### 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 chosen \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 chosen \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,us \end{align*} \end{thm} \section{CalculatingL(1,\chi)$} Let$\chibe 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 ($\mod N$),$N > 1. Then \begin{align*} L(1,\chi) = - \frac{1}{G} \sum_{n=1}^{N-1}\overline{\chi}(n) \log \sin \frac{\pi n}{N} + \frac{i \pi}{NG} \sum_{n=1}^{N-1}\overline{\chi}(n) n \end{align*} \end{thm} \begin{lem}[] LetD$be a fundamental discriminant and consider the gauss sum$G$of$\chi_D. Then \begin{align*} G = \left\{\begin{array}{ll} \sqrt{D} & \text{if } D > 0\\ i\sqrt{D} & \text{if } D < 0 \end{array} \right. \end{align*} \end{lem} So ifD > 0$, 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$D < 0, we have \begin{align*} h(D) = -\frac{w}{2 \abs{D}} \sum_{n=1}^{\abs{D} - 1} \chi_D(n) n \end{align*} wherew$is as in \ref{eq:w}. For$D > 0we have \begin{align*} h(D) = - \frac{1}{\log \epsilon_0} \sum_{n=1}^{D-1}\chi_D(n) \log \sin \frac{\pi n}{D} \end{align*} where\epsilon_0 > 1$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$w = 6, so \begin{align*} h(-3) = - \frac{6}{2 \cdot 3} \sum_{n=1}^{2} \chi_{-3}(n) n = - (1-2) = 1 \end{align*} \itemD = -4$: Here$w = 4, so \begin{align*} h(-4) = 1 \end{align*} \item ForD < -4$, we have$w = 2, so \begin{align*} h(-7) &= 1\\ h(-8) &= 1\\ h(-11) &= 1\\ h(-15) &= 2\\ h(-19) &= 1\\ h(-20) &= 2\\ h(-23) &= 3 \end{align*} \end{itemize} \end{xmp} If we check whenh(D) = 1$for$D < 0, we can find that this is the case for \begin{align*} D = -3, -4, -7, -8, -11, -19, -43, -67, -163 \end{align*} Gauss calculatedh(D)$for$D$up to$-10'000$and found no others and conjectured that theses were the only fundamental discriminants with class number$1. Furthermore, he conjectured that \begin{align*} h(D) \to \infty \quad \text{as} \quad D \to - \infty \end{align*} In 1934 it was proven that there could at most be a tenth fundamental discrimant withh(D) = 1$by Heilbronn and strenghed by Siegel where he shoed tha for$\epsilon > 0\begin{align*} h(D) > C \abs{D}^{\frac{1}{2} - \epsilon} (D < 0) \end{align*} for someC> 0$(depending on$\epsilon). Moreover, we can show that \begin{align*} \lim_{D \to -\infty} \frac{\log h(D)}{\log \abs{D}} = \frac{1}{2} \end{align*}  \section{Continued Fractions}  \section{CalculatingL(1,\chi)$} Let$\chibe a primitve character. \begin{align*} G = \sum_{n=1}^{N}\chi(n) e^{\frac{2 \pi i n}{N}} \end{align*} \begin{lem}[] Let\chi$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 ($\mod N$),$N > 1. Then \begin{align*} L(1,\chi) = - \frac{1}{G} \sum_{n=1}^{N-1}\overline{\chi}(n) \log \sin \frac{\pi n}{N} + \frac{i \pi}{NG} \sum_{n=1}^{N-1}\overline{\chi}(n) n \end{align*} \end{thm} \begin{lem}[] LetD$be a fundamental discriminant and consider the gauss sum$G$of$\chi_D. Then \begin{align*} G = \left\{\begin{array}{ll} \sqrt{D} & \text{if } D > 0\\ i\sqrt{D} & \text{if } D < 0 \end{array} \right. \end{align*} \end{lem} So ifD > 0$, 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$D < 0, we have \begin{align*} h(D) = -\frac{w}{2 \abs{D}} \sum_{n=1}^{\abs{D} - 1} \chi_D(n) n \end{align*} wherew$is as in \ref{eq:w}. For$D > 0we have \begin{align*} h(D) = - \frac{1}{\log \epsilon_0} \sum_{n=1}^{D-1}\chi_D(n) \log \sin \frac{\pi n}{D} \end{align*} where\epsilon_0 > 1$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$w = 6, so \begin{align*} h(-3) = - \frac{6}{2 \cdot 3} \sum_{n=1}^{2} \chi_{-3}(n) n = - (1-2) = 1 \end{align*} \itemD = -4$: Here$w = 4, so \begin{align*} h(-4) = 1 \end{align*} \item ForD < -4$, we have$w = 2, so \begin{align*} h(-7) &= 1\\ h(-8) &= 1\\ h(-11) &= 1\\ h(-15) &= 2\\ h(-19) &= 1\\ h(-20) &= 2\\ h(-23) &= 3 \end{align*} \end{itemize} \end{xmp} If we check whenh(D) = 1$for$D < 0, we can find that this is the case for \begin{align*} D = -3, -4, -7, -8, -11, -19, -43, -67, -163 \end{align*} Gauss calculatedh(D)$for$D$up to$-10'000$and found no others and conjectured that theses were the only fundamental discriminants with class number$1. Furthermore, he conjectured that \begin{align*} h(D) \to \infty \quad \text{as} \quad D \to - \infty \end{align*} In 1934 it was proven that there could at most be a tenth fundamental discrimant withh(D) = 1$by Heilbronn and strenghed by Siegel where he shoed tha for$\epsilon > 0\begin{align*} h(D) > C \abs{D}^{\frac{1}{2} - \epsilon} (D < 0) \end{align*} for someC> 0$(depending on$\epsilon). Moreover, we can show that \begin{align*} \lim_{D \to -\infty} \frac{\log h(D)}{\log \abs{D}} = \frac{1}{2} \end{align*}  ... ... @@ -26,6 +26,7 @@ to obtain a Formax^{2} + bxy + cy^{2}with \begin{align*} - \abs{a} < b \leq \abs{a} \leq \abs{c} \end{align*} \begin{dfn}[] A positive definite formax^{2} + bxy + cy^{2}is said to be \textbf{reduced}, if \begin{align*} ... ... @@ -49,27 +50,23 @@ First note that for a reduced formf$with$(x,y) \neq (0,0)$one has Let$f = ax^{2} + bxy + cy^{2}. We define an action on the set of forms \begin{align*} TF = S_n f \quad \text{for} \quad n \in \Z \text{ with }n > \overline{f} > n-1 TF = S_n f \quad \text{for} \quad n \in \Z \text{ with }n > \frac{b + \sqrt{D}}{2a} > n-1 \end{align*} where\overline{f} = \frac{b + \sqrt{D}}{2a}given by \begin{align*} S_n = \begin{pmatrix} n & 1\\ -1 & 0 \end{pmatrix} : ax^{2} + bxy + cy^{2} [a,b,c] \mapsto (an^{2} - bn + c)x^{2} + (2an - b)xy + ay^{2} [an^{2} - bn + c,2an - b,a] \end{align*} \end{dfn} \textbf{Notation:} We notate a formf(x,y) = ax^{2} + bxy + cy^{2}$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$[a,b,c]is \textbf{reduced}, if \begin{align*} a > 0, c > 0, b > a + c \end{align*} ... ... @@ -86,7 +83,19 @@ Every equivalence of reduced forms is obtained by iteration ofT$. 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} Letn_{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*}
Markdown is supported
0% or .
You are about to add 0 people to the discussion. Proceed with caution.
Finish editing this message first!