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elliptic-draft.tex
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elliptic-draft.tex
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\def\filepath{templates}
%\def\filepath{C:/Users/holden-lee/Dropbox/Math/templates}
%\def\filepath{G:/transfer/Math/templates}
\input{\filepath/packages_book.tex}
\input{\filepath/theorems_with_boxes.tex}
\input{\filepath/macros.tex}
\input{\filepath/formatting.tex}
\input{\filepath/other.tex}
\input{\filepath/theorem_num.tex}
\def\name{Elliptic Curves}
\input{\filepath/titlepage.tex}
\begin{document}
\maketitle
%\input{beginning/intro.tex}
\tableofcontents
\frontmatter
\mainmatter
\chapter*{Introduction}
\section*{Resources}
These are some notes (in progress) on elliptic curves that I've combined from various sources, including the following classes:
\begin{enumerate}
\item
Andrew Sutherland's course at MIT (18.783) from spring 2012~\url{http://co.mit.edu/18.783} (parts of these notes are from notes scribed by students in the class and edited by Sutherland),
\item
a reading course with Sug Woo Shin at MIT on class field theory and complex multiplication, and
\item
Tom Fisher's course at the University of Cambridge from autumn 2013 (lecture notes at~\url{https://dl.dropboxusercontent.com/u/27883775/math\%20notes/part_iii_elliptic.pdf}; alternate version at~\url{http://www.pancratz.org/notes/Elliptic.pdf}).
\end{enumerate}
I will draw heavily on the following books:
\begin{enumerate}
\item
Silverman, The arithmetic of elliptic cuves~\ref{Si86}.
\item
Silverman, Advanced topics in the arithmetic of elliptic curves, by Silverman~\ref{Si94}.
\item
Washington, Elliptic curves and cryptography~\ref{Wa08}.
\item
Cox, Primes of the form $x^2+ny^2$~\ref{Co89}.
\end{enumerate}
I would also like to put in material such as from the following:
\begin{enumerate}
\item
Silverman and Hindry, Diophantine Geometry: An Introduction~\ref{HS00}.
\item
Koblitz, Elliptic Curves and Modular Forms~\ref{Ko84}.
\item
Lang, Elliptic functions~\ref{La87}.
\item
Diamond and Shurman, A First Course in Modular Forms~\ref{DS05}.
\end{enumerate}
\section*{Using these notes}
I eventually want these to be a complete set of notes, but for the time being, see it more as a ``reading guide" or ``road map," to be used as a supplement to textbooks or course material.
When you see things like ``ADD a discussion on ...", take this as a sign that you should be able to discuss the topic in your studies on elliptic curves.
I would like the prerequisites for these notes to be minimal. However, I will assume familiarity with basic things such as exact sequences and equivalence of categories that make many theorems much more compact to state. Currently, I will have to refer a reader elsewhere for the basics of algebraic geometry.
\section*{Philosophy}
(I.e., what I'd like to do differently from texts already out there.)
In these notes I would like to focus on...
\begin{enumerate}
\item
intuition.
How to talk about the material in a non-rigorous way? A strong way of building intuition is connecting to previous topics, even if the analogies are imperfect. I also hope to add in discussion of big-picture questions such as ``why does geometry matter for arithmetic questions?" and ``what the heck do elliptic curves have to do with modular forms?" As a result, take everything outside of formal statements of theorems and proofs with a grain of salt.
\item motivations, and connections. How would someone come up with the statements or proofs? How are different subtopics related to one another?
\item
summaries and ``index-carding." What is the most compact way you can remember the topics? What is the big-picture?
\item
road maps. For topics where I do not have notes for yet, what resources are out there? What is the big picture?
\item
problem-solving based learning. Have problems before theorems to get the reader thinking, and possibly derive some of the ideas of the theorems on his/her own.
\item
fun problems. Stray away from the core material occasionally.
\item
algorithms. Elliptic curves is a very computational subject, so it is helpful to learn how to program algorithms involving EC. I'll try to include SAGE code with comments. (For more on SAGE, see~\ref{http://www.sagemath.org/}.)
\end{enumerate}
\section*{Collaboration}
Send me any comments or corrections at holdenlee@alum.mit.edu.
In particular, let me know if you'd like to collaborate on these notes.
%Add in link to philosophy of math notes on tiddlywiki
\pagestyle{fancy}
\input{ag-chapters/genus0.tex}
\input{ag-chapters/basics.tex}
\input{ag-chapters/ff.tex}
\input{ag-chapters/applications.tex}
\chapter{Modular forms}
(Placeholder)
\input{ag-chapters/C.tex}
\input{ag-chapters/formal_groups.tex}
\input{ag-chapters/local.tex}
\input{ag-chapters/global.tex}
\input{ag-chapters/computing_mw.tex}
\input{ag-chapters/integral_pts.tex}
%\input{af-chapters/chapter1.tex}
\input{ag-chapters/CM.tex}
\input{ag-chapters/modular_curves.tex}
\bibliographystyle{plain}
\bibliography{\filepath/refs}
\printindex
\printnomenclature
\end{document}