Elliptic Curves: Good books to get started

A few books for getting started in the subject of Elliptic Curves, each with a different perspective. I give detailed overviews and my personal take on each book.
0:00 Intro
0:41 McKean and Moll, Elliptic Curves: Function Theory, Geometry, Arithmetic
10:14 Silverman, The Arithmetic of Elliptic Curves
17:57 Silverman and Tate, Rational Points on Elliptic Curves
21:50 Washington, Elliptic Curves: Number Theory and Cryptography
24:23 Knapp, Elliptic Curves
30:02 Iwaniec and Kowlaski, Analytic Number Theory
Amazon Links to Books Mentioned
McKean and Moll, Elliptic Curves: Function Theory, Geometry, Arithmetic amzn.to/3i5QVV6
Silverman, The Arithmetic of Elliptic Curves amzn.to/3kHg2iN
Silverman and Tate, Rational Points on Elliptic Curves amzn.to/3ugIACX
Washington, Elliptic Curves: Number Theory and Cryptography amzn.to/2XTAUe0
Knapp, Elliptic Curves amzn.to/3zLga58
Iwaniec and Kowalski, Analytic Number Theory amzn.to/3AJd0jO
(I get a small commission from purchases made from these links.)
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[@anthonyymm511]

As an analysis student I get cold sweats when I see anything remotely related to algebraic geometry.

[@DanielRubin1]

I can understand that view. But all the major theorems in this subject require both algebra and analysis, and I say the hard, key steps are analytical.

[@theflaggeddragon9472]

As an aspiring number theorist, I have to say my absolute favorite math happens when different disciplines starting mixing. Analysis and algebra have some beautiful interactions all across mathematics (e.g. Lie groups, Hodge theory, Tate's thesis, etc.), and in algebraic geometry the results of their exquisite interplay comes in droves.

[@mathaleph8501]

There is also the second book of Silverman : Advanced topics in the arithmetic of elliptic curves. I worked on the complex multiplication chapter which is a wonderful subject, however Silverman often makes the confusion between elliptic curves and isomorphism classes of elliptic curves. Sometimes he works on isogenies between curves only defined up to isomorphism and it is quite confusing. Thanks for your video, my main references until here was Silverman's books, I am very interested in the different approaches of the other books you mentioned, I which someone showed me this at the beginning of my PhD ! In any case I will recommend your video to whoever is interested in the subject !

[@notjohn9]

Hi Dr. Rubin, I recently found your channel and wanted to express how much I enjoy the content. Your Tricky Parts of Calculus series is a real gem and a breath of fresh air in the KZbin Mathematics landscape -- thank you for that. From your informal chats with friends about life and movies, to the math curriculum and pedagogy experience conversations, I keep finding interesting content delivered through your unique and interesting perspective. I also look forward to checking out Fourier Analysis by S&S after the semester ends. Humbly and respectfully, John

[@DanielRubin1]

Thanks so much! Glad to hear you like the different types of videos on the channel.

[@charlottenewton6432]

I found this video very insightful! I'm completing my pure mathematics bachelors all the way in Australia (down under) and you've inspired me to learn more about elliptic curves. will be sharing with friends! Cheers Charlotte x

[@calencooney7448]

She shared it with me

[@ficsur86]

really good videos and really good channel! Keep it up. Cheers, Tamas.

[@DanielRubin1]

Hey, good to hear from you! Thanks a lot!

"Amazingly" I don't have the book by Iwaniec and Kowalski. All the books on elliptic curves mentioned here I have and they are the perfect selection. I have of course many more books on elliptic curves (Cassels, more Silverman and more which I don't remember) since I've been interested in the subject for some time. Thank you for presenting these books for future readers.

[@kono152]

im definitely getting mckean's book when i get the money, elliptic curves are absolutely beautiful (i learnt of my love for them from an oxford test that had a great elliptic curve question) and that book also talks about other stuff i wanna learn about. Plus, its affordable

[@azimuth4850]

Hecka cool!

[@AkamiChannel]

This video was non-trivial. Thank you.

[@Ouvii]

I'm very much not a learned mathematician, but I've always wonder if there was a non-calculus path to the mathematics of ellipse topics relevant to astronomy-some hack ancient astronomers might have happened upon to oppose epicycles. Definitely want to check out the prerequisites and eventually these books at some point.

[@DanielRubin1]

The name "elliptic curve" is a bit confusing, since elliptic curves have very little to do with the ellipses of the elliptical orbits of planets. (Their name stems from their connection to elliptic integrals, the integrals that come up when you want to find the arc-length of an ellipse.) But the story of how astronomy drove the development of mathematics is really interesting. It took a lot of steps to get to the theory of elliptical orbits. For a very good history of the scientific development up to the heliocentric model, check out T.S. Kuhn's "The Copernican Revolution." For Kepler's laws and Newton's explanation of them, there are lots of good treatments, including Toeplitz's "The Calculus: A Genetic Approach" which also covers the development of trig and log tables and their roles in astronomy, Bressoud's "Second Year Calculus" which begins with Kepler and Newton, and Arnold's "Huygens and Barrow, Newton and Hooke." I don't think there's any way to avoid calculus, though.

[@BB-mr3vy]

One of these about PDE books maybe at some point?

[@DanielRubin1]

That's definitely coming. For now, you can check out my comments about PDE in my Math Major Guide video (link in description above). Now, I only have expertise in a certain kind of PDE, but I'll try to be of help to as many people as I can.

[@miltonma212]

May you recommend any book about high school math ?

[@DanielRubin1]

This is a great question! I've been looking into this, and I think I'll make a video about high school math books at some point, but I don't have any completely solid recommendations yet. Keep in mind, I've never taught at the high school level. I haven't found quite what I'm looking for yet, but I'll mention a few books that are out there. Serge Lang has a book called Basic Mathematics and Felix Klein wrote several volumes of Elementary Mathematics from an Advanced Standpoint; both give a mathematician's perspective on high school math. I've heard great things about the Art of Problem Solving series, and I also like Zetiz's The Art and Craft of Problem Solving; very good for training for contest math, but also for learning.

[@juanserratos2596]

A book I found recently is "Elliptic Curves, Modular Forms, and Their L-functions" by Alvaro Lozano Robledo which I think serves as a great introduction at the undergraduate level

[@100timezcooler]

What would you recommend for a computer scientist looking to brush up/reinforce their knowledge on cryptography?

[@DanielRubin1]

I've gotten similar questions to this before, so at some point I may look at some resources on cryptography and review the best ones. I'm sorry I don't have a great book recommendation at hand; I never took any courses in cryptography, and I learned what I need for my work in cryptography from papers and chapters from many books. Note that I don't have a background in computer science, so I don't know what CS people come to the subject with. It certainly helps to start with as much number theory as you can get, and it's probably useful to focus on Computational Number Theory.

[@danielzapata9428]

@@DanielRubin1 One question I already finished computer science and on my own I study mathematics (almost on par with the curriculum of said career) I am in doubt to study mathematics or do a postgraduate degree. (Seeing 2 topics can be number theory or computational complexity theory)

[@user-xf6ig9ur2y]

Check out Elliptic Tales by Ash.

[@SkyFoxTale]

"You can tell it's an important theorem because it's an inequality." Spoken like a true analyst LOL

[@infiniteseries6210]

Unfortunately the last book isn't available as hardcover anymore. And you can't read such thick books when they are softcover.

[@spazmoidectomorf6209]

All these books have no solution to the exercises which is so frustrating.

[@miltonma212]

Because I don't have the good basic knowledge to study the college math .