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\title{Modularity Conjecture for Calabi--Yau manifolds}
\institute{Instytut Matematyki Uniwersytetu Jagiello\'nskiego}
\author{S\l awomir Cynk}
%\subtitle{Effective methods in analytic and algebraic geometry}
\date{First Joint Meeting AMS--PTM\\
Warszawa 1 sierpnia 2007\\[2mm]\small joint works with D. van
Straten, Ch. Meyer, K. Hulek}
\begin{document}
\frame{\titlepage}
\transdissolve
%\begin{frame}
%\frametitle{Plan}
%\tableofcontents
%\end{frame}
%\section{Calabi--Yau manifolds}
\begin{frame}
\frametitle{Calabi--Yau manifolds}
\begin{definition}
\emph{\color{dm}Calabi--Yau manifold} is a complex, projective
(k\"ahler)
threefold $X$ satisfying
\begin{itemize}
\item $K_X=\mathcal O_X$
\item $H^1(\mathcal O_X)=0$.
\end{itemize}
\end{definition}
\pause
there are no global holomorphic 1--forms on $X$\\[3mm]
there exists a nowhere vanishing global holomorphic
3--forms $X$\\[1mm]\pause
\textbf{Another definition}:\\ Calabi--Yau manifold is a compact
riemannian
manifold with holonomy group in SU(3) (holonomy group equal
SU(3)).
\end{frame}
\begin{frame}
\frametitle{Numerical invariants of
    Calabi--Yau manifolds}
\begin{itemize}
\item Euler characteristic $e(X)$,
\item Hodge numbers $h^{i,j}=\dim H^j(\Omega^i_X)$ ($0\leq
i,j\leq3$, $i+i\leq6$),
\item Betti numbers $b_{0},\dots,b_{6}$,\
  ($b_{i}=\dim_{\mathbb
C}H^{i}(X,\mathbb C)=\sum\limits_{p+q=i}h^{ij}$).
\end{itemize}
\(\begin{array}{ccccccc}
 &&&\only<1>{h^{00}}\only<2->{1}&&&\\
&&\only<1>{h^{01}}\only<2->{0}&&\only<1>{h^{10}}\only<2->0&&\\
&\only<1>{h^{02}}\only<2->{0}&&h^{11}&&\only<1>{h^{20}}\only<2->{
0}\\
\only<1>{h^{03}}\only<2->{1}&&h^{12}&&\only<-2>{h^{21}}
\only<3->{h^{12}}&&\only<1> { h^ { 30 } } \only<2->{1}\\
&\only<1>{h^{13}}\only<2->{0}&&\only<-2>{h^{22}}
\only<3->{h^{11}}&&\only<1> { h^ { 31 } } \only<2-> { 0}\\
&&\only<1>{h^{23}}\only<2->{0}&&\only<1>{h^{32}}\only<2->{0}&&\\
&&&\only<1>{h^{33}}\only<2->{1}&&&\\
\end{array}
\only<4->{\qquad
\begin{array}{l}\\
e(X)=2(h^{11}-h^{12})\\
h^{11}=\rho(X)\text{\ Picard number}\\
h^{12}\text{\ dimension of deformation space}
\end{array}
}
\)

\end{frame}
\transwipe
\begin{frame}
 \frametitle{L--series of a Calabi--Yau manifold}
Let $X$ be a Calabi--Yau manifold defined over $\mathbb Q$ and
let $p$ be a prime of good reduction, i.e. the reduction $X_p$
of $X$ mod $p$ is smooth. By the Weil Conjecture the Zeta
function of $X_p$ can be written as
\[
\frac{P_{1,p}(t)P_{3,p}(t)P_{5,p}(t)}{P_{0,p}(t)P_{2,p}(t)P_{4,p}
(t)P_{6,p}(t)}
\]
where $P_{i,p}$ is a polynomial of degree $b_i$. \pause
We define the $i$--th cohomological L--series of $X$ as
\[
L(H^i_{\text{\'et}}(\bar X,\mathbb Q_l),s)=(*)\prod_{p\text{ good
prime} } \frac1{P_{i,p}(p^{-s})}
\]
where $(*)$ stands for the Euler factors corresponding to the
primes of bad reduction.\pause The most interesting is the third
L--series
\[L(X,s)=L(H^3_{\text{\'et}}(\bar X,\mathbb Q_l),s).\]
\end{frame}
\begin{frame}
 \frametitle{Frobenius morphism}
The L--series has expansion $L(X,s)=\sum_{k=1}^\infty
\frac{a_k(X)}{k^s}$.\pause

By the proof of Weil Conjecture \[
P_{i,p}(t)=det(1-t\operatorname{Frob}_p^*|H^i_{\text{\'et}}(\bar
X,\mathbb Q_l))\] and so
\[
a_p(X)=\operatorname{tr}(\operatorname{Frob}_p^*|H^i_{\text{\'et}
})\] for $p$ prime, and $a_k(X)$ can be recovered from $a_p(X)$
for $p$ prime factors of $k$.
\pause

Lefschetz fixed point formula
\[
\#X_{p^r}=\sum_{i=0}^6(-1)^i
\operatorname{tr}(\operatorname{Frob}_p^*|H^i_{\text{\'et}
})=1+p^3+\operatorname{tr}(\operatorname{Frob}_p^*|H^2_{\text{
\'et}
})(1+p)-a_p
\]
\end{frame}

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

An \emph{\color{dm}{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{\color{dm}modular form} iff $c_{n}=0$ for
$n<0$. \\ \pause If
moreover $c_{0}=0$, it is a \emph{\color{dm}cusp form}.
\end{frame}
%\section{Modularity Conjecture}
\begin{frame}
 \frametitle{Modularity Conjecture for Calabi--Yau manifolds}
\emph{\color{dm}Every Calabi--Yau manifold is modular in the
sense that its L--series is L--series of some modular form}.

One dimensional Calabi--Yau manifolds are elliptic curves, in
that case the modularity conjecture is Taniyama--Shimura--Weil
Conjecture.

\pause
Two dimensional Calabi--Yau manifolds are K3 surfaces, in this
case middle cohomology has rank 20, but we consider only its
sublattice of transcendental cycles. In the case of maximal
Picard number (singular K3 surfaces) Livn\'e showed that the
L--series is the one of some weight 3 modular form.
\end{frame}
%\subsection{Rigid Calabi--Yau manifolds}
\begin{frame}
Modularity Conjecture has particulary explicit form for
\emph{\color{dm}rigid
Calabi--Yau manifolds}, i.e. Calabi--Yau manifolds with $b_3=2$.

\begin{conj}[Modularity Conjecture for rigid Calabi--Yau
manifolds]
 Let $X$ be a rigid Calabi--Yau manifold defined over $\mathbb
Q$. Then there exists a weight four cusp form $f$ for
$\Gamma_0(N)$ such that $L(f,s)=L(X,s)$, where $N$ is an integer
divisible only by bad primes of $X$.
\end{conj}
{\tiny J.-M.~Fontaine and B.~Mazur
\emph{Geometric Galois
  representations}\/, Elliptic curves, modular forms, \& Fermat's
last
  theorem (Hong Kong, 1993),  41--78, Ser. Number Theory, I,
  Internat. Press, Cambridge, MA, 1995.\\
N.~Yui, \emph{Update on the modularity of
Calabi-Yau varieties}\/,
With an appendix by Helena Verrill. Fields Inst. Commun., 38,
Calabi-Yau varieties and mirror symmetry (Toronto, ON, 2001),
307--362,
Amer. Math. Soc., Providence, RI, 2003.\\
K.~Hulek, R.~Kloosterman, M.~Sch\"utt, \emph{Modularity of
Calabi--Yau varieties}, preprint math.AG/0601238\\
C.~Meyer, \emph{Modular Calabi-Yau threefolds},
Fields Institute Monograph \textbf{22} (2005), AMS.

}

\pause
Modularity Conjecture for rigid Calabi--Yau manifold was proved
by Dieulefait and Manoharmayum
under very mild assumptions on the primes of bad reduction

{\tiny L.~Dieulefait and
J.~Manoharmayum,  \emph{Modularity of rigid
  Calabi-Yau threefolds over $\mathbb Q$}.  Calabi-Yau varieties
and
  mirror symmetry (Toronto, ON, 2001),  159--166, Fields
  Inst. Commun., 38, Amer. Math. Soc., Providence, RI, 2003.

}
\end{frame}

%\subsection{First examples}
\begin{frame}
 \frametitle{First examples of (modular) rigid Calabi--Yau
manifolds}
\textbf{Schoen's quintic:} in $\mathbb P^4$ given by the
equation \[x_0^5+x_1^5+\dots+x_4^5-x_0x_1x_2x_3x_4=0.\]

\pause\textbf{Fiber product of elliptic fibrations (Schoen):}

Let $p_1:S_1\lra \mathbb P^1$ and $p_2:S_2\lra \mathbb P^1$ be
rational elliptic surfaces with sections, the fiber product
$X=S_1\times_{\mathbb P^1} S_2$ is a singular Calabi--Yau
manifold. Singularities of $X$ are located on the product of
singular fibers of $S_1$ and $S_2$. If both fibers are
semistable then the only singularities are nodes.

\pause
The easiest case is of a self fiber product ($S_1=S_2=S$) and
$S$ semistable. Then $X$ has a smooth model $\hat X$ defined over
$\mathbb Q$ which is a calabi--Yau manifold with $h^{12}(\hat
X)$  equals to four minus number of singular fibers.
\end{frame}

\begin{frame}
 \frametitle{Self fiber products of Beauville surfaces}
There are 6 semistable, rational elliptic surfaces with four
singular fibers (Beauville surfaces). They give six modular
Calabi--Yau threefolds
\[ \begin{array}{||c|c|c|c||}
  \hline \Gamma&\text{singular fibers}&\text{weight 4
form}&\text{level}\\\hline
\Gamma(3)&I_3,I_3,I_3,I_3&\eta(3\tau)^8&9\\
\hline
\Gamma_1(4)\cap\Gamma(2)&I_4,I_4,I_2,
I_2&\eta(2\tau)^4\eta(4\tau)^4&8\\
\hline
\Gamma_1(5)&I_5,I_5,I_1,
I_1&\eta(\tau)^4\eta(5\tau)^4&5\\
\hline
\Gamma_1(6)&I_6,I_3,I_2,
I_1&\eta(\tau)^2\eta(2\tau)^2\eta(3\tau)^2\eta(6\tau)^2&6\\
\hline
\Gamma_0(8)\cap\Gamma_1(4)&I_8,I_2,I_1,
I_1&\eta(4\tau)^{16}\eta(2\tau)^{-4}\eta(8\tau)^{-4}&16\\
\hline
\Gamma_0(9)\cap\Gamma_1(3)&I_6,I_3,I_2,
I_1&\eta(3\tau)^8&8\\
\hline \end{array}
\]
where $\eta(\tau)=q^{\frac1{24}}\prod_{n\ge1}(1-q^n)$
\pause
This construction was generalized in several ways (f.i.
M.~Sch\"utt). \pause

There are several proofs of modularity, one
based on the Dieulefait--Manoharmayum theorem another on the
Shimura isomorphism and based on the Faltings--Serre--Livn\'e
method.
\end{frame}
%\subsection{Faltings--Serre--Livn\'e method}
\begin{frame}
\frametitle{The Faltings--Serre--Livn\'e method}
\begin{Theorem}
 Let $S$ be a finite set of primes and let
$\rho_1,\rho_2:\operatorname{Gal}(\bar{\mathbb Q}/\mathbb Q)\lra
\operatorname{Gl}_2(\mathbb Q_2)$ be continuous Galois
representations, unramified outside $S$ and satisfying
\begin{enumerate}
 \item $\operatorname{tr}\rho_1=\operatorname{tr}\rho_2=0\mod 2$
and $\operatorname{det}\rho_1=\operatorname{det}\rho_2\mod 2$
\item There exists a finite set $T$ of primes, disjoint from $S$
for which the image of the set $\{\operatorname{Frob}_t:\;t\in
T\}$ in $\operatorname{Gal}(\mathbb Q_S/Q)$ is non--cubic and
\[
\operatorname{tr}\rho_1(\operatorname{Frob}_t)=\operatorname{tr}
\rho_2(\operatorname{Frob}_t)
\text{ and
}\operatorname{det}\rho_1(\operatorname{Frob}_t)=\operatorname{
det}\rho_2(\operatorname{Frob}_t)
\]
for all $t\in T$.
\end{enumerate}
Then $\rho_1$ and $\rho_2$ have isomorphic
semi--simplifications. In particular
$\operatorname{tr}\rho_1(\operatorname{Frob}_t)=\operatorname{tr}
\rho_2(\operatorname{Frob}_t)$ for all $t\not\in S$.
\end{Theorem}
\end{frame}
\begin{frame}
 \frametitle{Faltings--Serre-Livn\'e method}
\begin{block}{Corollary}
 Let $X$ be a rigid Calabi--Yau manifold defined over $\mathbb
Q$ and let \[f(q)=\sum\nolimits_{m=1}^\infty b_m q^m\]
be a cusp form of weight 4 for $\Gamma_o(N)$.Let $S$ be a set of
primes containning bad primes for $X$ and prime divisors of $N$.
Suppose that \[a_p=b_p=0\mod 2\] for all primes $p\not\in S$ and
\(a_p(X)=b_p\)for all $p\in T$.
Then $L(X,s)=L(f,s)$, in particular
\(a_p(X)=b_p\) for $p\not]in S$.
\end{block}
The set $T $is easy to describe for a given $S$.
\end{frame}
\begin{frame}
 \frametitle{Double octic Calabi--yau manifolds}
Let $D\subset \mathbb P^3$ be a smooth octic surface, $\pi:X\lra
\mathbb P^3$ double covering branched along $D$. Then $X$ is a
Calabi--yau manifold. If $D$ is singular then $X$ is singular as
well (singulrities of $X$  coresponde to those of $D$).
However, $D$ are ``nice'' there can exists a Calabi--yau
resolution of singularities.

I have studied the case of $D$ being an octic arrangement
(locally looks like intersection of plane) with no sixfold point
nor a fourfold curve. Then we have a nice description of
resolution of singularities and formula for Hodge numbers.
Also using the Lefschetz fixed point formula it is easy to
compute the trace of Frobenius hence to prove modularity in
rigid cases.
\end{frame}
\begin{frame}
 \frametitle{Modularity of non--rigid Calabi--Yau manifolds}
There are two explicit formulations for modularity of non--rigid
Calabi--Yau manifolds. One is (due to Livn\'e and Yui)
\[L(X,s)=L(f_2\otimes f_3)\text{ or }L(X,s)=L(f_2^1\otimes
f_2^2\otimes f_2^3).\]Examples
come from a singular K3 surface $S$ with involution $i_S$
and elliptic curve $E$ with involution $i_E$. Then on $E\times S$
we have diagonal involution $i$ and the quotient $(E\times S)/i$
has a Calabi--Yau resolution of singularities.

Similarly, if $E_j$ is an elliptic curve with involution $i_j$
($j=1,2,3$), then $(E_1\times E_2\times E_3)/D_2$
(where $D_2=\{1, (i_1,i_2,1),(i_1,1,i_3),(1,i_2,i_3)\}$) has a
Calabi--Yau resolution of singularities.

The Calabi--Yau manifolds have L--series as above.
\end{frame}
\begin{frame}
 \frametitle{Modularity of non--rigid Calabi--Yau manifolds}
Another formularion for modularity was suggested by Hulek and
Verrill.
Let $S_i,i=1, \dots,r$
($r=h^{21}(X)$) be a birational ruled surface over an elliptic
curve $E_i$. Assume that the map \[H^3(X,\mathbb C)\lra
\bigoplus_{i=1}^rH^3(S_i,\mathbb C)\]is surjective.

Under the same assumptions on bad primes as in Dieulefait and
Manoharmayum we get \[L(X,s)=L(g_4,s)\prod_jL(g_2^j,s-1)\]
where $g_2^i$ is the weiight 2 modular form for elliptic curve
$E_i$ and $g_4$ is some weight 4 modular form.
\end{frame}
\begin{frame}
 \frametitle{Modularity of non--rigid Calabi--Yau manifolds}
If $L(X,s)=L(g_4,s)\prod_jL(g_2^j,s-1)$ then
\[a_p(X)=b_p+p(c_p^1+\dots+c_p^r),\]
where $b_p$ (resp. $c_p^j$) is a Fourier coefficient of $g_4$
(resp. $g_2^j$).
In particular $a_p(X)\cong b_p\mod p$.

\pause C.~Meyer run an enormous computer search for double
octic Calabi--Yau manifolds with $h^{12}=0$ or $1$ satisfying
the above ``numerical evidence'' of modularity. Modularity of
all 11 rigid examples can be proved using the
Faltings--Serre--Livn\'e method, the only delicate point is to
correctly count points added in resolution of singularities.
\end{frame}
\begin{frame}
 \frametitle{Modularity of non--rigid Calabi--Yau manifolds}
We proved modularity of 17 (out of 18) examples with $h^{12}=1$
using the methods\pause
\begin{itemize}
 \item existence of elliptic ruled surface
(Hullek--Verrill)\pause
\item correspondence with a product of a K3 surface and elliptic
curve (Livn\'e--Yui)\pause
\item correspondence with a (modular) rigid Calabi--Yau
manifold\pause
\item existence of an involution giving a splitting of the
fourdimensional representation into twodimensional
subrepresentations\pause
\end{itemize}
In fact all proofs were based on finding a correspondence with a
fiber product of elliptic fibrations, all studied examples were
given by K3 fibrations with large Picard number, a K3 surface
with large Picard number is related to a product of elliptic
curves.
\end{frame}
\begin{frame}
 \frametitle{Corespondences between modular Calabi--Yau
manifolds}
There are many known modular Calabi--Yau manifolds, among them 15
rigid with the unique weight 4 newform of level 8. According to
the tate Conjecture this should be explained by an existence of
correspondences between them.\pause

With C.~Meyer we found explicit examples of correspondences for
most pairs, in particular we join ten of them in a cycle.\pause

We were able to construct only a few examples with some ad hoc
methods, in most cases we proceed by constructing first
correspondences with some fibers products of rational elliptic
fibrations.
\end{frame}
\begin{frame}
 \frametitle{Higher dimensions}
The first known modular example in dimension bigger then 3 was
due to Ahlgren. Let $X$ be the variety given in $\mathbb C^6$ by
\[
w^2=x(x-1)(x-\lambda)y(y-1)(y-\lambda)z(z-1)(z-\lambda)t(t-1)(t-
\ lambda)\]
\begin{Theorem}[Ahlgren]
 \[\#X_p=p^5+2p^3-4p^2-9p-1-b_p,\]
where $b_p$ is the unique weight 6 level 4 cusp form (which is
$\eta(2\tau)^{12}$).
\end{Theorem}
\pause

With K. Hulek we proved that the projective closure (which is a
double cover of $\mathbb P^5$ branched along twelve hyperplanes)
has a Calabi--Yau smooth model $\tilde X$ with $b_5(\tilde X)=2$
and $a_p(\tilde X)=b_p$.
\end{frame}
\begin{frame}
 \frametitle{Higher dimensions}
Let $E$ (resp. $F$) be the elliptic curve given by the
Weierstrass--ype equation
\[y^2=x^3-D \text{ (resp. }y^2=x^3-x\text{)}.\]
Then $E$ has action of $\mathbb Z_3$ and $F$ has an action of
$\mathbb Z_4$.

\pause
The quotients $X^n=E^n/\mathbb Z_3^{n-1}$ and $Y^n=F^n/\mathbb
Z_4^{n-1}$ have  Calabi--Yau resolutions of singularities
$\tilde Y^n$such that for $n$ odd $b_n(\tilde X^n)=b_n(\tilde
Y_n)=2$ and the L--series of $\tilde X^n$ (resp. $\tilde Y^n$)
equals the L--series of the modular form corresponding to the
$n$--th power of the Hecke character corresponding to the curve
$E$ (resp. $F$).
\end{frame}

\end{document}