Commit 1afedc6a authored by Han-Miru Kim's avatar Han-Miru Kim
Browse files

Seminar L-functions stuff

parent b01024de
\section{Binary quadratic forms}
\section{Summary of previous talks about Binary quadratic forms}
\begin{dfn}[]
A \textbf{binary quadratic form} is an expression of the form
\begin{empheq}[box=\bluebase]{align*}
......@@ -10,11 +9,16 @@ where $a,b,c$ are fixed \emph{coefficients} and $x,y$ are variables.
We will assume that $a,b,c \in \Z$ and that the number of variables $x,y$ is always two, so we will drop the word ``binary''.
We will also omit the word ``quadratic'' aswell and speak of ``forms'', meaning binary quadratic forms.
\begin{rem}[]
We will use the notation where $[a,b,c]$ denotes the bqf $f(x,y) = ax^{2} + bxy + cy^{2}$.
\end{rem}
The main question is whether the equation
\begin{align}\label{eq:bqf}
f(x,y) = ax^{2} + bxy + cy^{2} = n \quad \text{for} \quad x,y \in \Z
f(x,y) = ax^{2} + bxy + cy^{2} = n, \quad n \in \Z
\end{align}
has solutions or not.
has solutions for $x,y \in \Z$ or not.
If we plot the graph of $z = f(x,y)$, we would expect that transformations such as flipping $x,y$ or doing $90$-degree rotations should preserve the structure of the set of solutions to \ref{eq:bqf}.
......@@ -62,7 +66,7 @@ equation \ref{eq:bqf} turns into
\\
&= a'x^{2} + b'xy + c'y^2
\end{align*}
where $a',b',c'$ is given by
where $a',b',c'$ are given by
\begin{align*}
a' &= a \alpha^{2} + b \alpha \gamma + c \gamma^{2}\\
b' &= 2 a \alpha \beta + b(\alpha \delta + \beta \gamma) + 2 c \gamma \delta\\
......@@ -70,7 +74,7 @@ where $a',b',c'$ is given by
\end{align*}
\begin{dfn}[]
Two quadratic forms $f(x,y) =ax^{2} + bxy + cy^{2}$ and $f'(x,y) = a'x^{2} + b'xy + c'y^{2}$ are \textbf{equivalent}, if there exist $\begin{pmatrix}
Two forms $[a,b,c]$, $[a',b',c']$ are \textbf{equivalent}, if there exist $\begin{pmatrix}
\alpha & \beta\\
\gamma & \delta
\end{pmatrix}
......@@ -85,7 +89,7 @@ One quickly verifies that this defines an equivalence relation on all binary qua
\begin{prop}[]
Given a form $ax^{2} + bxy + cy^{2}$, the \textbf{discriminant}
Given a form $[a,b,c]$, the \textbf{discriminant}
\begin{empheq}[box=\bluebase]{align*}
D = b^{2} - 4ac
\end{empheq}
......@@ -117,20 +121,20 @@ Moreover, every form $ax^{2} + bxy + cy^{2}$ is equivalent to a form $a'x^{2} +
\begin{rem}[]
The theorem is even true if $D \neq 0$ is a square, but we won't consider forms with square discriminant as they reduce to linear factors.
\end{rem}
\begin{proof}
We provide an terminating algorithm.
\begin{enumerate}
\item Substitute $(a,b,c)$ with
\begin{align*}
(a, b- 2na, c - nb + n^{2}a)
\end{align*}
where we chose $n \in \Z$ such that $- \abs{a} < b - 2na \leq \abs{a}$.
\item If $\abs{c} \geq \abs{a}$ we're done.
\item Else, substitute $(a,b,c)$ with $(c,-b,a)$ and start over.
\end{enumerate}
After every cycle, $\abs{a}$ decreases.
\end{proof}
%\begin{proof}
%We provide an terminating algorithm.
%\begin{enumerate}
%\item Substitute $(a,b,c)$ with
%\begin{align*}
%(a, b- 2na, c - nb + n^{2}a)
%\end{align*}
%where we chose $n \in \Z$ such that $- \abs{a} < b - 2na \leq \abs{a}$.
%\item If $\abs{c} \geq \abs{a}$ we're done.
%\item Else, substitute $(a,b,c)$ with $(c,-b,a)$ and start over.
%\end{enumerate}
%After every cycle, $\abs{a}$ decreases.
%\end{proof}
We wish to compute and study the number of equivalence classes with discriminant $D$.
......@@ -257,113 +261,6 @@ $\epsilon_0 = \frac{1}{2}(t + u \sqrt{D})$, for $t^{2} -Du^{2} = 4$ with $t,u$ s
\end{align*}
\end{thm}
\section{Calculating $L(1,\chi)$}
Let $\chi$ be a primitve character.
\begin{align*}
G = \sum_{n=1}^{N}\chi(n) e^{\frac{2 \pi i n}{N}}
\end{align*}
\begin{lem}[]
Let $\chi$ be a primitive dirichlet character ($\mod N$) and $G$ as above. Then
\begin{itemize}
\item $\sum_{n=1}^{N}\chi(n) e^{\frac{2 \pi i k n}{N}} = \overline{\chi(k)}G$ for all $k \in \Z$.
\item $\abs{G} = \sqrt{N}$.
\end{itemize}
\end{lem}
\begin{lem}[]
For $0 < \theta < 2 \pi$
\begin{align*}
\sum_{n=1}^{\infty} \frac{e^{in \theta}}{n} = - \log(2 \sin \tfrac{\theta}{2}) + i (\tfrac{\pi}{2} - \tfrac{\theta}{2})
\end{align*}
\end{lem}
\begin{thm}[]
Let $\chi$ be a primitive dirichlet character ($\mod N$), $N > 1$.
Then
\begin{align*}
L(1,\chi) =
- \frac{1}{G}
\sum_{n=1}^{N-1}\overline{\chi}(n)
\log \sin \frac{\pi n}{N}
+
\frac{i \pi}{NG}
\sum_{n=1}^{N-1}\overline{\chi}(n) n
\end{align*}
\end{thm}
\begin{lem}[]
Let $D$ be a fundamental discriminant and consider the gauss sum $G$ of $\chi_D$.
Then
\begin{align*}
G =
\left\{\begin{array}{ll}
\sqrt{D} & \text{if } D > 0\\
i\sqrt{D} & \text{if } D < 0
\end{array} \right.
\end{align*}
\end{lem}
So if $D > 0$, then $G$ is real and in the previous equation for $L(1,\chi_D)$, the imaginary part vanishes.
Likewise, if $D <0$, the real part vanishes.
\begin{thm}[]
Let $D$ be a fundamental discriminant. For $D < 0$, we have
\begin{align*}
h(D) = -\frac{w}{2 \abs{D}} \sum_{n=1}^{\abs{D} - 1} \chi_D(n) n
\end{align*}
where $w$ is as in \ref{eq:w}.
For $D > 0$ we have
\begin{align*}
h(D) = - \frac{1}{\log \epsilon_0} \sum_{n=1}^{D-1}\chi_D(n) \log \sin \frac{\pi n}{D}
\end{align*}
where $\epsilon_0 > 1$ is the fundamental unit of $f$.
\end{thm}
\begin{xmp}[]
With the previous theorem, we can compute the class numbers.
\begin{itemize}
\item $D = -3$: Here $w = 6$, so
\begin{align*}
h(-3) = - \frac{6}{2 \cdot 3} \sum_{n=1}^{2} \chi_{-3}(n) n = - (1-2) = 1
\end{align*}
\item $D = -4$: Here $w = 4$, so
\begin{align*}
h(-4) = 1
\end{align*}
\item For $D < -4$, we have $w = 2$, so
\begin{align*}
h(-7) &= 1\\
h(-8) &= 1\\
h(-11) &= 1\\
h(-15) &= 2\\
h(-19) &= 1\\
h(-20) &= 2\\
h(-23) &= 3
\end{align*}
\end{itemize}
\end{xmp}
If we check when $h(D) = 1$ for $D < 0$, we can find that this is the case for
\begin{align*}
D = -3, -4, -7, -8, -11, -19, -43, -67, -163
\end{align*}
Gauss calculated $h(D)$ for $D$ up to $-10'000$ and found no others and conjectured that theses were the only fundamental discriminants with class number $1$.
Furthermore, he conjectured that
\begin{align*}
h(D) \to \infty \quad \text{as} \quad D \to - \infty
\end{align*}
In 1934 it was proven that there could at most be a tenth fundamental discrimant with $h(D) = 1$ by Heilbronn and strenghed by Siegel where he shoed tha for $\epsilon > 0$
\begin{align*}
h(D) > C \abs{D}^{\frac{1}{2} - \epsilon} (D < 0)
\end{align*}
for some $C> 0$ (depending on $\epsilon)$.
Moreover, we can show that
\begin{align*}
\lim_{D \to -\infty} \frac{\log h(D)}{\log \abs{D}} = \frac{1}{2}
\end{align*}
\section{Calculating $L(1,\chi)$}
Let $\chi$ be a primitve character.
\begin{align*}
G = \sum_{n=1}^{N}\chi(n) e^{\frac{2 \pi i n}{N}}
\end{align*}
\begin{lem}[]
Let $\chi$ be a primitive dirichlet character ($\mod N$) and $G$ as above. Then
\begin{itemize}
\item $\sum_{n=1}^{N}\chi(n) e^{\frac{2 \pi i k n}{N}} = \overline{\chi(k)}G$ for all $k \in \Z$.
\item $\abs{G} = \sqrt{N}$.
\end{itemize}
\end{lem}
\begin{lem}[]
For $0 < \theta < 2 \pi$
\begin{align*}
\sum_{n=1}^{\infty} \frac{e^{in \theta}}{n} = - \log(2 \sin \tfrac{\theta}{2}) + i (\tfrac{\pi}{2} - \tfrac{\theta}{2})
\end{align*}
\end{lem}
\begin{thm}[]
Let $\chi$ be a primitive dirichlet character ($\mod N$), $N > 1$.
Then
\begin{align*}
L(1,\chi) =
- \frac{1}{G}
\sum_{n=1}^{N-1}\overline{\chi}(n)
\log \sin \frac{\pi n}{N}
+
\frac{i \pi}{NG}
\sum_{n=1}^{N-1}\overline{\chi}(n) n
\end{align*}
\end{thm}
\begin{lem}[]
Let $D$ be a fundamental discriminant and consider the gauss sum $G$ of $\chi_D$.
Then
\begin{align*}
G =
\left\{\begin{array}{ll}
\sqrt{D} & \text{if } D > 0\\
i\sqrt{D} & \text{if } D < 0
\end{array} \right.
\end{align*}
\end{lem}
So if $D > 0$, then $G$ is real and in the previous equation for $L(1,\chi_D)$, the imaginary part vanishes.
Likewise, if $D <0$, the real part vanishes.
\begin{thm}[]
Let $D$ be a fundamental discriminant. For $D < 0$, we have
\begin{align*}
h(D) = -\frac{w}{2 \abs{D}} \sum_{n=1}^{\abs{D} - 1} \chi_D(n) n
\end{align*}
where $w$ is as in \ref{eq:w}.
For $D > 0$ we have
\begin{align*}
h(D) = - \frac{1}{\log \epsilon_0} \sum_{n=1}^{D-1}\chi_D(n) \log \sin \frac{\pi n}{D}
\end{align*}
where $\epsilon_0 > 1$ is the fundamental unit of $f$.
\end{thm}
\begin{xmp}[]
With the previous theorem, we can compute the class numbers.
\begin{itemize}
\item $D = -3$: Here $w = 6$, so
\begin{align*}
h(-3) = - \frac{6}{2 \cdot 3} \sum_{n=1}^{2} \chi_{-3}(n) n = - (1-2) = 1
\end{align*}
\item $D = -4$: Here $w = 4$, so
\begin{align*}
h(-4) = 1
\end{align*}
\item For $D < -4$, we have $w = 2$, so
\begin{align*}
h(-7) &= 1\\
h(-8) &= 1\\
h(-11) &= 1\\
h(-15) &= 2\\
h(-19) &= 1\\
h(-20) &= 2\\
h(-23) &= 3
\end{align*}
\end{itemize}
\end{xmp}
If we check when $h(D) = 1$ for $D < 0$, we can find that this is the case for
\begin{align*}
D = -3, -4, -7, -8, -11, -19, -43, -67, -163
\end{align*}
Gauss calculated $h(D)$ for $D$ up to $-10'000$ and found no others and conjectured that theses were the only fundamental discriminants with class number $1$.
Furthermore, he conjectured that
\begin{align*}
h(D) \to \infty \quad \text{as} \quad D \to - \infty
\end{align*}
In 1934 it was proven that there could at most be a tenth fundamental discrimant with $h(D) = 1$ by Heilbronn and strenghed by Siegel where he shoed tha for $\epsilon > 0$
\begin{align*}
h(D) > C \abs{D}^{\frac{1}{2} - \epsilon} (D < 0)
\end{align*}
for some $C> 0$ (depending on $\epsilon)$.
Moreover, we can show that
\begin{align*}
\lim_{D \to -\infty} \frac{\log h(D)}{\log \abs{D}} = \frac{1}{2}
\end{align*}
......@@ -26,6 +26,7 @@ to obtain a Form $ax^{2} + bxy + cy^{2}$ with
\begin{align*}
- \abs{a} < b \leq \abs{a} \leq \abs{c}
\end{align*}
\begin{dfn}[]
A positive definite form $ax^{2} + bxy + cy^{2}$ is said to be \textbf{reduced}, if
\begin{align*}
......@@ -49,27 +50,23 @@ First note that for a reduced form $f$ with $(x,y) \neq (0,0)$ one has
Let $f = ax^{2} + bxy + cy^{2}$.
We define an action on the set of forms
\begin{align*}
TF = S_n f \quad \text{for} \quad n \in \Z \text{ with }n > \overline{f} > n-1
TF = S_n f \quad \text{for} \quad n \in \Z \text{ with }n > \frac{b + \sqrt{D}}{2a} > n-1
\end{align*}
where $\overline{f} = \frac{b + \sqrt{D}}{2a}$
given by
\begin{align*}
S_n = \begin{pmatrix}
n & 1\\
-1 & 0
\end{pmatrix}
:
ax^{2} + bxy + cy^{2}
[a,b,c]
\mapsto
(an^{2} - bn + c)x^{2} + (2an - b)xy + ay^{2}
[an^{2} - bn + c,2an - b,a]
\end{align*}
\end{dfn}
\textbf{Notation:} We notate a form $f(x,y) = ax^{2} + bxy + cy^{2}$ as a triple $[a,b,c]$
Warning: the notation $\overline{f}$ is not used in the book.
\begin{dfn}[]
An indefinite form $ax^{2} + bxy + cy^{2}$ is \textbf{reduced}, if
An indefinite ($D > 0$) form $[a,b,c]$ is \textbf{reduced}, if
\begin{align*}
a > 0, c > 0, b > a + c
\end{align*}
......@@ -86,7 +83,19 @@ Every equivalence of reduced forms is obtained by iteration of $T$.
In particular, two reduced forms are equivalent if and only if they belong to the same cycle.
\end{thm}
\begin{xmp}[]
Use \texttt{bqf.hs} to show cycle of $f = x^{2} - 6 xy + 3y^{2}$.
Use \texttt{bqf.hs} to show cycle of $f = [-1,6,-3]$.
\end{xmp}
\begin{proof}[Proof Sketch]
\begin{enumerate}
\item Show there are only finitely many reduced forms.
\item The transformation $T$ is injective on the set of reduced forms, i.e. if $f_1,f_2$ are reduced forms and $Tf_1 = Tf_2$, then $f_1 = f_2$.
\item Every form can be reduced after finitely many applications of $T$.
\item If $f$ is reduced, then $Tf$ is.
\item If $f,f'$ are equivalent, then application of $T$ can transform one into the other.
\end{enumerate}
From (a),(b),(d), (e) it follows that the reduced forms split into disjoint cycles of equivalence classes.
\end{proof}
......@@ -8,8 +8,6 @@
#+LATEX_HEADER: \input{header.tex}
\input{./files/01}
\input{./files/03}
\input{./files/binary-quadratic-forms}
\input{./files/10}
\input{./files/reduction-theory}
\input{./files/continued-fractions}
% Created 2021-10-19 Tue 12:20
% Created 2021-12-07 Tue 08:51
% Intended LaTeX compiler: pdflatex
\documentclass[11pt]{article}
\input{header.tex}
......@@ -18,9 +18,7 @@
\maketitle
\tableofcontents
\input{./files/01}
\input{./files/03}
\input{./files/binary-quadratic-forms}
\input{./files/10}
\input{./files/reduction-theory}
\input{./files/continued-fractions}
\end{document}
\ No newline at end of file
......@@ -150,6 +150,8 @@ The transformation is done by applying the transformation ~T = S_n = [[n,1],[-1,
| x == y = [x]
| otherwise = [x] ++ aux xs (Just y)
-- using point-free notation
-- reduction' f = flip aux Nothing . iterate reduceStep
reduce :: BQF -> BQF
reduce = head . (dropWhile (not . isReduced)) . (iterate reduceStep)
......
......@@ -83,6 +83,8 @@ reduction f = aux (iterate reduceStep f) Nothing
| x == y = [x]
| otherwise = [x] ++ aux xs (Just y)
-- using point-free notation
-- reduction' f = flip aux Nothing . iterate reduceStep
reduce :: BQF -> BQF
reduce = head . (dropWhile (not . isReduced)) . (iterate reduceStep)
......@@ -98,6 +100,8 @@ areEquivalent f g
| (d /= d') = False
| d <= 0 = error "not definite forms"
| otherwise = (reduce f) `elem` (reduction g)
-- point-free form:
-- otherwise = (. reduction) . elem . reduce
where
d = discriminant f
d' = discriminant g
......
\section{Continued Fractions}
\begin{frame}[fragile]\frametitle{Continued Fractions}
\begin{block}{Definition}
Let $n_{0}, n_1,n_2,\ldots$ be a sequence of integers, with $n_1,n_2,\ldots \geq 2$.
We define
\begin{align*}
[[n_0,n_1,\ldots,n_s]] := n_0 - \frac{1}{n_1 - \frac{1}{n_2 - \frac{}{\ddots - \frac{1}{n_s}}}}
\end{align*}
as well as $[[n_0,n_1,n_2\ldots]] := \lim_{s \to \infty} [[n_0,n_1,\ldots,n_s]]$.
\end{block}
One can show that for all $w \in \R$, there exists a sequcene $(n_i)_{i \in \N},n_i \in \Z,n_1,\ldots \geq 2$ such that $[[n_0,n_1,n_2,\ldots] = w$.
Moreover, there exists a bijective corresopncence of real numbers and such sequences.
In particular, one has
\begin{align*}
w \in \Q \iff \exists N \in \N: n_i = 2 \forall i \geq N
\end{align*}
\end{frame}
......@@ -76,10 +76,6 @@
\end{block}
\end{frame}
\begin{frame}[fragile]\frametitle{More Definitions}
\begin{block}{Representation number}
......
......@@ -11,8 +11,13 @@
\headcommand {\beamer@framepages {5}{5}}
\headcommand {\slideentry {1}{0}{4}{6/6}{}{0}}
\headcommand {\beamer@framepages {6}{6}}
\headcommand {\beamer@partpages {1}{6}}
\headcommand {\beamer@subsectionpages {2}{6}}
\headcommand {\beamer@sectionpages {2}{6}}
\headcommand {\beamer@documentpages {6}}
\headcommand {\gdef \inserttotalframenumber {5}}
\headcommand {\beamer@subsectionpages {2}{6}}
\headcommand {\sectionentry {2}{Continued Fractions}{7}{Continued Fractions}{0}}
\headcommand {\slideentry {2}{0}{1}{7/7}{}{0}}
\headcommand {\beamer@framepages {7}{7}}
\headcommand {\beamer@partpages {1}{7}}
\headcommand {\beamer@subsectionpages {7}{7}}
\headcommand {\beamer@sectionpages {7}{7}}
\headcommand {\beamer@documentpages {7}}
\headcommand {\gdef \inserttotalframenumber {6}}
......@@ -5,7 +5,9 @@
#+LATEX_CLASS: org-plain-beamer
#+LATEX_CLASS_OPTIONS: [aspectratio=169]
#+OPTIONS: toc:t
#+LATEX_HEADER: \input{header}
\input{files/intro}
\input{files/confrac}
% Created 2021-11-16 Tue 12:03
% Created 2021-11-17 Wed 21:13
% Intended LaTeX compiler: pdflatex
\documentclass[aspectratio=169]{beamer}
\input{header}
\author{Han-Miru Kim}
\author{Han-Miru Kim, Alexander}
\date{\today}
\title{Dirichlet L-functions\\\medskip
\large Reduction Theory}
\hypersetup{
pdfauthor={Han-Miru Kim},
pdfauthor={Han-Miru Kim, Alexander},
pdftitle={Dirichlet L-functions},
pdfkeywords={},
pdfsubject={},
......@@ -19,4 +19,5 @@
\tableofcontents
\input{files/intro}
\end{document}
\input{files/confrac}
\end{document}
\ No newline at end of file
\frametitle{More Definitions}
\begin{block}{Representation number}
\frametitle{Continued Fractions}
\begin{block}{Definition}
Let $n_{0}, n_1,n_2,\ldots$ be a sequence of integers, with $n_1,n_2,\ldots \geq 2$.
We define
\begin{align*}
[[n_0,n_1,\ldots,n_s]] := n_0 - \frac{1}{n_1 - \frac{1}{n_2 - \frac{}{\ddots - \frac{1}{n_s}}}}
\end{align*}
as well as $[[n_0,n_1,n_2\ldots]] := \lim_{s \to \infty} [[n_0,n_1,\ldots,n_s]]$.
\end{block}
One can show that for all $w \in \R$, there exists a sequcene $(n_i)_{i \in \N},n_i \in \Z,n_1,\ldots \geq 2$ such that $[[n_0,n_1,n_2,\ldots] = w$.
Moreover, there exists a bijective corresopncence of real numbers and such sequences.
In particular, one has
\begin{align*}
w \in \Q \iff \exists N \in \N: n_i = 2 \forall i \geq N
\end{align*}
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