Sharedwww / 168 / midterm / midterm.texOpen in CoCalc
Author: William A. Stein
1\documentclass[11pt]{article}
2\voffset=-0.1\textheight
3\textheight=1.2\textheight
4%\newcommand{\zmod}[1]{\Z/#1\Z}
5\newcommand{\zmod}[1]{\F_{#1}}
7
8\title{Math 168A: MIDTERM}
9\author{William Stein}
10\date{\bf Due: Wednesday, Oct 26, 2005}
11\begin{document}
12\maketitle
13
14\noindent{\em The problems have equal point value, and parts of multi-part
15  problems are of the same value.
16{\bf You may not talk to anybody about these problems.}  You are allowed
17to use your notes, computer software, books and web pages.
18}
19
20\begin{enumerate}
21\item Make a conjecture about the set of primes $p$
22such that $y^2 = x^3 + x$ has exactly $p+1$ points
23modulo $p$.
24You do not have to prove your conjecture.
25You might find the following table helpful:
26%
27% E = EllipticCurve([1,0])
28% t = [(p, E.Np(p)) for p in primes(100)]
29% sage: ' & '.join([str(a) for a, _ in t])
30% _7 = '2 & 3 & 5 & 7 & 11 & 13 & 17 & 19 & 23 & 29 & 31 & 37 & 41 & 43 & 47 & 53 & 59 & 61 & 67 & 71 & 73 & 79 & 83 & 89 & 97'
31%sage: ' & '.join([str(b) for _, b in t])
32% _9 = '3 & 4 & 4 & 8 & 12 & 20 & 16 & 20 & 24 & 20 & 32 & 36 & 32 & 44 & 48 & 68 & 60 & 52 & 68 & 72 & 80 & 80 & 84 & 80 & 80'
33%sage: '|'.join(['c' for _, b in t])
34%_10 = 'c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c'
35\begin{center}
36\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}\hline
37$p$ & 2 & 3 & 5 & 7 & 11 & 13 & 17 & 19 & 23 & 29 & 31 & 37 & 41 \\\hline
38$N_p$ & 3 & 4 & 4 & 8 & 12 & 20 & 16 & 20 & 24 & 20 & 32 & 36 & 32 \\\hline
39\end{tabular}
40
41\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|}\hline
42$p$ & 43 & 47 & 53 & 59 & 61 & 67 & 71 & 73 & 79 & 83 & 89 & 97 \\\hline
43$N_p$ &44 & 48 & 68 & 60 & 52 & 68 & 72 & 80 & 80 & 84 & 80 & 80\\\hline
44\end{tabular}
45\end{center}
46
47\item The point $(1,1)$ is a solution to the equation
48\begin{equation}\label{eqn:0}
49  3 + 2y + y^2 = x^2 + 2x + 3x^3.
50\end{equation}
51Find three other solutions as follows:
52\begin{enumerate}
53\item Multiply both sides of the equation by $9$
54and let $X=3x$, $Y=3y$ to obtain an equation
55of the form
56\begin{equation}\label{eqn:1}
57  Y^2 + a_1 XY + a_3 Y = X^3 + a_2X^2 + a_4 X + a_6.
58\end{equation}
59\item Find the point $P$ on (\ref{eqn:1}) corresponding
60to the point $(1,1)$ on (\ref{eqn:0}).
61\item Use the group operation on (\ref{eqn:1}), e.g., via
62SAGE, to compute several multiples of $P$.
63
64\item Use the multiples of $P$ to recover points on (\ref{eqn:0}).
65
66\end{enumerate}
67
68
69\item The right triangle with rational side
70lengths $a=24/5$, $b=35/12$, $c=337/60$ has area $7$.
71Find another right triangle with rational side lengths
72and area $7$ as follows (note that none of the sides
73lengths of this second triangle are allowed to be the same
74as the side lengths of the above triangle):
75
76\begin{enumerate}
77\item  Find the point $P$ on $y^2 = x^3 - 49x$ corresponding
78to the triangle $(24/5, 35/12, 337/60)$ under the bijection
79from class (see notes from 2005-10-17).
80
81\item Compute $Q = P+P$ using the elliptic curve group law.
82
83\item Find the triangle corresponding to $Q$ under
84the bijection from class.
85
86\end{enumerate}
87
88\end{enumerate}
89\end{document}
90%%% Local Variables:
91%%% mode: latex
92%%% TeX-master: t
93%%% End:
94