1. Introduction & Purposes

This mini-course explores the arithmetic of elliptic curves over finite fields. Designed for students with a foundational background in abstract algebra, the curriculum transitions systematically from Weierstrass equations and group laws to isogenies, Vélu’s formulas, and curve classification. Our primary objective is to equip students with the essential mathematical foundations required for isogeny-based cryptography. This mini-course serves a preparatory course for the incoming NCTS lecture series “Elliptic Curves in Mathematical Cryptography” given by Professor Yusuke Aikawa.

2. Outline & Descriptions

This mini-course explores the arithmetic of elliptic curves over finite fields. Designed for students with a foundational background in abstract algebra, the curriculum transitions systematically from Weierstrass equations and group laws to isogenies, Vélu’s formulas, and curve classification. Our primary objective is to equip students with the essential mathematical foundations required for isogenybased cryptography.

Day One: Geometric Foundation and Group Structure

Lecture 1 Weierstrass Normal Form and Projective Spaces

l  Affine and projective spaces

l  Weierstrass normal form derivation

l  Smoothness and discriminant

l  The j-invariant

Lecture 2 The Group Law and Torsion Subgroup

l  Chrod-and-Tangent geometric construction

l  Associativity of group operation

l  Structure of the torsion subgroup

Day Two: Morphisms and Quotient computations

Lecture 3 Isogenies and Kernels

l  Isogenies between elliptic curves

l  Degree and separability

Lecture 4 Computing Quotient Maps

l  Quotient curves

l  Vélu’s formulas

Day Three: Elliptic curves over finite fields

Lecture 5 The Structure of the Endomorphism Ring

l  The Frobenius endomorphism and its characteristic polynomial

l  Classification of the endomorphism ring

Lecture 6 Ordinary and Supersingular Elliptic Curves

l  Ordinary and supersingular curves and equivalent condition

l  Application of supersingular elliptic curves

To complement the lectures, students are encouraged to read the corresponding material from Joseph H. Silverman’s The Arithmetic of Elliptic Curves [Sil09]. The course sections map to the textbook as follows:

Day One: For Lecture 1 and Lecture 2, read Chapters I.1, I.2, III.1, III.2, and III.6, which cover projective planes, Weierstrass equations, the group law, and the general torsion structure.

Day Two: For Lecture 3 and Lecture 4, focus on Chapters III.4 and III.5, which establish the geometry of morphisms, the definition of isogenies, and the role of the kernel. See [DF10, §8.1] and [Was08, §12.3] for Vélu’s formulas.

Day Three: For Lecture 5 and Lecture 6, read Chapters III.4, III.5, III.9, V.1, V.2, and V.3, which cover the endomorphism ring, the Frobenius endomorphism, Hasse’s theorem, and the classification of ordinary versus supersingular curves over finite fields. For isogeny-based cryptograpy, read [DF17] for a survey.

3. Grading

N/A

4. Prerequisites

Preliminary knowledge about algebra.

5. Registration

https://forms.gle/oY8cCsXrSziEnpuc8