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\title{Problems for the UML / Junior Seminar}

\author{Steven Miller}

%\date{\today}

\begin{document}

\maketitle %\centerline{Mathematics Department}
% \centerline{Princeton University}
% \centerline{Princeton, NJ 08544}

%\begin{abstract}
%The beginning of the abstract. The middle. How do you like it so
%far? Almost done. Done.
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\section{General Problems}
\setcounter{equation}{0}

$$ $$

\textbf{Problem 1: Rank of a Curve.} Building on investigations
from last year, can one see the rank of the elliptic curve by
investigating (smoothed) sums of $a_p$ and $a_p^2$? Let $\lambda_p
= \frac{a_p}{\sqrt{p}}$. Similar to the proof of the Prime Number
Theorem we can show

\begin{eqnarray}
\sum_\rho \frac{x^\rho}{\rho} = - \sum_{p \le x} \lambda_p \log p
- \sum_{p \le \sqrt{x}} \lambda^2_p \log p + 2x^{\frac{1}{2}} +
small,
\end{eqnarray}

where $\rho$ runs thru the zeros of the elliptic curve. Let $r_a$
be the multiplicity of the zero at the critical point $s = 1$. We
can show the above implies

\begin{eqnarray}
\frac{1}{x^{\frac{1}{2}}} \sum_{p \le x} \lambda_p \log p =
(1-2r_a) + O\Big(\sum_{\rho \neq 1} \frac{x^{\rho-\frac{1}{2}}}
{\rho} \Big).
\end{eqnarray}

Instead of having a sharp cut-off, we could instead study a
smoothed version of the above. If we assume the Generalized
Riemann Hypothesis, we might expect the sum to oscillate around
$1-2r_a$.


$$ $$

\textbf{Problem 2: The Parity Conjecture.} We can analytically
continue the $L$-function of an elliptic curve, and we get a
functional equation $\Lambda(s,E) = \epsilon_E \Lambda(2-s,E)$,
where $\epsilon_E = \pm 1$. The Parity Conjecture asserts that (in
the limit) $\epsilon_E = 1$ and $\epsilon_E = -1$ are equally
likely. Alternatively, one could see if the Parity Conjecture
still holds if, instead of considering the family of all elliptic
curves, we instead consider a one-parameter family $y^2 = x^3 +
A(t)x + B(t)$.



$$ $$

\textbf{Problem 3: Conductors.} $\Lambda(s,E) = (2 \pi)^{-s}
N^{\frac{s}{2}} \Gamma(s)L(s)$, where $N$ is the conductor, and
has the same prime factors as the discriminant. Study how the
conductor varies in a family (say the family of all elliptic
curves, or maybe a one-parameter family). In many applications
(for example, trying to bound the average rank or investigating
density functions) what enters our equations is not $N$ but $\log
N$. For example, consider the family $\mathcal{F}_N = \{E_{a,b}:
y^2 = x^3 + ax + b, a \in [N^2, 2N^2], b \in [N^3,2N^3] \}$. Then
$\Delta_{a,b}$ is of size $N^6$ for all curves in this family. How
does $\log N_{a,b}$ vary?



$$ $$

\textbf{Problem 4: Excess Rank.} Consider one-parameter families
of elliptic curves. By construction, we can often force points of
infinite order on our curves. Let us assume we have a family where
we have forced $r$ points on each curve $E_t$. By the Birch and
Swinnerton-Dyer Conjecture, the geometric rank of the group of
rational points should equal the order of vanishing of the
associated $L$-function at the critical point. Assuming the Parity
Conjecture, we might expect half the curves to have rank $r$, and
half to have rank $r+1$. (The functional equation being odd forces
another zero). Previous investigations (see, for example,
Fermigier) for many curves with small values for $t$ have shown
that this is not the case; ie, that for small $t$, for many
families about $32\%$ have rank $r$, $48\%$ have rank $r+1$,
$18\%$ have rank $r+2$, and $2\%$ have rank $r+3$. Is this a
general phenomenon that persists in the limit, or the result of
small numbers?





$$ $$

\textbf{Problem 5: Elliptic Surfaces.} Let $\mathcal{F}$ be a
parametrized family of Elliptic Curves. For a curve $E \in
\mathcal{F}$, denote the curve by $y^2 = f_E(x)$. For a prime $p$
and a curve $E \in \mathcal{F}$, let

\begin{eqnarray}
a_E(p) = \sum_{x=0}^{p-1} \js{f_E(x)} = O(\sqrt{p})
\end{eqnarray}

Silverman and Rose have shown that, if our family is parametrized
by a variable $t$, summing $a_E(p) = a_{E_t}(p)$ is related to the
rank over $\mathbf{Q}(T)$. Specifically, let $\mathcal{E}$ be the
elliptic surface $y^2 + a_1(T)xy + a_3(T) = x^3 + a_2(T)x +
a_6(T)$, $a_i(T) \in \mathbf{Z}[T]$, $\Delta(T) \neq 0$. (Note: in
this case, our family $F$ will often be $\{E_t: t \in [N,2N]\}$.
Let $A_{\mathcal{E}}(p) = \frac{1}{p} \sum_{t=0}^{p-1}
a_{E_t}(p)$. (Note: $A_{\mathcal{E}}(p)$ is bounded independently
of $p$). Silverman and Rosen show (assuming Tate's conjecture for
Elliptic Surfaces, including the weak non-vanishing assumption
that $L_2(\mathcal{E},s)$ does not vanish on the line $Re(s) = 2$)

\begin{eqnarray}
\lim_{X \to \infty} \sum_{p \leq X} -A_\mathcal{E}(p) \log p =
\textbf{rank}\ \mathcal{E}(\mathbf{Q}(T)).
\end{eqnarray}

For curves of the form $y^2 = x^3 + A(t)x + B(t)$, we can often
prove the necessary assumptions. It is possible to cleverly chose
families so that $A_{\mathcal{E}}(p)$ is constant. One can also
(by numerical experimentation) find families where
$A_{\mathcal{E}}(p)$ varies, but is an integer (and always in some
range $[-N,M]$). Since we are dividing by $p$, it is not at all
immediate that we should get an integer, and in fact, there are
plenty of curves where we don't. Could these be the coefficients
of an interesting form (say, something associated to Dirichlet
$L$-series, or Galois representations, ...)?


$$ $$

\textbf{Problem 6: Torsion Groups.} For any elliptic curve, we can
calculate its torsion group; moreover, only a few groups may
appear as the result. In the family of all elliptic curves, what
are the relative frequencies of each group? If we specialize to
interesting one-parameter families, what are the relative
frequencies here?




$$ $$

\textbf{Problem 7: Points of Low Height.} For any elliptic curve,
we can try and find points of small height. How does the height of
the lowest point (lowest infinite point?) vary in a family? Do
curves in a given family with a low point have higher than
predicted rank?




$$ $$

\textbf{Problem 8: Birch and Swinnerton-Dyer Conjecture.} In its
simplest form, the B-SD Conjecture states the geometric rank of an
elliptic curve should equal the analytic rank (the order of
vanishing of the associated $L$-function at the critical point).
For small rank curves, numerically verify.



$$ $$

\textbf{Problem 9: Sato-Tate.} We know that $|a_p| \le 2\sqrt{p}$;
we can therefore write $a_p$ as $2\cos \theta_p$. How does
$\theta_p$ vary with $p$? Fix a one-parameter family. Write
$a_{E_t}(p)$ as $2 \cos \theta_{E_t}(p)$. For large primes, how do
the $\theta_{E_t}(p)$ vary?



$$ $$

\textbf{Problem 10: Lowest Zero.} We expect the zeros of an
elliptic curve to follow the GUE distribution. To see this may be
beyond the computational ability of the UML (to see the
convergence may require more computations than we can do on our
machines; many such investigations use large amounts of time on
supercomputers). We can, however, look at the first few zeros near
the critical point. If we are (again) looking at a family, we can
study the distribution of the lowest (or second lowest) zero. We
may want to normalize with respect to the logarithm of the
conductor.





\end{document}
