MATH 131 CALCULUS I Spring 2000

MATH 410 – Topics in Number Theory

Instructor: Lucian Ionescu; Contact Info: STV 312J 438-7167;[email protected]; Office Hours: TBA, or by appointment at other times. Main info sources: 1) A Classical Introduction to Modern Number Theory, K. Ireland and M. Rosen, 2 nd ed.; 2) The SAGE Tutorial (available online); 3) The course’s web site: www.ilstu.edu/~lmiones/mat410.html Content: The course integrates research on the Hasse-Weil zeta function zeros of elliptic curves over finite fields and teaching advanced topics of Number Theory. We focus on concepts, while testing the students’ ability to work with them concretely via SAGE as a computer interface to computations. The theory will focus on Gauss sums and Gauss periods, with their relation with Jacobi sums and number of points of elliptic curves over finite fields. For computations and to check our mastery of the concepts, at a practical level, we will use SAGE, which is a free, online mathematical software. Prerequisites: MAT 330 or consent of instructor. Objectives: The general goals of the course are 1) To develop some of the advanced topics of number theory, building on top of those learned in MAT 330; 2) To introduce the student to applications of number theory in other areas of mathematics; 3) To train the student with the use of an advanced mathematics software while learning advanced topics in mathematics; 4) To emphasize the connections with algebraic structures, after a review of groups, rings and fields, with their geometric significance at the applications / user’s level; 5) To integrate teaching and research, by researching a specific problem: the significance and structure of Hasse-Weil zeta function zeros, both theoretically and computationally, through computer explorations. In general it is the aim of the course to provide students with an opportunity to enhance their ability to learn, understand and present topics of mathematics; to give students an experience of being an abstract mathematician; to increase their understanding of how mathematical products are built (proofs), and to communicate abstract ideas in written and oral form, via in class exercises and short presentations at the board. Course Format: You are responsible for reading assignments, and this is not a trivial task. Reading assignments are to be completed before the next scheduled meeting of the class. During the class, we will focus on understanding the concepts and methods involved. After the class, write one page outline of the corresponding section, before attempting the homework, which consists in writing SAGE worksheets implementing the computations corresponding to the concepts studied. Attempt to complete the homework / problems before the next class period, when you should turn in the outline for one point per submission. Before discussing the new assigned material, we will discuss your difficulties or questions regarding the homework. You may choose to present your solutions to the homework problems, using the projector, for class participation points. Attendance and active participation are expected. Homework: will be assigned weekly; check the course’s web site for updates. Please keep up with the syllabus. Exams: In class quizzes will replace the usual exams during the semester. Turn in your SAGE programs documented by your explanations, as a final project / culminating experience, in place of a final exam. Evaluation: Grades will be based on the following points: Homework 200 Class activities (Attendance 100, outlines 100, and participation 100): 300 Final Project 200 Total 700 points. The grading scale is based on: A [90-100%], B [80-90%), C [70-80%), D [60,70%), F [0,60%).

N.B.: Students who believe they may need accommodations in this class/program are encouraged to contact the Disability Access Center as soon as possible to better ensure that such accommodations are implemented in a timely fashion. “Warning: Plagiarism and cheating are serious offenses. Penalties can range from a minimum of a zero grade on the invalid instrument to expulsion from the University”

Suggestions for Learning (Mathematics)

Make an outline: For each assigned section of the text, make an outline containing a description of the major concepts and procedures. Also write down a brief definition of all technical terms. Find a typical example of each concept; find a non- example. Sketch a picture representing the concepts, if appropriate. Show relationships between concepts. As you are making this outline, make a list of questions you want to ask your study partner or instructor.

Do the assignments: Work problems and do other tasks assigned. Develop the list of questions to ask your study partner or instructor.

To have a good head start, the topics for the first few weeks are listed below. The sequel topics will depend on the adopted pace, student’s background and SAGE programming abilities, which will be learned during the first part of the course.

Week & Topic 1) Introduction: a) Crash recall on congruence arithmetic and finite abelian group and fields; b) Using SAGE to tell the computer to do the computations for us (Opening an account; elementary NT with SAGE); c) Relations between Number Theory, Discrete Mathematics and Abstract Algebra. 2) Finite fields (FF), Elliptic Curves (EC) and Counting Points a) Programming and plotting with SAGE; b) Finite fields and their multiplicative structure (Klein geometry); c) Counting points of EC over FF; how Jacobi sums result out of this; 3) Gauss sums and Jacobi sums; relations with Gauss periods 4) Characters: additive and multiplicative (Dirichlet characters) 5) The generating zeta function: a) Hasse-Weil; b) Weil conjectures 6) Exploring Weil zeros 7) Exploring Gauss periods 8) Recap & the “big picture” 9) – 16) A more in depth study and research: a. Gauss sums, Gauss periods, Jacobi sums and Weil zeros (periods) revisited. b. Studying the dependence on the prime number, and the corresponding structure of the finite field (multiplicative group / factors of p-1). c. … and whatever interesting we may find together during the first weeks!