Author: William A. Stein
1\documentclass[11pt]{article}
2\voffset=-0.1\textheight
3\textheight=1.2\textheight
4\include{macros}
5
6\title{Math 168A: FINAL}
7\author{William Stein}
8\date{\bf Due: Friday, Dec 9, 2005, at 5pm}
9\begin{document}
10\maketitle
11
12\noindent{\em The problems have equal point value, and parts of
13  multi-part problems are of the same value.  {\bf You may not talk to
14  anybody about these problems.}  You {\em are} allowed to use your
15  notes, computer software (show complete session logs), books and web
16  pages.   There are 8 problems. This exam is worth 25\% of your grade.}
17
18\begin{enumerate}
19
20\item Find $13$ pairs $x,y$ of rational numbers such that
21$y^2 + xy = x^3 +1$.   You will only receive half credit
22if you only find $12$ solutions.
23
24\item Let $E$ be an elliptic curve and $P\in E(\Q)$ a point.
25Prove from the definition of the group law that $(P+P) + P = P + (P+P)$.
26(You may use either the geometric definition involving
27the chord and tangent procedure or the algebraic formula
28for the group law from class.)
29
30\item The right triangle with side lengths $33$, $56$, and $65$ has
31area $924$.  Find two more right triangles with positive rational
32sides and area $924$.
33
34\item Let $a$ be a positive integer.
35\begin{enumerate}
36\item  For any prime $p\nmid 6a$, prove that the (projective)
37curve $E_a$ associated to the equation $y^2 = x^3 + a$
38is an elliptic curve. (I.e. check that a certain discriminant is
39nonzero modulo $p$.)
40\item Suppose $p\nmid 6a$ is a prime with $p\con 2\pmod{3}$.
41Prove that $E_a$ has $p+1$ points modulo $p$, i.e.,
42$\#E_a(\F_p) = p+1$.
43\end{enumerate}
44
45\item Consider the following elliptic curve ElGamal cryptosystem (as in
46the textbook) over the finite field $\F_{97}$ of order $97$:
47\begin{align*}
48   E &: y^2  = x^3 + x + 3\\
49   B &= (51,3) \text{\qquad (base point)}\\
51\end{align*}
52\begin{enumerate}
53\item Compute the public key $(p, E, B, nB)$.
54\item There is a point $P$ on $E$ whose $x$-coordinate
55encodes a very special day in December.   For some
56random $e$, the point
57$P$ encrypts as
58$$59 (rB, P+r(nB)) = ((28, 62), (63, 85)). 60$$
61Show how to use knowledge of $E,B,n$ to find~$P$,
62then find~$P$.
63\item Find a point $P \in E(\F_p)$ with $x$-coordinate
64equal to the day of the month when you were born.
65Then encrypt this point by computing
66$(rB, P+r(nB))$
67for some random value of $r$.
68\end{enumerate}
69
70\item
71\begin{enumerate}
72\item Compute the matrix of $T_2$ on the following (Victor Miller) basis
73for $M_{32}(\SL_2(\Z))$:
74{\small
75\begin{align*}
76f_1 = 1 + 2611200q^{3}& + 19524758400q^{4} + 19715347537920q^{5} + 5615943999897600q^{6} + \cdots,\\
77f_2 = q + 50220q^{3}& + 87866368q^{4} + 18647219790q^{5} + 965671206912q^{6} + \cdots,\\
78f_3 = q^{2} + 432q^{3}& + 39960q^{4} - 1418560q^{5} + 17312940q^{6} + \cdots
79\end{align*}
80}
81\item Factor the characteristic polynomial of the matrix of $T_2$.
82\item Let $n$ be a positive integer.  How many of the coefficients of~$f_1$,~$f_2$,
83and~$f_3$ are needed in order for you to compute the matrix of~$T_n$
84using the formula for the action of~$T_n$ on $q$-expansions.
85\end{enumerate}
86
87\item Compute with $\sM_2(\Gamma_0(5))$ explicitly:
88\begin{enumerate}
89\item Write down the $6$ Manin symbols for $\Gamma_0(5)$.
90\item Write down a few $2$ and $3$ term relations.
91\item Using dimension formulas one can show that
92$\sM_2(\Gamma_0(5))$ has dimension $1$ (you do not have
93to prove this).  Find a Manin symbol $(c,d)$ so that every
94other Manin symbol is a multiple of $(c,d)$ modulo the
95$2$-term and $3$-term relations.
96\item Compute the $1\times 1$ matrix of $T_2$ acting
97on  $\sM_2(\Gamma_0(5))$.
98\item Compute the cuspidal subspace $\sS_2(\Gamma_0(5))$.
99\end{enumerate}
100
101\item There is a basis for $\sS_2(\Gamma_0(23))$ such
102that the Hecke operators~$T_2$ and~$T_3$ with respect
103to this basis are given by the following matrices:
104$$105T_2 = \left(\begin{array}{rrrr} 1060&1&-1&0\\ 1070&1&-1&1\\ 108-1&2&-2&1\\ 109-1&1&0&-1 110\end{array}\right), 111\qquad 112T_3 = 113\left(\begin{array}{rrrr} 114-1&-2&2&0\\ 1150&-3&2&-2\\ 1162&-4&3&-2\\ 1172&-2&0&1 118\end{array}\right) 119$$
120Using this show how to write down a basis for $S_2(\Gamma_0(23))$.
121(Here each basis element should be a $q$-expansion of the form
122$a_1 q + a_2 q^2 + a_3 q^3 + \cdots$.  Also, I expect you to
123use the algorithm I described in class that relates
124$S_2(\Gamma_0(23))$ to homomorphisms $\T \to \C$.)
125
126
127
128\end{enumerate}
129
130
131\end{document}
132%%% Local Variables:
133%%% mode: latex
134%%% TeX-master: t
135%%% End:
136