recently he has been working with students in Ecuador.
 

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