# Before you read Sliverman's The Arithmetic of Elliptic Curves Book V0.1

# Yao Wang

Joseph H. Silverman's "The Arithmetic of Elliptic Curves" is a wonderful self-contained book for people interesed in elliptic curve. However, this book is tough even for those who have a sold maths background.

I (a former Maths Phd) have self-studied this book for a few months and here I share with you a self-made guidaince and I will provide a selected list of chapters for crypto nerds.

## Preparation and Required Mathematics.

1. Abstract Algebra including groups, rings, fields and relations between them.
2. Galois theory. (for advanced topics)
3. Commutative algebra and basic algebraic geometry

Abstract algebra is bread and butter for evething about EC.
Galois theory is for understanding relations of curves based on different base field.

The third one is for those who want to have a rigorous understanding for divisors and pairing.

## A Selected List of Chapters for Crypto Nerds

I would divide the book into three parts.

1. General theory applied to both char-0 fields including **C,Q**, and char-p fields including **F_p and F_q** etc.
2. ECs on char-p field
3. ECs on char-0 field.

Most topic which crypto people find interesting are in part 2.

| Chapter | Picked Sections | Comments |char of field|
| -------- | -------- | -------- |----|
| ChI Algebraic Varieties | 1,2,3 | This chapter is warm-up Ignores exmaples based on **Q** | 0, p|
|Ch II Algebraic Curves|1,2,3,4| basics|0,p|
|Ch II The Geometry of Elliptic Curves|1,2,3,4,5|Ignores exmaples based on **Q**. Read isogeny related sections if you are interested in isogeny-based cryptography|0,p|
| Ch V Elliptic Curves over Finite Fields |1,2,3| crypto-related elliptic curves lives here |p|
|Ch X Computing the Mordell–Weil Group| TBD| twisting is here |0,p|
| Ch XI Algorithmic Aspects of Elliptic Curves|all |all important algorithms are here|p|

I believe you do not need to read the rest of the book except you are a number theory ethusiast.

## Disclaimer

I am not an expert in elliptic curve and this note may contain errors or inaccurate knowledge.

## Cheers
Feel free to leave a comment. share your own experience about this book. Learning is endless and let's do it together.