Elliptic curve basics + p-adics

latex/elliptic-curve-basics.tex

@@ -0,0 +1,29 @@
+\chapter{Elliptic Curve Basics}%
+\label{sec:elliptic-curves}
+
+% Weierstrass equation
+\begin{defn}
+Let $K$ be a field. An $\textbf{elliptic curve $E$ over $K$}$ is defined by an equation:
+$$E : y^2 + a_1xy + a_3y = x^3 + a_2x^2 + a_4x + a_6$$
+where $a_1, a_2, a_3, a_4, a_6 \in K$ and the $\textbf{discriminant}$ $\Delta$ is non-zero. This equation is called a $\textbf{Weierstrass equation}$.
+\end{defn}
+
+% L-rational points
+\begin{defn}
+With $K$ and $E$ defined as above, the set of $\textbf{L-rational points}$ on $E$ for any extension $L$ of $K$ is the set of pairs $(x, y) \in L \times L$ that
+satisfy $E$, together with $\OO$, the point at infinity.
+
+The set of L-rational points is denoted $E(L)$.
+\end{defn}
+
+% Trace of Frobenius
+\begin{defn}
+Let $E$ be an elliptic curve over a finite field $\finfield$. The $\textbf{trace of Frobenius}$ t is defined by:
+$$ \#E(\finfield) = q + 1 - t, $$
+where $\#E(\finfield)$ is the number of elements in $E(\finfield)$.
+\end{defn}
+
+\begin{rmk}
+The trace of Frobenius is equal to one if and only if $E(\finfield)$ has exactly $q$ elements. This has important implications
+for cryptography, as we will see.
+\end{rmk}

latex/formal-log.tex

@@ -0,0 +1,12 @@
+\chapter{Formal Power Series and Formal Logarithm}%
+\label{sec:formal-log}
+
+\section{Formal Power Series}
+
+%% Defintion of 
+\begin{defn}
+\end{defn}
+
+
+\section{Formal Logarithm}
+Hello

latex/p-adics.tex

@@ -0,0 +1,35 @@
+\chapter{P-adic numbers}%
+\label{sec:p-adics}
+
+\section{The p-adics}
+
+\begin{defn}
+For a rational number $a$ and a prime number $p$, separate out all factors of $p$ from $a$ and write: $$ a = p^r \dfrac{m}{n} $$ where $r$, $m$ and $n$ are integers, and $p$ does not divide $m$ or $n$. The exponent $r$ is called the $\textbf{p-adic ordinal}$ of $a$, denoted $\text{ord}_p(a)$.
+\end{defn}
+
+% The absolute value
+\begin{defn}
+For a prime $p$, we define a function $|.|_p : \Q \to \Q_{\ge 0}$ where for $a \in \Q$:
+$$
+|a|_p = \left\{
+        \begin{array}{ll}
+            p^{-\text{ord}_p(a)} & \quad a \neq 0 \\
+            0 & \quad a = 0.
+        \end{array}
+    \right.
+$$
+The function $|.|_p$ is called the $\textbf{p-adic absolute value}$.
+\end{defn}
+
+\begin{prop}
+The p-adic absolute value is a norm on $\Q$, and induces a metric $$d_p(a, \ b) = | a - b |_p$$ for $a, b \in \Q$.
+\end{prop}
+
+\begin{defn}
+A p-adic number $a$ is called a $\textbf{p-adic integer}$ if $ord_p(a) \ge 0$. The set of all p-adic integers is
+denoted $\Z_p$.
+\end{defn}
+
+\begin{rmk}
+A p-adic integer is always of the form $$a_0 + a_1p + a_2p^2 + ... ,$$ i.e., all powers of $p$ are non-negative.
+\end{rmk}

latex/thesis-wrapper.tex

@@ -8,8 +8,26 @@
 \advisor{Christopher Towse}
 \reader{Second Reader}
 
-%%%  END HEADER %%%
+%%% Including new commands and packages.
+\usepackage{amsthm}
+\newtheorem{thm}{Theorem}[chapter]
+\newtheorem{lem}[thm]{Lemma}
+\newtheorem{prop}[thm]{Proposition}
+\newtheorem{cor}[thm]{Corollary}
+
+\theoremstyle{definition}
+\newtheorem{defn}[thm]{Definition}
+
+\theoremstyle{remark}
+\newtheorem*{rmk}{Remark}
 
+\newcommand{\OO}{\mathcal{O}}
+\newcommand{\Q}{\mathbb{Q}}
+\newcommand{\Z}{\mathbb{Z}}
+\newcommand{\NN}{\mathbb{N}}
+\newcommand{\finfield}{\mathbb{F}_q}
+
+%%%  END HEADER %%%
 
 \begin{document}
 
@@ -43,6 +61,13 @@
 \mainmatter
 
 \include{intro}
+\include{elliptic-curve-basics}
+\include{formal-log}
+\include{p-adics}
+%\include{}
+%\include{}
+%\include{}
+%\include{}
 
 %%% End of the main matter.