# Complex analysis prerequisites for Silverman's The Arithmetic of Elliptic Curves

I would like to take a course on elliptic curves using Silverman's The Arithmetic of Elliptic Curves next year. I would be taking complex analysis concurrently, but it was listed as a formal prerequisite, so I was planning to learn some complex analysis beforehand. What topics from complex analysis (esp. chapters from Stein's book) would I need to know to do elliptic curves over $$\mathbb{C}$$?

If you are referring to Stein's Complex Analysis (Princeton Lectures in Analysis, vol 2) by Elias M. Stein, Rami Shakarchi, then I'd suggest the following chapters/topics:

• Chapter 1: Preliminaries to Complex Analysis
• Chapter 2: Cauchy's Theorem
• Chapter 3: Meromorphic Functions and the Logarithm
• Chapter 5: Entire Functions
• Chapter 6: The Gamma and Zeta Functions
• Chapter 7: The Zeta Function and Prime Number Theorem
• Chapter 9: An Introduction to Elliptic Functions

To begin on this topic I highly recommend Lectures on Elliptic Curves, J. W. S. Cassels. London Mathematical Society. Student Texts 24. This book of Professor Cassels is indeed bright and decorated with a truly extraordinary didactic sense. You can get it free by internet.

• It isn't clear to me how this answers the OP's question about the complex analysis needed for elliptic curves, but maybe I am missing something. Does Cassels's book contain the requisite complex analysis?
– J W
Commented Oct 13, 2022 at 12:08
• @J W: Elliptic curves (e.c) involved many branches of mathematics.Respecting complex variables there is mainly the identification of the e.c. defined over $\mathbb C$ with a complex torus via an analytical isomorphism (also there is the complex multiplication). All point is showed by professor Cassels with a very good didactic. Commented Oct 14, 2022 at 15:51
• Thanks for the clarification. I am aware of the points you mention regarding complex variables and elliptic curves, but didn't know whether they were treated in Cassels's book.
– J W
Commented Oct 14, 2022 at 18:11