\documentclass[a4paper,12pt]{article}
\usepackage[screen,nopanel,blue,sectionbreak]{pdfscreen}
\usepackage{color}
\usepackage[display]{texpower}
\usepackage{fancybox}
\input epsf
\newcommand{\heading}[1]{%
  \begin{center}
    \large\bf
    \shadowbox{\color{DarkMagenta}#1}%
  \end{center}
  \vspace{1ex minus 1ex}}

\parindent=0pt
\parskip=10pt plus 5pt minus 5pt
\hypersetup{pdfpagemode=FullScreen}
\usepackage{tpslifonts}
\let\proofname\undefined
\usepackage{amsthm}
\usepackage{amsmath}
\DeclareMathOperator{\f}{Frob}
%\usepackage{qtimes}
%\usepackage{colortab}
\bottombuttons
\input{rgb}
\bottombuttons
 \margins{.65in}{.65in}{.65in}{.65in}
 \screensize{6.25in}{8in}
\newtheorem{thm}{\color{green1}Twierdzenie}
\theoremstyle{definition}
\newtheorem{defi}{\color{green1}Definicja}
\newtheorem{examp}{\color{green1}Przyk�ad}
\definecolor{green1}{rgb}{0,.45,.2}
\definecolor{blue1}{rgb}{0,.2,.45}
\overlay{overlay6}
\begin{document}
\large





{\Huge
\color{blue1}
\title{\color{blue1}
\Huge\bf Modularity of rigid Calabi--Yau 3--folds\rule[-15mm]{0cm}{0cm}}
\author{\huge\bf S\csname l\endcsname awomir Cynk\\[25mm]}
\date{\em Effective Methods in Algebraic and Analytic Geometry\\[5mm]
Brenna, September 2003}}
\maketitle

\clearpage
\overlay{overlay9}



\def\refname{\small\color{green1}Joint work with Duco van Straten and
  Christian Meyer (Mainz)}
\begin{thebibliography}{aa}
\color{DarkMagenta}
\bibitem{Cynk}{\em Double coverings of octic arrangements with
  isolated singularities}, Adv. Theor. Math. Phys. \textbf{3} (1999),
  217--225. 
\color{DarkBlue}
\bibitem{CS}\mbox{S.~C.,~D.~van Straten}, 
{\em Infinitesimal deformations of double covers of smooth
  algerbaic varieties}, preprint math.AG/0303329.
\color{DarkRed}
\bibitem{CM}S.~C.,~C.~Meyer, {\em Geometry and Arithmetic of certain 
    Double Octic Calabi-Yau Manifolds}, preprint math.AG/0304121, (to
  appear in Canadian Mathematical Bulletin)
\end{thebibliography}
%\end{slide}


\pagedissolve{Split /D 1 /Dm /V /M /O}
\clearpage
\overlay{overlay1}

\pause


\stepwise{
\begin{defi}
\color{DarkSlateBlue}
\emph{\color{blue} {Calabi--Yau}}  manifold is a smooth, projective
(k\csname"\endcsname ahler) threedimensional variety $X$ such that
\step{
\begin{enumerate}
\item $b_{1}(X)=0$,
\step{\item $\mathcal K_{X}=0.$}
\end{enumerate}}
\step{
\color{SlateBlue4}
there are no global holomorphic 1--forms on $X$\\[3mm]}
\step{there exists a nowhere vanishing global holomorphic 3--forms $X$}
\end{defi}
\medskip
}
\pause
Equivalently:  {\sl\color{blue}Calabi--Yau} is a compact riemannian
manifold with holonomy group in SU(3), \emph{Ricci flat}.

\medskip\rightline{\small(Calabi Conjecture (1954) proved by  Yau (1976)) }
 \pagebreak
\clearpage  

%\end{slide}
%\begin{slide}
\color{SlateBlue4}
\textbf{\centerline{\color{DarkBlue}Numerical invariants of
    Calabi--Yau manifolds}}
\begin{itemize}
\item Euler characteristic $e(X)$, 
\item Hodge numbers $h^{i,j}$ ($0\leq i,j\leq3$, $i+i\leq6$),
\item Betti numbers $b_{0},\dots,b_{6}$,\
  ($b_{p}=\sum\limits_{i+j=p}h^{ij}=\dim_{\mathbb C}H^{p}(X,\mathbb C)$).
\end{itemize}\pause
For a  Calabi--Yau manifold we have
\begin{eqnarray*}
  &&h^{0,0}=h^{0,3}=h^{3,0}=h^{3,3}=1,\\
  &&h^{1,0}=h^{0,1}=h^{2,0}=h^{0,2}=h^{1,3}=h^{3,1}=h^{2,3}=h^{3,2}=0\\
  &&e(X)=2(h^{1,1}-h^{1,2}).
\end{eqnarray*}\pause
Moreover the only non--trivial Hodge numbers have the following interpretation
\begin{itemize}
\item []$h^{1,1}$ (=$h^{2,2}$) equals the Picard number,
\item []$h^{1,2}$ (=$h^{2,1}$) -- number of deformations.
\end{itemize}
\eject

  

%\end{slide}
%\begin{slide}
\color{SlateBlue4}
\vspace{-5mm}


\textbf{\centerline{\color{DarkBlue}$l$--adic cohomology}}
{\baselineskip=14.1pt

Let $X$ be a variety over an algebraic closed field of characteristic
$p\geq0$ and $l$ a prime number $l\not=p$. There is a nice cohomology
theory for $X$, the \emph{$l$--adic cohomology}, which is a good
replacement for complex cohomology
\vspace*{-5mm}
\liststepwise{
  \begin{itemize}
  \item[] \step{$\bullet$ The groups $H^{i}(X,\mathbb Q_{l})$ are
      vector spaces, $H^{i}(X,\mathbb Q_{l})=0$ for $i>2\dim X$,}
    \item[]\step{$\bullet$ $H^{i}(X,\mathbb Q_{l})$ is a contravariant
      functor in $X$,}
  \item [] \step{$\bullet$ If $X$ is smooth compact manifold then 
    \[H^{i}(X,\mathbb Q_{l})\rm{\otimes}_{\mathbb Q_{l}}\mathbb C\cong
    H^{i}(X,\mathbb C),\]}
\item [] \step{$\bullet$ Poincare duality holds, i.e. there is a
    cup--product \[H^{i}(X,\mathbb Q_{l})\rm {\otimes} H^{j}(X,\mathbb Q_{l})\longrightarrow
    H^{i+j}(X,\mathbb Q_{l}),\] $H^{2n}(X,\mathbb Q_{l})$ is one
    dimensional and \[H^{i}(X,\mathbb Q_{l})\rm {\otimes} H^{2n-i}(X,\mathbb Q_{l})\longrightarrow
    H^{2n}(X,\mathbb Q_{l})\] is a perfect pairing.}
  \end{itemize}
}}

%\end{slide}
\clearpage

%\begin{slide}
\color{SlateBlue4}
\textbf{\centerline{\color{DarkBlue}Frobenius map}}
If $X$ is a projective variety defined over a finite field $k=\mathbb
F_{p}$ and $\bar X=X\rm{�}_{k}\bar k$ is the corresponding variety
over an algebraic closure of $k$, we define a \emph{Frobenius
  morphism}
\[\f_{p}:\bar X\ni(x_{i})\mapsto (x_{i}^{p})\in\bar X.\]\pause
For a point $P\in\bar X$ we have $P\in\bar X_{p^{r}}$ iff $P$ is a fixed
point of $\f_{p}^{r}.$ If $X$ is smooth then using the Lefschetz
fixed--point formula we get
\[N_{r}:=\#X_{p^{r}}=\sum_{i=0}^{2n}(-1)^{i}{\rm Tr}(\f_{p}^{r*}:H^{i}(\bar
X,\mathbb Q_{l})).\]\pause

Using the numbers $N_{r}$ we form the \emph{zeta function} 
\[Z(t)=Z(X;t)=\exp\left(\sum_{r=1}^{\infty}N_{r}\dfrac{t^{r}}r\right).\]\pause
\textbf{\color{DarkBlue}Example} For $X=\mathbb P^{n}$ we have $Z(t)=\dfrac1{(1-t)(1-pt)\dots(1-p^{n}t)}$.
\clearpage
\textbf{\centerline{\color{DarkBlue}Weil Conjecture}}
Let $X$ be a smooth $n$--dimensional projective variety define over $\mathbb F_{p}$.
\pause
\vspace{-7mm}

\stepwise{
  \begin{itemize}
    \item \emph{Rationality. } $Z(t)$ is a rational function.
  \step{\item \emph{Functional equation. } $Z(t)$ satisfies  the following functional equation
    \[Z\left(\dfrac1{p^{n}t}\right)=\pm p^{nE/2}t^{E}Z(t),\] where $E$
    is the top Chern class of the tangent bundle of $X$.}
  \step{\item \emph{Analogue of the Riemann hypothesis. } We can write
  \[Z(t)=\dfrac{P_{1}(t)P_{3}(t)\dots
    P_{2n-1}(t)}{P_{0}(t)P_{2}(t)\dots}P_{2n}(t),\]
where $P_{0}(t)=1-t$, $P_{2n}(t)=1-p^{n}t$ and $P_{i}(t)$ is a
polynomial with integer coefficients,
$P_{i}(t)=\prod(1-\alpha_{ij}t)$, where $\alpha _{ij}$ is an algebraic
integer with $|\alpha_{ij}|=p^{i/2}$.}
  \end{itemize}
}
\clearpage
The proof of the Weil Conjecture (due to Deligne) is based on
\[P_{i}(t)=\det\left(1-\f_{p}^{*}:H^{i}(\bar X,\mathbb Q_{l})\right).\]
\stepwise
{\step{\textbf{Example:} the zeta function for an elliptic curve equals
\[Z(t)=\dfrac{1-at+pt^{2}}{(1-t)(1-pt)},
\text{ where } |a|\leq2\sqrt p.\]}%
\step{Integer
$a$ is the trace of Frobenius map $\f_{p}^{*}$ on $H^{1}(\bar
X,\mathbb Q_{l})$. (As $a$ depends on $p$ we shall denote it by $a_{p}$.)}
\step{Taking into account the Lefschetz fixed point formula we get
$1-a_{p}+p=\#\bar X_{p}$, and so $|\#\bar X_{p}-1-p|\leq2\sqrt p$ which
is the Hasse theorem.}

\step{
\textbf{Taniyama--Shimura Conjecture:}  every elliptic
curve $E$ is \emph{modular}, i.e. there exists a weight 2 and level
$N$ (where $N$ is the conductor of $E$) cusp form such that the above
coefficients $a_{p}$ equal coefficient of the Fourier series of that
modular form. 
}

}
\clearpage
\textbf{\centerline{\color{DarkBlue}Modular forms.}}
We call $\Gamma:=\text{SL}(2,\mathbb Z)$ the \emph{full modular
  group}. The group
\[\Gamma_{o}(N):=\left\{\left(
    \begin{array}[c]{cc}
a&b\\c&d
    \end{array}
\right)\in\Gamma\,|\,c\equiv0 (\text{mod } N)\right\}, \text{\ for
}N\in\mathbb N\]
having a finite index in $\Gamma$ is called a Hecke subgroup of
$\Gamma$.
\pause

An \emph{unrestricted modular form} of weight $k$ and level $N$ is a
holomorphic function $f$ on the upper half plane $\mathbb H$ such that
\[f\left(\frac {a\tau+b}{c\tau+d}\right)=(c\tau+d)^{k}f(\tau)\quad
\text{for }\left(
  \begin{array}[c]{cc}
a&b\\c&d
  \end{array}\right)\in\Gamma_{o}(N),\quad\tau\in\mathbb H.\]\pause

Since $f(z+N)=f(z)$ we have Fourier expansion
\[f(\tau)=\sum\limits_{n=-\infty}^{\infty}c_{n}q^{n},\quad\text{with }q=\exp(2\pi i\tau/N).\]
\pause
$f$ is called a \emph{modular form} iff $c_{n}=0$ for $n<0$. If
moreover $c_{0}=0$, it is a \emph{cusp form}.
\clearpage
\textbf{\centerline{\color{DarkBlue}Modularity conjecture 
for Calabi--Yau manifolds}}
Let $X$ be a Calabi--Yau manifold defined over $\mathbb Q$. 
Denote  $t_{i}:=\text{tr}(\f_{p}^{*}:H^{i}(X,\mathbb Q_{l}))$. We have
$t_{0}=1, t_{1}=0, t_{5}=0, t_{6}=p^{3}$. By the Poincare duality
$t_{4}=pt_{2}$.  Lefschetz fixed--point formula gives
$\#X_{p}=1+p^{3}+t_{2}(1+p)-t_{3}$. \pause

Modularity conjectures for
Calabi--Yau manifolds asserts that $a_{p}:=t_{3}$ are coefficients of
of a Fourier series of a modular form. There are special cases when
this conjecture has a more explicit form (and some evidence for being
true. One of them is when $b_{2}(X)=2$. It is equivalent to $X$ being
rigid. Then $a_{p}$ are conjectured to be coefficients of the Forier
series of a weight 4 and level $N$ cusp form, $N$ being divisible only
by the primes of bad reduction.
\pause

Modularity conjecture for rigid Calabi--Yau manifold was almost proved
by L. Dieulefait, J. Manoharmayum. Their proof do not determine the
cusp form. There are only quite few known rigid Calabi--Yau manifolds,
for all of them the cusp form was computed using a method due to
Serre, Faltings and Livn� (it is enough to verify the equality of a
well-described finite set of coefficients.)

\clearpage
\textbf{\centerline{Double octic Calabi--Yau manifolds.}} 
Let $D$ be a sum of eight planes in $\mathbb P^{3}$, no six through a
point, no four through a line. Then the double covering of $\mathbb
P^{3}$ branched along $D$ has a smooth model $X$ being a Calabi--Yau
manifold. There is a simple formula for the euler characteristc of $X$. 
There is also a method to compute the Hodge numbers.
\pause 

There are nine
rigid examples in the described family (one has two variants). All
examples have the following additional properties:
\pause

\stepwise{
  \begin{itemize}
  \item they are defined over $\mathbb Q$,
\step{  \item their Picard groups are generated by divisors defined
  over $\mathbb Q$,}
\step{
\item there is a nice explicit description of a resolution of
  singularities (there are only eight types of singularities),}
\step{
\item the reduction mod $p$ is smooth for all primes $p\geq 5$ (in fact
  for most of them reduction mod 3 is also smooth).}
  \end{itemize}
}
\clearpage
\overlay{overlay7}
\color{DarkRed}
%\begin{slide}
\heading{Arrangement no. 2} may be defined by the equation
\[xyzt(x+y)(y+z)(z+t)(t+x),\]
it consists of the faces of a tetrahedron and additional four planes going trough
four vertices of the tetrahedron and intersecting in one point.
%\vspace*{-4mm}
\begin{picture}(0,0)%
\includegraphics{octic_2.pdf}%
\end{picture}%
\setlength{\unitlength}{4144sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined%
\gdef\SetFigFont#1#2#3#4#5{%
  \reset@font\fontsize{#1}{#2pt}%
  \fontfamily{#3}\fontseries{#4}\fontshape{#5}%
  \selectfont}%
\fi\endgroup%
\begin{picture}(4592,3512)(0,-2942)
\end{picture}%
\clearpage
%\end{slide}

%\begin{slide}
\heading{Arrangement no. 6} may be defined by the equation
\[xyz(x-t)(y-t)(z-t)(x+y-t)(x+y+z-t),\]
it consists of the faces of a cube and additional two planes, one through
three vertices and the other through four vertices of the cube and
intersecting along the diagonal of a face.\\
\begin{picture}(0,0)%
\includegraphics{octic_6.pdf}%
\end{picture}%
\setlength{\unitlength}{4144sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined%
\gdef\SetFigFont#1#2#3#4#5{%
  \reset@font\fontsize{#1}{#2pt}%
  \fontfamily{#3}\fontseries{#4}\fontshape{#5}%
  \selectfont}%
\fi\endgroup%
\begin{picture}(4592,3512)(0,-2942)
\end{picture}%
%\end{slide}

%\begin{slide}
\heading{Arrangement no. 23} may be defined by the equation
\[xyz(x-t)(y-t)(z-t)(x+y-t)(x-y+z-t),\]
it consists of the faces of a cube and additional two planes, one through
three vertices and the other through four vertices of the cube, having
only one of the vertices of the cube in common.

\begin{picture}(0,0)%
\includegraphics{octic_23.pdf}%
\end{picture}%
\setlength{\unitlength}{4144sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined%
\gdef\SetFigFont#1#2#3#4#5{%
  \reset@font\fontsize{#1}{#2pt}%
  \fontfamily{#3}\fontseries{#4}\fontshape{#5}%
  \selectfont}%
\fi\endgroup%
\begin{picture}(4592,3512)(0,-2942)
\end{picture}%

%\end{slide}

%\begin{slide}
  \heading{Arrangement no. 29}
 may be defined by the equation
\[xyzt(x+y+z+t)(y+z+t)(x-z+t)(x+y+2t),\]
\vfill

\heading {Arrangement no. 84} may be defined by the equation
\[xyzt(x+y+z+t)(2x+2z+t)(2y+2z+t)(x+y+2z+2t),\]
\bigskip

\centerline{\color{black}Sorry -- no pictures}
%\end{slide}
\clearpage



%\begin{slide}
\heading{Arrangement no. 44} may be defined by the equation
\[xyz(x-t)(y-t)(z-t)(x+y+z-t)(x-y+z-t),\]
it consists of the faces of a cube and additional two planes through
three vertices of the cube intersecting along the diagonal of a face.\\ 
\begin{picture}(0,0)%
\includegraphics{octic_44.pdf}%
\end{picture}%
\setlength{\unitlength}{4144sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined%
\gdef\SetFigFont#1#2#3#4#5{%
  \reset@font\fontsize{#1}{#2pt}%
  \fontfamily{#3}\fontseries{#4}\fontshape{#5}%
  \selectfont}%
\fi\endgroup%
\begin{picture}(4592,3512)(0,-2942)
\end{picture}%

  
%\end{slide}

%\begin{slide}
\heading{Arrangement no. 62}may be defined by the equation
\[xyz(x-t)(y-t)(z-t)(x+y+z-2t)(x+y),\]
{\small it consists of the faces of a cube and additional two planes, one plane
through an edge of the cube and parallel to a diagonal of the cube,
and one plane through three vertices of the cube not belonging to the
first plane.}\\
\begin{picture}(0,0)%
\includegraphics{octic_62.pdf}%
\end{picture}%
\setlength{\unitlength}{4144sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined%
\gdef\SetFigFont#1#2#3#4#5{%
  \reset@font\fontsize{#1}{#2pt}%
  \fontfamily{#3}\fontseries{#4}\fontshape{#5}%
  \selectfont}%
\fi\endgroup%
\begin{picture}(4592,3512)(0,-2942)
\end{picture}%

%\end{slide}
\clearpage
%\begin{slide}
\heading{Arrangement no. 86} 
may be defined by the equation
\[(x-t)(x+t)(y-t)(y+t)(z-t)(z+t)(x+y+z+t)(x+y+z-3t),\]
\begin{minipage}[b]{6cm}
{\small   it consists of the faces of a cube and additional two parallel planes, one through
three vertices of the cube and the second through one. The 4--fold points
are: four vertices, three points at infinity which are the intersection of parallel
edges of the cube, and three points of intersection at infinity of a pair of
parallel faces of the cube and the additional two planes.}
\end{minipage}
%\begin{picture}(0,0)%
\includegraphics[scale=1]{octic_86.pdf}%
%\end{picture}%
\setlength{\unitlength}{4144sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined%
\gdef\SetFigFont#1#2#3#4#5{%
  \reset@font\fontsize{#1}{#2pt}%
  \fontfamily{#3}\fontseries{#4}\fontshape{#5}%
  \selectfont}%
\fi\endgroup%
%\begin{picture}(4592,3512)(0,-2942)
%\end{picture}%

%\end{slide}
\clearpage
%\begin{slide}
\heading{Arrangement no. 86$^{a}$} with the same numerical data as
arrangement 86 may be defined by the equation
\[(x-t)(x+t)(y-t)(y+t)(z-t)(z+t)(x+y+z-t)(x+y+z-3t),\]
\begin{picture}(0,0)%
\includegraphics{octic_86a.pdf}%
\end{picture}%
\setlength{\unitlength}{4144sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined%
\gdef\SetFigFont#1#2#3#4#5{%
  \reset@font\fontsize{#1}{#2pt}%
  \fontfamily{#3}\fontseries{#4}\fontshape{#5}%
  \selectfont}%
\fi\endgroup%
\begin{picture}(4592,3512)(0,-2942)
\end{picture}%

%\end{slide}

%\begin{slide}
\heading{Arrangement no. 87} may be defined by the equation
\[(x-t)(x+t)(y-t)(y+t)(z-t)(z+t)(x+y+z+t)(x+y+z-t),\]
\small it consists of the faces of a cube and additional two parallel planes through
three vertices. The 4--fold points are: six vertices, three points at infinity
which are the intersection of parallel edges of the cube, and three points of
intersection at infinity of a pair of parallel faces of the cube and the
additional two planes.\\
\begin{picture}(0,0)%
\includegraphics{octic_87.pdf}%
\end{picture}%
\setlength{\unitlength}{4144sp}%
%
\begingroup\makeatletter\ifx\SetFigFont\undefined%
\gdef\SetFigFont#1#2#3#4#5{%
  \reset@font\fontsize{#1}{#2pt}%
  \fontfamily{#3}\fontseries{#4}\fontshape{#5}%
  \selectfont}%
\fi\endgroup%
\begin{picture}(4592,3512)(0,-2942)
\end{picture}%
%\end{slide}

\clearpage
\overlay{overlay4}
\color{DarkGreen}
Since the Picard groups are generated by divisors defined over
$\mathbb Q$ we have $t_{2}=ph^{1,1}$, $t_{4}=p^{2}h^{1,1}$ and so
\[a_{p}=1+p^{3}+(p+p^{2})h^{1,1}-\#\bar X_{p}.\]
\pause
So to compute the coefficient $a_{p}$ we have to find number of points
on $X$ modulo $p$.\pause
First we count points on the (\emph{singular}) double covering of
$\mathbb P^{3}$ -- if the equation of the branch locus if
$f(x,y,z,t)=0$, we count points on $\mathbb P^{3} (\mathbb F_{p})$ for
which value of $f$ is zero, square or not-square mod $p$. It is very
easy (for a computer).
\pause

Now we have to take into account resolution of singularities, for
seven types of singularities it is very easy the number of added
points depends only on the number of those types of singularities. 
Unfortunately if four planes intersects, but no three of them contains
a line, blow--up of that point replace one point by a double covering
of $\mathbb P^{2}$ branched along an arrangement of four lines. We
count the points in a similar manner as before. 
\clearpage
\color{black}
\stepwise{
\begin{center}
A C++ program gave the following table.
\bigskip
\begin{tabular}{||r|r|r|r|r|r|r|r|r||}
\hline\hline
p&5&7&11&13&17&19&23&73\\\hline\hline
\multicolumn{5}{||c|}{Arrangements 2, 87}&
\multicolumn{4}{|c||}{\step{\color{magenta}8k4A}}
\\\hline
$a_{p}$&-2&24&-44&22&50&44&-56&154\\\hline
\multicolumn{5}{||c|}{Arrangement 6}&
\multicolumn{4}{|c||}{\step{\color{magenta}32k4C}}\\\hline
$a_{p}$&-10&-16&40&-50&-30&-40&-48&-630
\\\hline
\multicolumn{5}{||c|}{Arrangement 23}&
\multicolumn{4}{|c||}{\step{\color{magenta}64k4A}}\\\hline
$a_{p}$&-22&0&0&18&-94&0&0&1098
\\\hline
\multicolumn{5}{||c|}{Arrangements 29, 44}&
\multicolumn{4}{|c||}{\step{\color{magenta}16k4A}}\\\hline
$a_{p}$&-2&-24&44&22&50&-44&56&154\\\hline
\multicolumn{5}{||c|}{Arrangement 62}&
\multicolumn{4}{|c||}{\step{\color{magenta}64k4C}}\\\hline
$a_{p}$&2&-24&-44&-22&50&44&56&154\\\hline
\multicolumn{5}{||c|}{Arrangements 84, 86}&
\multicolumn{4}{|c||}{\step{\color{magenta}6k4A}}\\\hline
$a_{p}$&6&-16&12&38&-126&20&168&218\\\hline
\multicolumn{5}{||c|}{Arrangement 86$^{a}$}&
\multicolumn{4}{|c||}{\step{\color{magenta}12k4A}}\\\hline
$a_{p}$&-18&8&36&-10&18&-100&72&26\\\hline\hline
\end{tabular}
\end{center}
\vspace*{-1cm}
Comparing above with the database of modular form (by W. Stein) we
can recognize the following modular forms}
\end{document}
\textbf{\centerline{\color{DarkBlue}L series of a Calabi--Yau manifold
  defined over $\mathbb Q$}}


%%% Local Variables: 
%%% mode: latex
%%% TeX-master: t
%%% End: 
