Sharedwww / 168 / final / final.texOpen in CoCalc
Author: William A. Stein
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\title{Math 168A: FINAL}
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\author{William Stein}
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\date{\bf Due: Friday, Dec 9, 2005, at 5pm}
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\begin{document}
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\maketitle
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\noindent{\em The problems have equal point value, and parts of
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multi-part problems are of the same value. {\bf You may not talk to
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anybody about these problems.} You {\em are} allowed to use your
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notes, computer software (show complete session logs), books and web
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pages. There are 8 problems. This exam is worth 25\% of your grade.}
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\begin{enumerate}
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\item Find $13$ pairs $x,y$ of rational numbers such that
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$y^2 + xy = x^3 +1$. You will only receive half credit
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if you only find $12$ solutions.
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\item Let $E$ be an elliptic curve and $P\in E(\Q)$ a point.
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Prove from the definition of the group law that $(P+P) + P = P + (P+P)$.
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(You may use either the geometric definition involving
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the chord and tangent procedure or the algebraic formula
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for the group law from class.)
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\item The right triangle with side lengths $33$, $56$, and $65$ has
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area $924$. Find two more right triangles with positive rational
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sides and area $924$.
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\item Let $a$ be a positive integer.
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\begin{enumerate}
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\item For any prime $p\nmid 6a$, prove that the (projective)
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curve $E_a$ associated to the equation $y^2 = x^3 + a$
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is an elliptic curve. (I.e. check that a certain discriminant is
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nonzero modulo $p$.)
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\item Suppose $p\nmid 6a$ is a prime with $p\con 2\pmod{3}$.
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Prove that $E_a$ has $p+1$ points modulo $p$, i.e.,
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$\#E_a(\F_p) = p+1$.
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\end{enumerate}
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\item Consider the following elliptic curve ElGamal cryptosystem (as in
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the textbook) over the finite field $\F_{97}$ of order $97$:
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\begin{align*}
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E &: y^2 = x^3 + x + 3\\
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B &= (51,3) \text{\qquad (base point)}\\
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n &= 17 \text{\qquad\qquad (secret)}
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\end{align*}
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\begin{enumerate}
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\item Compute the public key $(p, E, B, nB)$.
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\item There is a point $P$ on $E$ whose $x$-coordinate
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encodes a very special day in December. For some
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random $e$, the point
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$P$ encrypts as
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$$
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(rB, P+r(nB)) = ((28, 62), (63, 85)).
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$$
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Show how to use knowledge of $E,B,n$ to find~$P$,
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then find~$P$.
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\item Find a point $P \in E(\F_p)$ with $x$-coordinate
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equal to the day of the month when you were born.
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Then encrypt this point by computing
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$(rB, P+r(nB))$
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for some random value of $r$.
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\end{enumerate}
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\item
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\begin{enumerate}
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\item Compute the matrix of $T_2$ on the following (Victor Miller) basis
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for $M_{32}(\SL_2(\Z))$:
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{\small
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\begin{align*}
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f_1 = 1 + 2611200q^{3}& + 19524758400q^{4} + 19715347537920q^{5} + 5615943999897600q^{6} + \cdots,\\
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f_2 = q + 50220q^{3}& + 87866368q^{4} + 18647219790q^{5} + 965671206912q^{6} + \cdots,\\
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f_3 = q^{2} + 432q^{3}& + 39960q^{4} - 1418560q^{5} + 17312940q^{6} + \cdots
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\end{align*}
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}
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\item Factor the characteristic polynomial of the matrix of $T_2$.
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\item Let $n$ be a positive integer. How many of the coefficients of~$f_1$,~$f_2$,
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and~$f_3$ are needed in order for you to compute the matrix of~$T_n$
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using the formula for the action of~$T_n$ on $q$-expansions.
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\end{enumerate}
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\item Compute with $\sM_2(\Gamma_0(5))$ explicitly:
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\begin{enumerate}
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\item Write down the $6$ Manin symbols for $\Gamma_0(5)$.
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\item Write down a few $2$ and $3$ term relations.
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\item Using dimension formulas one can show that
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$\sM_2(\Gamma_0(5))$ has dimension $1$ (you do not have
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to prove this). Find a Manin symbol $(c,d)$ so that every
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other Manin symbol is a multiple of $(c,d)$ modulo the
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$2$-term and $3$-term relations.
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\item Compute the $1\times 1$ matrix of $T_2$ acting
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on $\sM_2(\Gamma_0(5))$.
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\item Compute the cuspidal subspace $\sS_2(\Gamma_0(5))$.
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\end{enumerate}
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\item There is a basis for $\sS_2(\Gamma_0(23))$ such
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that the Hecke operators~$T_2$ and~$T_3$ with respect
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to this basis are given by the following matrices:
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$$
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T_2 = \left(\begin{array}{rrrr}
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0&1&-1&0\\
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0&1&-1&1\\
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-1&2&-2&1\\
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-1&1&0&-1
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\end{array}\right),
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\qquad
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T_3 =
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\left(\begin{array}{rrrr}
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-1&-2&2&0\\
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0&-3&2&-2\\
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2&-4&3&-2\\
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2&-2&0&1
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\end{array}\right)
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$$
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Using this show how to write down a basis for $S_2(\Gamma_0(23))$.
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(Here each basis element should be a $q$-expansion of the form
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$a_1 q + a_2 q^2 + a_3 q^3 + \cdots$. Also, I expect you to
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use the algorithm I described in class that relates
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$S_2(\Gamma_0(23))$ to homomorphisms $\T \to \C$.)
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\end{enumerate}
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\end{document}
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