Elliptic Curves And Modular Forms | The Proof Of Fermat’s Last Theorem - Info and Reading Options

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Elliptic curves, modular forms, and the Taniyama-Shimura Conjecture: the three ingredients to Andrew Wiles’ proof of Fermat’s Last Theorem.<br /><br />This is by far the hardest video I've ever had to make: both in terms of learning the content and explaining it. So there a few questions I don't have answers for. If you're up for it, feel free to answer these as a YouTube comment or on Twitter (@00aleph00)!<br /><br />QUESTIONS:<br /><br />1. The Taniyama-Shimura Conjecture seems really contrived. We made a weirdly specific sequence from elliptic curves. We made a weirdly specific sequence from modular forms. And behold, the sequences match! It seems manufactured to work. What’s profound about it?<br /><br />2. Why do we care about elliptic curves of all things? It’s described by, again, a weirdly specific equation: why is it the darling child of number theory?<br /><br />3. Does the Taniyama-Shimura conjecture also guarantee uniqueness? That is, does it say that for every elliptic curve there is a *unique* modular form with the same sequence as it?<br /><br />4. We defined how a matrix from the group SL2Z “acts” on a complex number. Does anyone have a geometric picture for this? Does a matrix act on a complex number just like how it would act on a vector in R^2 (i.e: by rotating it)? <br /><br />5. This is a more advanced question. Most elliptic curve books encode the sequence m_n of a modular form using something called a Dirichlet L-function, a generalization of the Reimann Zeta function. More precisely, instead of associating a modular form to a *sequence*, we associate it to a modified version of the Riemann Zeta Function, where the n_th coefficient of the series is the term m_n. (This is sometimes called the Hasse-Weil L-function of a modular form). This seems unnecessary. What is the benefit of doing this? <br /><br />6. Does anyone understand Andrew Wiles’ paper? LOL<br /><br />SOURCES I USED TO STUDY:<br /><br />Keith Conrad’s Lectures on Modular Forms (8 part video series): <br />https://www.youtube.com/watch?v=LolxzYwN1TQ<br /><br />Keith Conrad’s Notes on Modular Forms: <br />https://ctnt-summer.math.uconn.edu/wp-content/uploads/sites/1632/2016/02/CTNTmodularforms.pdf<br /><br />“Elliptic Curves, Modular Forms, and their L-Functions” by A. Lozano-Robledo. <br />(The above book is very accessible! You only need basic calculus to understand it. You also need to know the definition of a group, but that’s pretty much it.)<br /><br />“The Arithmetic of Elliptic Curves” by Joseph Silverman<br /><br />HOMEWORK IDEA CREDIT goes to Looking Glass Universe! <br /><br />SAGE RESOURCES:<br /><br />“Sage for Undergraduates”: Gregory Bard’s Free Online Book on SAGE : http://www.gregorybard.com/Sage.html <br />Download SAGE: https://www.sagemath.org/download.html<br /><br />Proof of the Hasse-Weil Bound on Terry Tao’s Blog: https://terrytao.wordpress.com/2014/05/02/the-bombieri-stepanov-proof-of-the-hasse-weil-bound/<br /><br />OTHER VIDEOS ON THESE TOPICS:<br /><br />Numberphile Playlist: https://www.youtube.com/playlist?list=PLt5AfwLFPxWLD3KG-XZQFTDFhnZ3GHMlW<br /><br />Elliptic Curves and Modular Forms: https://www.youtube.com/watch?v=A8fsU97g3tg<br /><br />SOFTWARE USED TO MAKE THIS VIDEO:<br /><br />SAGE for the code and the graphs<br />https://github.com/hernanat/dcolor for domain coloring<br />Adobe Premiere Elements For Video Editing<br /><br />MUSIC:<br />Music Info: Documentary - AShamaluevMusic.<br />Music Link: https://www.ashamaluevmusic.com<br /><br />Follow me!<br /><br />Twitter: https://twitter.com/00aleph00<br />Instagram: https://www.instagram.com/00aleph00<br /><br />Intro: (0:00)<br />Elliptic Curves: (0:58)<br />Modular Forms: (3:26) <br />Taniyama Shimura Conjecture: (7:26)<br />Fermat's Last Theorem: (8:02) <br />Questions for you!: (8:51)

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