Notes from the International Autumn School on Computational Number Theory

This volume collects lecture notes and research articles from the International Autumn School on Computational Number Theory, which was held at the Izmir Institute of Technology from October 30th to November 3rd, 2017 in Izmir, Turkey. Written by experts in computational number theory, the chapters cover a variety of the most important aspects of the field. By including timely research and survey articles, the text also helps pave a path to future advancements. Topics include:Modular formsL-functionsThe modular symbols algorithmDiophantine equationsNullstellensatzEisenstein seriesNotes from the International Autumn School on Computational Number Theory will offer graduate students an invaluable introduction to computational number theory. In addition, it provides the state-of-the-art of the field, and will thus be of interest to researchers interested in the field as well.

Part I: Lecture Notes.- An Introduction to Modular Forms.- Computational Arithmetic of Modular Forms.- Computational Number Theory in Relation with L-functions.- Exponential Diophantine Equations.- Part II: Research Contributions.- Nullstellensatz via Nonstandard Methods.- On the (1 + u2 + u3)-Constacyclic and cyclic codes over the finite ring F2 + uF2 + u2F2 + u3F2 + vF2.- On higher congruences between cusp forms and Eisenstein series.- Lucas Numbers which are Products of Two Balancing Numbers.