Order of study (books) modular forms/elliptic curves/Iwasawa theory

can you please tell me a order to study these books?

1. Algebraic number theory  (Neukirch)

2. Cohomology of number fields (Neukirch)

3. A first course in modular forms (Diamond)

4. The Arithmetic of elliptic curves (Silverman)

5. Advanced topics on the arithmetic of elliptic curve (Silverman)

P.S. May I ask if these books cover the path to understand this book of papers? —-> Elliptic curves, modular forms and Iwasawa theory (Loeffler)

• Does Neukirch really have an algebraic geometry book? Presumably you mean his algebraic number theory book. Commented Sep 6, 2023 at 16:31
• Yes, I mean algebraic number theory
– user1201101
Commented Sep 6, 2023 at 16:44
• FWIW: the book you refer to, edited by me (jointly with Sarah Zerbes), is not a textbook; it a collection of research papers, which are not intended to be read sequentially, each one being addressed at a narrow audience of experts in a specific subarea of number theory. [...] Commented Sep 14, 2023 at 6:55
• I'm not sure there's anyone in the world who could read the whole thing cover-to-cover without having to consult numerous other books and papers as they go along. Certainly I couldn't do that myself! (Perhaps the book's dedicatee, our great teacher John Coates, could have done so; but he is sadly no longer with us.) Commented Sep 14, 2023 at 6:57
• Other books you should read, if you want to understand this material better, include Coates and Sujatha's "Cyclotomic Fields and Zeta Values". Commented Sep 14, 2023 at 7:06

hunter's answer is a reasonable one, so I will just elaborate on some details. I'll name the books respectively Neukirch $$1$$, Neukirch $$2$$, Diamond Shurman, Silverman, and Advanced Silverman.

First, I am assuming you mean Neukirch's "Algebraic Number Theory"? I don't think he has written a book with the title "Algebraic Geometry".

Now, there are two clear sequences in that list of books, namely $$\text{Neukirch 1} \longrightarrow \text{Neukirch 2}$$ $$\text{Silverman} \longrightarrow \text{Advanced Silverman}$$ By this I mean that the book on the right presumes knowledge from, and is generally more advanced than, the book on the left. This doesn't mean you have to finish the book on the left from cover to cover to begin the one on the right, but I would advise getting through some introductory chapters of the left before opening the one on the right.

Diamond Shurman is a great book, and gives an introduction to most of the advanced concepts they use. The story they tell is, in very coarse terms, how rational elliptic curves correspond to modular forms. One of the very important ingredients of this is $$L$$-functions.

Note that there are also intersections among the books. to mention a few; modular forms and functions enter both Silverman and Advanced Silverman, while having a major role in Diamond Shurman. Varieties, including elliptic curves, are also part of Diamond Shurman while being the central topic of both Silvermans. Neukirch 1 devotes much of the latter part on class field theory, which is involved in (at least) the chapter of complex multiplication of Advanced Silverman.

However, in my experience none of these intersections are too serious - whenever you will encounter a topic for the second time, it would be with a different angle, and you will be able to skip past parts you know.

Just reiterating what hunter said, in math it is often wise having an eye on where you are going and trying to figure out what you need, in contrast to just reading textbooks from cover to cover. Opening an advanced book also often gives perspective and direction when studying the more basic material. That being said, here is my reading guide:

1. Start Neukirch 1.
2. Start Diamond Shurman and Silverman (cpt. 3+4, skim back in 1+2 when needed).
3. Finish Neukirch 1 cpt. 1+2, Diamond Shurman cpt. 1++, Silverman cpt 3+4.
4. Read Silverman cpt. 1+2, tied together with Neukirch 1 cpt. 3.
5. One does not have to see the proofs of class field theory, but it is important to know the results. Look at the relevant chapters of Neukirch. Also continue reading Silverman, the chapters you find relevant.
6. Start advanced Silverman, and open Neukirch 2 (important: you should swap between the algebraic and arithmetic parts as needed, do not read it chronologically). Finish Diamond Shurman cpt. 1-5, and consider diving into the later chapters, in combination with the final Neukirch 1 chapter on zeta- and $$L$$-functions.

From there on you will have a solid grasp on the basic theory, with views into the advanced theory. You should by then also have looked at where you want to go, and be able to take informed decisions as to what to read next from then on.

All at once, reading a little bit of each at a time and doing the exercises, moving on from one to the other as you get bored. Also start right away on the collection of papers which is your goal; that will motivate you to learn some of the more intricate details in the textbook. Be sure you are understanding at least something and doing exercises every day, not just skimming for vocabulary. You will eventually have to understand everything, but there's no rush.

• I disagree with "start right away on the collection of papers...": for someone not already familiar with basic number theory (at the level of Neukirch's first book), starting on the JHC70 proceedings straight away is simply too steep a learning curve – anyone who doesn't posess superhuman patience is likely to either give up, or (worse) be lulled into believing they understand things that they really don't, since research papers will frequently omit or gloss over details which are easy for experts but deeply nonobvious for beginners. Commented Sep 14, 2023 at 7:04
• i think you would know the level of difficulty of the collection of papers, so i cheerfully withdraw that suggestion :-) Commented Sep 14, 2023 at 11:55