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\title{Rank of Elliptic Curves and The Birch Swinnerton-Dyer Conjecture}%
\author{J. Hsu, S. Miller}%
%\address{}%
%\email{}%
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\begin{document}
\maketitle \centerline {\small Department of Mathematics}
\centerline{\small Princeton University}
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\thispagestyle{empty}
\begin{abstract}
We numerically verify the Conjecture of Birch and Swinnerton-Dyer
concerning the analytic and geometric rank of an elliptic curve.
An algorithm (based on the work of Cremona) is developed in the
PARI/GP language for computing the order of vanishing of the
$L$-function for any (non-singular) curve. The analytic rank
outputs for several families of curves are compared with readily
available data on geometric ranks. Some related results on excess
rank are also presented.
\end{abstract}
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\newpage
\section{Introduction}
\par The conjecture of Birch and Swinnerton-Dyer relating the
analytic and geometric rank of an elliptic curve is perhaps one of
the less debated and widely accepted conjectures in the field. The
conjecture is now known to hold for elliptic curves with rank 0
and 1 due to the work of Kolyvagin in 1990. Nevertheless, the
general case still remains to be proved (or disproved) after it's
first proposal in 1965 \cite{BSD2}.
\\
\par In this project we study the analytic ranks of one-parameter
families with forced rank over the rational numbers. In
particular, an algorithm is developed to calculate the order of
vanishing of the $L$-function associated to an elliptic curve. The
results can then be compared with data of geometric ranks of the
same curves in verification of the Birch and Swinnerton-Dyer
conjecture.

\section{Generalities}
\par A polynomial relation $f(x,y)=0$ in two variables defines a curve, and
the general cubic curve is of the form:
\begin{equation}
ax^3+bx^2y+cxy^2+dy^3+ex^2+fxy+gy^2+hx+iy+j=0
\end{equation}
If all the coefficients are rational numbers then the cubic is
said to be rational. In particular, (non-singular) rational cubic
curves can be transformed into a simpler and more manageable form
using suitable change of variables and some projective geometry
\cite{SilTat}. This is the Weierstrass normal form, generally
expressed as $y^2=x^3+ax^2+bx+c$. If the discriminant $\Delta
=-4a^3c+a^2b^2+18abc-4b^3-27c^2$ is non-zero then the curve is an
elliptic curve\footnote{The more familiar case is when $a=0$,
$\Delta =-4b^3-27c^2$.}. Instead of completing the square and
changing variables fully, we can also write the elliptic curve in
the alternate form\footnote{See Appendix A}
\begin{equation}
y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6
\end{equation}
and many computational programs takes the 5-component vector
$[a_1,a_2,a_3,a_4,a_6]$ as the representation of the curve.
\\
\par We will restrict our investigation to rational elliptic curves,
where we can ask for rational solutions to a curve.  The set of
all such points for an elliptic curve $E$ will be denoted
$E(\Rat)$.

\section{Defining the $L$-function for elliptic curves}
\par We begin by recalling the Riemann zeta function defined by
$\zeta(s) = \sum_{n=1}^{\infty} n^{-s}$. Using the theorem of
unique factorisation and some elementary analysis \cite{Kob} one
can show that an equivalent form for the function is $\zeta(s) =
\prod_{p} (1-p^{-s})^{-1}$ where the product is over all positive
primes $p$. This is the form we will use.
\\
\par If we restrict an elliptic curve $E$ to a finite field of
integers modulo $p$ (prime) $\mathbb{F}_p$, then it is known that
$\#\{E(\Rat) \mod p\}=p+1-a_p$, where $a_p \leq 2\sqrt{p}$ is a
result due to Hasse. With this definition we can defined a local
zeta function for the elliptic curve reduced modulo $p$
\begin{equation}
\zeta(E_p,s)=\frac{1-a_pp^{-s}+p^{1-2s}}{(1-p^{-s})(1-p^{1-s})}
\end{equation}
which reduces to $\zeta(E_p,s)=\frac{1}{(1-p^{-s})(1-p^{1-s})}$ if
$p$ $| \Delta$. The global zeta function is simply defined as the
product of local zeta functions over all primes: $\zeta(E,s)
=\prod_{p} \zeta(E_p,s)$. It is easy to see that
$\zeta(E,s)=\zeta(s)\zeta(s-1) L(E,s)^{-1}$ if
\begin{equation}
L(E,s)=\prod_{p} (1-a_pp^{-s}+p^{1-2s})^{-1}
\end{equation}

\par We will take this as the definition for the $L$-function
associated with an elliptic curve. This product converges for
$\Re(s) > \frac{3}{2}$ and can be analytically continued
\cite{Kob} to all of $\mathbb{C}$. Thus we can speak of the
behaviour of the function at $s=1$. If the order of
vanishing\footnote{That is if $L^{(r)}(E,1) \neq 0$ but
$L^{(k)}(E,1)=0$ $\forall k<r$.} of $L(E,1)$ is $r$, then we say
the elliptic curve $E$ has analytic rank $r$ and we denote it
$r_a$.
\\
\par We are now ready to state the conjecture of Birch and
Swinnerton-Dyer.

\section{The Birch and Swinnerton-Dyer Conjecture}
\par By the theorem of Mordell, it is known that for an elliptic
curve $E$ over the rationals $\Rat$, the set $E(\Rat)$ is finitely
generated. More explicitly:
\begin{equation}
E(\Rat) \cong \mathbb{Z}^{r} \oplus \mathbf{T}
\end{equation}
for some non-negative integer $r$, and $\mathbf{T}$ a finite
abelian group. The integer $r$ is called the geometric rank of
$E$, and we shall denote it $r_g$.
\\
\par The Birch and Swinnerton-Dyer Conjecture can then be stated
simply as: for any elliptic curve $E$ over $\Rat$, the analytic
and geometric rank equal\footnote{A weaker form of the conjecture
states that $r_a=r_g \mod 2$, but we shall only consider the
general case.}; that is, $r_a=r_g$. An equivalent way of stating
the conjecture is as follows: the Taylor expansion of $L(E,s)$
about $s=1$ has the form
\begin{equation*}
L(E,s)=c(s-1)^{r}+\text{higher order terms}
\end{equation*}
with $c \neq 0$ and $r(=r_a)=r_g$.

\section{Modular forms}
\par The algorithm used for computing the analytic rank of elliptic
curves requires the theory of modular functions, we describe some
basic properties of these functions before we discuss the
algorithm itself.
\\
\par Consider the upper half complex plane; that is the set of
complex numbers $z$ with $\Im(z)>0$.  For a matrix $\gamma
=\begin{bmatrix} a & b \\ c & d \end{bmatrix}$; $a,b,c,d \in
\mathbb{Z}$ and $\det \gamma =1$, define its action on a complex
number $z$ as $\gamma z=\frac{az+b}{cz+d}$. Note that
\begin{equation*}
\Im(\gamma z)=\frac{\Im(az+b)(c\overline{z}+d)}{\abs{cz+d}^2}
=\frac{\Im(adz+bc\overline{z})}{\abs{cz+d}^2}
=\frac{\Im(ad-bc)iy}{\abs{cz+d}^2}=\frac{\Im(z)}{\abs{cz+d}^2}
\end{equation*}
Thus $\Im(z)>0 \Rightarrow \Im(\gamma z)>0$.
\\
\par A function $f$ is said to be modular of weight $2k$ if it satisfies
the relation
\begin{equation}
f(\gamma z)=(cz+d)^{-2k} f(z)
\end{equation}
In particular, it is called a modular form if the function is
holomorphic everywhere. An important property of modular functions
is the existence of a fundamental domain for the group of matrices
acting on the half plane.  This is the region above the
semi-circle $x^2+y^2=1$ bounded on the left by $x=-\frac{1}{2}$
and on the right by $x=\frac{1}{2}$ \cite{Ser}. Under the action
of the two matrices $S=\begin{bmatrix} 0 & -1 \\ 1 & 0
\end{bmatrix}$, $T=\begin{bmatrix} 1 & 1 \\ 0 & 1
\end{bmatrix}$ every point on the complex half plane can be transformed
into the fundamental domain. It is easily seen that $Tz=z+1$ and
$Sz=-\frac{1}{z}$.
\\
\par Two properties following immediately from the definitions above
are:
\begin{equation}
\begin{split}
f(z+1)=f(Tz)=f(z) \\
f(-1/z)=f(Sz)=z^{2k} f(z)
\end{split}
\end{equation}
The first relation implies the periodicity of $f$, and therefore
we can express a modular form as a series
$f(z)=\sum_{n=0}^{\infty} a_n \exp(2\pi inz)$ which converges for
all $z$ in the upper half complex plane and satisfies the identity
as given in the second relation. It is called a cusp form if
$a_0=0$.

\section{Evaluating $L(E,1)$}
\par Due to the work of Taylor-Wiles and Wiles, it is now known that
all elliptic curves are modular. Hence all the properties
mentioned in the previous section are valid for elliptic curves.
We will derive an explicit expression for $L(E,1)$ in this
section; the method used for higher derivatives is a
generalisation of this technique and will be addressed in the
following section.

\par Since an elliptic curve $E$ is modular, we can express it as a
series
\begin{equation}
f(iy)=\sum a_n \exp(-2\pi ny)
\end{equation}
which can then be used for the Mellin transform\footnote{Using the
substitution $t=2\pi ny$.} as follows:
\begin{equation}
\begin{split}
\int_{0}^{\infty} f(iy) y^{s-1} dy &=\sum a_n \int_{0}^{\infty}
\exp(-2\pi ny)y^{s-1}dy \\
&=\sum a_n \int_{0}^{\infty} \exp(-t) (2\pi n)^{-s} t^{s-1} dt \\
&=(2\pi)^{-s} \sum a_n n^{-s} \int_{0}^{\infty} \exp(-t)t^{s-1}dt \\
&=(2\pi)^{-s}L(E,s) \Gamma(s)
\end{split}
\end{equation}
where $\Gamma(s)=\int_{0}^{\infty} \exp(-y) y^{s-1} dy$ is the
usual gamma function defined for all complex numbers. Now modular
forms satisfy $f(-\frac{1}{Nz})=\omega Nz^2 f(z)$ with $\omega =
\pm 1$ and we can evaluate the integral in two parts and apply
this equality.
\begin{equation}
\begin{split}
(2\pi)^{-1} \Gamma(1) L(E,1) &=\int_{0}^{\infty} f(iy) dy \\
&=\int_{0}^{1/ \sqrt{N}} f(iy)
dy+\int_{1/ \sqrt{N}}^{\infty} f(iy) dy \\
&=\int_{1/ \sqrt{N}}^{\infty} f(iy) dy+\int_{0}^{1/ \sqrt{N}} f(i/Nt)(-1/Nt^2) dt \\
&=\int_{1/ \sqrt{N}}^{\infty} f(iy) dy+\omega \int_{1/ \sqrt{N}}^{\infty} f(it) dt \\
&=(1+\omega) \int_{1/ \sqrt{N}}^{\infty} f(iy) dy \\
&=(1+\omega) \left[\sum \frac{a_n \exp(-2\pi ny)}{-2\pi
n}\right]_{1/ \sqrt{N}}^{\infty}
\end{split}
\end{equation}
So we have the following explicit expression for $L(E,1)$:
\begin{equation}
L(E,1)=(1+\omega) \sum_{n=1}^{\infty} \frac{a_n \exp(-2\pi n/
\sqrt{N})}{n}
\end{equation}

\section{Computing derivatives of the $L$-function}
\par Define the Mellin transform $\int_{0}^{\infty} f(iy/ \sqrt{N})
y^{s-1}dy=N^{s/2}(2\pi)^{-s} \Gamma(s) L(f,s)$ as a function of
$f$ and $s$, which we denote $\Lambda(f,s)$. For any elliptic
curve $E$, this satisfies the functional equation
\begin{equation}
\Lambda(E,s)=\omega (-1)^k \Lambda(E,k-s)
\end{equation}
for $\omega=\pm 1$. (A specific case of this is
$\Lambda(E,s)=\omega \Lambda(E,2-s)$.) If $\omega=-1$ we say the
sign of the functional equation is odd, and it is even otherwise.
It is known that the parity of the analytic rank of an elliptic
curve is the same as that for the functional equation. This is
particularly useful as we only need to compute the odd/even
derivatives of the $L$-function in calculating $r_a$ for odd/even
sign.
\\
\par We now generalise the integral transforms for derivatives of
the $L$-function. As before, we split the integral into two parts
and re-combine to obtain a single integral over a slightly
different interval:
\begin{equation}
\begin{split}
\Lambda(f,s) &=\int_{0}^{\infty} f(iy/ \sqrt{N})y^{s-1} dy \\
&=\int_{0}^{1} f(iy/ \sqrt{N})y^{s-1} dy+\int_{1}^{\infty} f(iy/
\sqrt{N})y^{s-1} dy \\
&=\int_{\infty}^{1} f(iy/ \sqrt{N})cy^{1-s} dy+\int_{1}^{\infty}
f(iy/ \sqrt{N})y^{s-1} dy \\
&=\int_{1}^{\infty} f(iy/ \sqrt{N})(y^{s-1}-cy^{1-s}) dy
\end{split}
\end{equation}
Differentiating $k$ times with respect to $s$ gives
\begin{equation}
\Lambda ^{(k)}(f,s)=\int_{1}^{\infty} f(iy/ \sqrt{N}) (\log y)^k
(y^{s-1}-c(-1)^k y^{1-s}) dy
\end{equation}
and at $s=1$ this simplifies to $\Lambda ^{(k)}(f,1)=(1-(-1)^kc)
\int_{1}^{\infty} f(iy/ \sqrt{N}) (\log y)^k dy$.
\\
\par Since the analytic rank $r_a$ is defined as the order of the
vanishing of the $L$-function, it is clear that we need only to
consider the case which yields a non-zero value.
\begin{equation}
\Lambda ^{(k)}(E,1)=2\sum_{n=1}^{\infty} a_n \int_{1}^{\infty}
\exp(-2\pi ny/ \sqrt{N}) (\log y)^k dy
\end{equation}
Notice that for $k=0$ this is equivalent to the expression for
$L(E,1)$ we obtained in the previous section since
$\Lambda(E,1)=(\sqrt{N}/2\pi)L(E,1)$.
\\
\par For $k \geq 1$ we can use integration by parts and get
\begin{equation}
\Lambda ^{(k)}(f,1)=\frac{\sqrt{N}}{\pi} \sum_{n=1}^{\infty}
\frac{a_n}{n} \int_{1}^{\infty} \exp(-2\pi ny/ \sqrt{N}) (\log
y)^{k-1} \frac{dy}{y}
\end{equation}
and obtain the explicit expression for $L^{(r)}(E,1)$ as follows:
\begin{equation}
L^{(r)}(E,1)=2r! \sum_{n=1}^{\infty} \frac{a_n}{n} G_r
\left(\frac{2\pi n}{\sqrt{N}}\right)
\end{equation}
where $G_r(x)=\frac{1}{(r-1)!} \int_{1}^{\infty} \exp(-xy)(\log
y)^{r-1} \frac{dy}{y}$.
\\
\par We can therefore compute the derivatives of the $L$-functions
provided we can compute the functions $G_r(x)$.  For the case of
$r=1$ it is simply an exponential integral and we have
\begin{equation}
G_1(x)=\log \frac{1}{x}-\gamma-\sum_{n=1}^{\infty}
\frac{(-x)^n}{n!n}
\end{equation}
where $\gamma=0.577216$ is the Euler constant. The general case
can be obtained by using the relation $G_0(x)=\exp(-x)$ and
$G'_r(x)=P_r(\log \frac{1}{x})+\sum_{n=1}^{\infty}
\frac{(-x)^{n-r}}{n!n^r}$ where $P_r$ is the $r$-degree polynomial
satisfying $P'_r(t)=P_{r-1}(t)$ and $P_0(t)=0$.
\\
\par These are precisely the functions we construct in the
algorithm, the values for which can be calculated to very high
precisions. The sign of the functional equation can be easily
obtained, and we can then proceed to calculate the odd/even
degrees of the derivatives. The number of terms considered in
these summations varies depending on the elliptic curve, and is of
order $\sqrt{N}$ where $N$ is the conductor of the curve.  The
tolerance for determining whether $L(E,1)=0$ was set at
$\abs{0.001}$ which seems to work quite well in practice, as any
non-zero values are very large in comparison. The analytic rank is
computed as the order at which the sequence of odd/even
derivatives does not vanish.

\section{Results}
\par The program was first tested with 4 curves of known geometric
rank:
\begin{tabbing}
\hspace{35pt} $E_1,r_g=3$: \hspace{25pt} \= $y^2+y=x^3-7x+6$ \\
\hspace{35pt} $E_2,r_g=4$: \> $y^2+xy=x^3-x^2-79x+289$ \\
\hspace{35pt} $E_3,r_g=5$: \> $y^2+y=x^3-79x+342$ \\
\hspace{35pt} $E_4,r_g=6$: \> $y^2+xy=x^3-7077x+235516$ \\
\end{tabbing}
All of them returned the same analytic rank as outputted from the
computation.
\\
\par Two other curves of known geometric rank was also tested much
later (due to the time consuming nature of the computations for
curves with large conductor):
\begin{tabbing}
\hspace{35pt} $E_5,r_g=7$: \hspace{25pt} \= $y^2=x^3-12979x+405826$ \\
\hspace{35pt} $E_6,r_g=8$: \> $y^2=x^3-14752493461692x$ \\
\end{tabbing}
Once again, the analytic and geometric ranks agree.
\\
\par We then proceeded to investigate one-parameter families of
elliptic curves; in particular those with forced rank\footnote{See
Appendix C} over $\Rat(t)$. Three families were tested, as
follows:
\begin{tabbing}
\hspace{35pt} $E_1(t),r=2$: \hspace{25pt} \= $y^2=x^3-t^2x+t^4$ \\
\hspace{35pt} $E_2(t),r=3$: \> $y^2=x^3+5x^2-16t^2x+64t^2$ \\
\hspace{35pt} $E_3(t),r=4$: \> $y^2=x^3+41x^2+(184-16t^2)x+144$ \\
\end{tabbing}
Among them, those particular curves within each family with
analytic rank smaller than the forced rank is selected and ran
through another program to search for rational points and
determine if they are linearly dependent. Since there are no
readily available data on the geometric rank of the curves in
these families only the exceptional cases have been used in the
verification of the Birch and Swinnerton-Dyer conjecture.
\\
\par We summarise the distribution of the (analytic) ranks in the
following tables.
\\
\par The first 300 members in $E_1(t)$, $t>0$~
\begin{center}
\begin{tabular}[b]{c|c}
Analytic Rank & Number of Curves \\ \hline
0 & 0 \\
1 & 1 \\
2 & 112 \\
3 & 156 \\
4 & 31 \\
\end{tabular} \end{center}

\addvspace{12pt}
\par The first 100 members in $E_2(t)$, $t>0$~
\begin{center}
\begin{tabular}[b]{c|c}
Analytic Rank & Number of Curves \\ \hline
0 & 0  \\
1 & 1  \\
2 & 4  \\
3 & 38  \\
4 & 53  \\
5 & 4  \\
\end{tabular} \end{center}

\addvspace{12pt}
\par The first 50 members in $E_3(t)$, $t>0$~
\begin{center}
\begin{tabular}[b]{c|c}
Analytic Rank & Number of Curves \\ \hline
0 & 0 \\
1 & 0 \\
2 & 3 \\
3 & 6 \\
4 & 30 \\
5 & 10 \\
6 & 1 \\
\end{tabular} \end{center}

\par A family of rank 0 over $\Rat(t)$; $y^2=x^3+t$; was also
tested, and provided the largest number of raw data. However, a
large proportion of the curves in this family have analytic rank 0
or 1, for which the BSD conjecture is already known to be true.
Nonetheless, there are quite a number of curves with rank 2 or 3;
for which comparison with known data on geometric rank once again
confirms the validity of the conjecture.
\\
\par Rank distribution for $y^2=x^3+t$, $t=1 \dotsc 1000$ and $t=-1 \dotsc -1000$
\begin{center}
\begin{tabular}[b]{c|c}
Analytic Rank & Number of Curves \\ \hline
0 & 690  \\
1 & 965  \\
2 & 314  \\
3 & 31  \\
\end{tabular} \end{center}
\par Computations have also been performed for a number of families
presented by Fermigier, and all the families tested reaffirm the
results published \cite{Fer}. The families tested are:
\begin{tabbing}
\= Forced Rank \hspace{10pt} \= Curve $E(t)$ \hspace{125pt} \= Values of $t \in \mathbb{Z}$ \\
\> $r=0$ \> $y^2+y=x^3+x^2+x+t$ \> $-363 \dotsc 612$ \\
\> $r=0$ \> $y^2+xy+3y=x^3+tx+1$ \> $-150 \dotsc 105$ \\
\> $r=0$ \> $y^2+xy-2y=x^3+tx+1$ \> $-94 \dotsc 183$ \\
\> $r=0$ \> $y^2+xy+y=x^3+2x+t$ \> $-244 \dotsc 576$ \\
\> $r=1$ \> $y^2+xy+3y=x^3+x^2+tx$ \> $-131 \dotsc 295$ \\
\> $r=2$ \> $y^2+xy+7y=x^3+tx^2+(-t-1)x$ \> $-220 \dotsc 368$ \\
\end{tabbing}

\section{Technicalities and developments}
\par Since the program is written in the PARI/GP language, it runs
slower in comparison to an identical program written in C
developed independently in a similar project. Nevertheless, the
two programs concur on the analytic rank of all the curves tested;
a result which is rather reassuring. The lengthy run-time for
curves with high conductor remains problematic for rapid
generation of data, and one possible improvement would be to
calculate and store all the $a_n$ values in a database in advance
and have the program read off values instead of recalculating
during each computation.
\\
\par It is possible, even desirable, to incorporate this program
with one which calculates the geometric rank of a given elliptic
curve. That way, we can readily verify the Birch and
Swinnerton-Dyer conjecture for large number of curves in a family.
Yet another possible use of the program would be in the study of
the Gross-Zagier formula relating $L'(E,1)$ to the height of
Heegner points on an elliptic curve:
\begin{equation*}
\exists \eps \ni L'(E,1)=\eps h(x_E)
\end{equation*}
where $h(x)$ is the canonical height of a point and $x_E$ a
Heegner point. Last but not least, as mentioned before, it can be
useful for investigations on excess rank for families of curves.

\appendix
\section{Weierstrass normal form}
\par We outline here the steps in arriving at the Weierstrass normal
form of a cubic curve.  Begin with a cubic curve $C$ in the
projective plane and assume a rational point $O_a$ on $C$ is
given. Take $Z=0$ to be the tangent line to $C$ at $O_a$. This
line intersects $C$ at one other point $O_b$, and take the $X=0$
axis to be the tangent to $C$ at $O_b$. Finally choose $Y=0$ to be
any line except $Z=0$ going through $O_a$.
\\
\par Having chosen the axes in this fashion let $x=\frac{X}{Z}$, $y=\frac{Y}{Z}$.  By projective
transformation (with linear conditions in the new coordinates) the
resulting equation for $C$ will be of the form
\begin{equation*}
xy^2+(ax+b)y=cx^2+dx+e
\end{equation*}
Multiply through by $x$ and rename $xy$ as $y$ again we obtain
\begin{equation*}
y^2+(ax+b)y=\text{cubic in } x
\end{equation*}
Replace $y$ by $y-\frac{1}{2}(ax+b)$ and completing the square on
the left-hand side of the equation finally yields
\begin{equation*}
y^2=\text{cubic in } x
\end{equation*}
(We can always adjust the cubic to be monic by changing variables
$x \rightarrow \lambda x$, $y \rightarrow \lambda^2 y$ where
$\lambda$ is the leading coefficient of the cubic.)

\section{Algorithm in PARI/GP}
\begin{verbatim}
\\ Computing derivatives of L-series

default(primelimit,20000000);

\\ Prime number function
pith(x) = {
  local(l,r,lx);
  if (x <= 2, return (x==2));
  lx = log(x);
  l = floor(x/lx);
  r = floor(l * (1 + 3/(2*lx)));
  while (r-l>1,
    m = (l+r)>>1;
    if (prime(m)<=x, l=m, r=m));
  l;
}

\\ Series expansion
default(seriesprecision,30);
S=-Euler*s+sum(n=2,30,(-1)^n*zeta(n)/n*s^n); SE=exp(S);
SV=vector(25,i,polcoeff(SE,i-1));

SP(r,t)=local(k);sum(k=0,r,SV[r-k+1]*t^k/factorial(k));

{G(r,x)=
   local(ss,ss0,n,sn);
   if(x>30,return(0));
   ss=SP(r,log(1/x));
   ss0=ss-1;
   n=1;
   sn=-((-1)^r);
   while(abs(ss-ss0)>1e-15,
     ss0=ss;
     ss=ss+sn*x^n/(n^r*factorial(n));
     n=n+1;
     sn=-sn;
   );
  2*ss;
}

\\ Buhler-Gross recursion
getap(r)=if(r<=pn,aplist[r],ellap(F,prime(r)))
getp(r)=if(r<=pn,ellplist[r],prime(r))

{ BGadd(n,i,a,b)=
    local(j,j0,a1);
    if(a==0,j0=i,ES=ES+a*G(Er,X*n)/n;j0=1;);
    j=j0;
    while(j<=i && getp(j)*n<=Ebnd,
        a1=a*getap(j);
        if(j==i && N%getp(j)!=0,
            a1=a1-getp(j)*b);
        BGadd(getp(j)*n,j,a1,a);
        j=j+1;
    )
}

\\ Derivative function for the L-series
{ elllsD(E,r,bnd)=
  local(pmax,loop,N,F);
  F=ellinit(E);
  pmax=sqrt(bnd);
  pn=pith(pmax);
  loop=pith(bnd);
  ellplist=primes(pn);
  aplist=vector(pn,i,ellap(F,ellplist[i]));
  aplist[1]=ellak(F,2);
  N=ellglobalred(E)[1];
  X=2*Pi/sqrt(N);
  Er=r;
  Ebnd=bnd;
  ES=G(r,X);
  for(i=1,loop,BGadd(getp(i),i,getap(i),1));
  ES;
}

\\ Computing the analytic rank and writing results to a file
{
  ellarank(E)=
  local(sfeq,E1,p,cond,Lbnd,Lr1);
    E=ellchangecurve(E,ellglobalred(E)[2]);
  print("Minimised Curve=",[E[1],E[2],E[3],E[4],E[5]]);
  cond=ellglobalred(E)[1];
  Lbnd=round(2*sqrt(cond));
  print("Summing ",Lbnd," a_n terms");
  sfeq=ellrootno(E);
  if(sfeq==-1,print("Odd"),print("Even"));
  d=(1-sfeq)/2;
  Lr1=0;
  while(abs(Lr1)<0.001,
    Lr1=elllsD(E,d,Lbnd);
    print("L^(",d,")(E,1)=",Lr1);
    d=d+2);
  print("Rank=",d-2);
  print(" ");
\end{verbatim}

\section{One parameter families with forced rank}
\par The construction of one-parameter families of curves with
forced rank over $\Rat(t)$ is quite involved, we outline here only
two simple cases of rank 0 and rank 1 family.
\\
\par \textit{C1: Preliminaries}
\par Recall the non-zero elements of the multiplicative group
$(\mathbb{Z} /p \mathbb{Z})^*$ for $p$ prime forms a cyclic group
with generator $g$ such that
$\{1,g,g^2,\dotsc,g^{p-2}\}=\{{1,2,3,\dotsc,p-1}\}$. Define the
Legendre symbol $(\frac{x}{p})$ as 0 if $x=0$, 1 if $x$ is
equivalent to a non-zero square modulo $p$, and -1 if $x$ is not
equivalent to a square modulo $p$. The following properties are
immediate:
\begin{itemize}
    \item $(\frac{xy}{p})=(\frac{x}{p})(\frac{y}{p})$
    \item $(\frac{x+np}{p})=(\frac{x}{p})$
    \item $(\frac{x^2}{p})=(\frac{x^{-1}}{p})=(\frac{x}{p})$,\text{
if $x \neq 0$}
\end{itemize}

\par Consider now the case of $(\frac{f(x)}{p})$ when $f(x)=ax+b$ is
a linear polynomial. We abbreviate $\sum_{x=0}^{p-1}
(\frac{x}{p})$ by $\sum_{x(p)} (\frac{x}{p})$. The case when $a=0$
is easily seen as $\sum_{x(p)} (\frac{b}{p})=p(\frac{b}{p})$. If
$a \neq 0$ we can change variables $t=a^{-1}x$, $u=t+b$ and the
sum becomes $\sum_{x(p)} (\frac{ax+b}{p})=\sum_{t(p)}
(\frac{t+b}{p})=\sum_{u(p)} (\frac{u}{p})$. As $u$ (and hence $x$
also) progresses through the cyclic group each non-zero square
contributes +1 and each non-square contributes -1. There are as
many non-zero squares as non-squares, so we obtain the following
result:
\begin{equation*}
\sum_{x(p)} \left(\frac{ax+b}{p}\right)=\left\{%
\begin{array}{ll}
    0, & \hbox{$a=0$} \\
    p(\frac{b}{p}), & \hbox{$a \neq 0$} \\
\end{array}%
\right.
\end{equation*}
Now for the curve $y^2=x^3+ax+b$ the Legendre sum $\sum_{x(p)}
(1+(\frac{x^3+ax+b}{p}))$ counts the number of solutions modulo
$p$ if we ignore the point at infinity, and so we obtain a
Legendre expansion for $a_p=p+1-\#\{E(\Rat) \mod p\}$ as
\begin{equation*}
a_p=-\sum_{x=0}^{p-1} \left(\frac{x^3+ax+b}{p}\right)
\end{equation*}

\par \textit{C2: The Silverman-Rosen Theorem}
\par For simplicity, consider a family of elliptic curves $F$ in
Weierstrass form $y^2=x^3+A(T)x+B(T)$, $T \in \mathbb{Z}$. For
every choice $t$ of $T$ we get an elliptic curve and denote it by
$E_t$. We can regard $F$ as an elliptic curve over $\Rat(t)$. By
the Specialisation Theorem of Silverman, all but finitely many
$E_t$ will have rank at least $r$ if the family $F$ has rank $r$
over $\Rat(t)$. This requires a way of determining the rank of the
family of curves over $\Rat(t)$. We begin by defining the $a_p$'s
for the family as the average of the sum of the $a_p$'s:
$A_F(p)=\frac{1}{p} \sum_{t(p)} a_{E_t}(p)$. The Silverman-Rosen
Theorem states that for $F$ with $\deg A(T) \leq 3$, $\deg B(T)
\leq 5$ we have the equality
\begin{equation*}
\lim \frac{1}{X} \sum_{p \leq X} -A_F(p) \log p=rank(F(\Rat(t)))
\end{equation*}
Now the prime number theorem states that $\sum_{p \leq X} \log p=X
+ \text{lower order terms}$ and so the rank of the family over
$\Rat(t)$ is constant if we can show $A_F(p)$ is constant. We will
show this for two families.
\\
\par \textit{C3: Example of a rank 0 family}
\par Consider $y^2=x^3+t$. Now we calculate the Legendre sum $
\sum_{t=0}^{p-1} \sum_{x=0}^{p-1} (\frac{t+x^3}{p})=\sum_{x=0}
\sum_{t=0}^{p-1} (\frac{t+x^3}{p})+ \sum_{x \neq 0}
\sum_{t=0}^{p-1} (\frac{t+x^3}{p})=\sum_{t(p)}
(\frac{t}{p})+\sum_{x \neq 0} \sum_{t=0}^{p-1}
(\frac{t+x^3}{p})=0+\sum_{x \neq 0} \sum_{t(p)} (\frac{t+x}{p})$
since $x \rightarrow x^3$ is an automorphism, and the sum equals 0
for all $t$ so the rank of the family is $-A_F(p)=0$.
\\
\par \textit{C4: Example of a rank 1 family}
\par Consider $y^2+xy+3y=x^3+x^2+tx$ which in Weierstrass normal
form is $y^2=x^3+5x^2+16tx+24x+144$. The Legendre sum
$\sum_{t=0}^{p-1} \sum_{x=0}^{p-1}
(\frac{x^3+5x^2+16tx+24x+144}{p})=\sum_{x=0} \sum_{t(p)} (
\frac{x^3+5x^2+16tx+24x+144}{p} )+\sum_{x \neq 0} \sum_{t(p)}
(\frac{x^3+5x^2+16tx+24x+144}{p})=(\frac{144}{p})$ and so the rank
of the family is $-A_F(p)=\frac{p}{p}=1$.
% ----------------------------------------------------------------
\newpage
\bibliographystyle{plain}
\begin{thebibliography}{}

\bibitem{AkiTan}
{\bf Akiyama, Tanigawa}, {\it Calculation of values of
$L$-functions associated to elliptic curves}, Mathematics of
Computation Vol. 68 No. 227 (1999)

\bibitem{BSD1}
{\bf Birch, Swinnerton-Dyer}, {\it Notes on elliptic curves I},
Journal Reine u. angewandte Math 212 (1963)

\bibitem{BSD2}
{\bf Birch, Swinnerton-Dyer}, {\it Notes on elliptic curves II},
Journal Reine u. angewandte Math 218 (1965)

\bibitem{Cre}
{\bf Cremona}, {\it Algorithms for modular elliptic curves},
Cambridge University Press (1992)

\bibitem{Fer}
{\bf Fermigier}, {\it \'{E}tude exp\'{e}rimentals du rang de
familles de courbes elliptiques sur $\mathbb{Q}$}, Experimental
Math Vol. 5 (1996)

\bibitem{GouMaz}
{\bf Gouvea, Mazur}, {\it The square-free sieve and the rank of
elliptic curves}, Journal of the American Mathematical Society
Vol. 4 No. 1 (1991)

\bibitem{Kob}
{\bf Koblitz}, {\it Introduction to elliptic curves and modular
forms}, Springer-Verlag New York (1984)

\bibitem{Ser}
{\bf Serre}, {\it A course in arithmetic}, Springer-Verlag New
York (1973)

\bibitem{SilTat}
{\bf Silverman, Tate}, {\it Rational points on elliptic curves},
Springer-Verlag New York (1992)

\bibitem{Ste}
{\bf Stephens}, {\it The diophantine equation $X^3+Y^3=DZ^3$ and
the conjectures of Birch and Swinnerton-Dyer}, Journal fur
Mathematik Vol. 231 (1968)

\end{thebibliography}
\end{document}
% ----------------------------------------------------------------
