One Hat Cyber Team

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\author{Anton Mellit}
\title{Research plan for stay at MPI during May, 2007~--- August, 2007.}
\begin{document}
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\maketitle
\section{Higher Green's functions and conjectures on special values}
Let $\HH$ be the upper half plane.
Let $\Gamma$ be a congruence subgroup of $SL_2(\Z)$ and $k>1$. The Green's function on $\HH/\Gamma$ of weight $2k$ is the unique function $G_k^{\HH/\Gamma}$ with the following properties:

\begin{enumerate}
\item $G_k^{\HH/\Gamma}$ is a smooth function on $\HH\times\HH-\{\z_1=\gamma\z_2\ \,|\, \gamma\in\Gamma\}$ with values in $\R$.
\item $G_k^{\HH/\Gamma}(\gamma_1 \z_1, \gamma_2 \z_2) = G_k^{\HH/\Gamma}(\z_1, \z_2)$ for all $\gamma_1, \gamma_2\in \Gamma$.
\item $\Delta_i G_k^{\HH/\Gamma} = k(1-k) G_k^{\HH/\Gamma}$.
\item $G_k^{\HH/\Gamma} = \log|\z_1-\z_2|^2 + O(1)$ when $\z_1$ tends to $\z_2$.
\item $G_k^{\HH/\Gamma}$ tends to $0$ when $\z_1$ tends to a cusp.
\end{enumerate}

We have the following formula:
\[
G_k^{\HH/\Gamma}(z_1, z_2) = \sum_{\gamma\in\Gamma} -2 \calQ_{k-1}\left(1 + 2\frac{(\z_1-\gamma \z_2)(\gamma \zc_2-\zc_1)}{(\z_1-\zc_1)(\gamma\z_2-\gamma\zc_2)}\right).
\]

A particular case of the conjecture formulated in \cite{GKZ} is as follows:
Let $z_1, z_2 \in \HH$ be two different complex multiplication points. Let $Q_1$, $Q_2$ be the corresponding positive definite primitive quadratic forms. Let $D_1$, $D_2$ be their discriminants. Let $k=2,3,4,5,7$. The conjecture is
\begin{conjecture}
There exists an algebraic number $f$, such that the value of the Green's function equals
\[
    \wt{G}_k^{\HH/PSL_2\Z}(z_1, z_2) = (D_1 D_2)^{\frac{1-k}2} \log f.
\]
\end{conjecture}

\section{My result}
In my PhD thesis I prove the conjecture for the case $k=2$, $z_1=\sqrt{-1}$. The proof is composed from the following steps:

\begin{enumerate}
\item Let $g$ be a function on $\HH$ which is invariant under a congruence subgroup $\Gamma$, satisfies equation $\Delta g = k(1-k) g$, has only logarithmic singularities and small growth at infinity. Then $\partial^k g$ is a meromorphic modular form for $\Gamma$ of weight $2k$, where $\partial$ denotes the nonholomorphic derivative. The modular forms $\partial^k g$ and $\partial^k \bar{g}$ determine $g$.
\item Let $\E\To X$ be a family of elliptic curves over $X$, a smooth quasi-projective variety over $\C$. Let $E_t$ denote the fiber over $t\in X$. Let $x_t$ for each $t\in X$ be an element of the group $CH^2(E_t\times E_t,1)$, that is a sequence of curves $W_i\subset E_t\times E_t$, rational functions $f_i\in \C(W_i)$ with $\sum \Div f_i = 0$. Suppose $x_t$ form a "family", which means that $W_i$ is the intersection of a fixed subvariety of $\E\times_X\E$ with $E_t\times E_t$, and $f_i$ is the restriction of some fixed rational function to $W_i$. Then the Abel-Jacobi map gives for each $t\in X$ an element
\[
\epsilon_t\in \frac{\Hom_\C(F^1 H^2(E_t\times E_t,\C), \C)}{H^2(E_t\times E_t,\Z)}.
\]
\item Take the canonical element $\theta_t\in F^1 H^2(E_t \times E_t,\C)$ given by the form
\[
\frac{\omega \otimes \bar\omega + \bar\omega \otimes \omega}{\int_{E_t} \omega\wedge\bar{\omega}},
\]
where $\omega$ is a holomorphic differential form on $E_t$. Evaluating $\epsilon_t$ on $\theta_t$ gives a multi-valued function on $X$. I prove that the operator $\partial^2$ applied to this function gives a meromorphic modular form on $X$. Moreover I obtain a formula for this modular form which allows explicitly compute it for any given family $x_t$
\item For the function $g(z)=G^{\HH/SL_2(\Z)}_2(\sqrt{-1}, z)$ I find a family $x_t$ such that $\partial^2 \langle\epsilon_t, \theta_t\rangle$ and $\partial^2 g$ give the same modular form (up to a multiple of $2\pi \sqrt{-1}$. I conclude that for any $t\in X$ $g(z)=Re \langle\epsilon_t, \theta_t\rangle$, where $z$ correspond to the elliptic curve $E_t$.
\item Let $S$ be a smooth projective variety, $n=\dim S$, $i\in \Z$. Take the Abel-Jacobi map 
\[
AJ^{i,1}:CH^i(S,1)_0\To \frac{\Hom_\C(F^{n-i+1}H^{2n-2i+2}(S,\C), \C)}{H^{2i-2}(S,\Z)}.
\]
I prove that the value of $AJ^{i,1}$ on a cycle defined over a number field evaluated at a cohomology class representable by an algebraic subvariety defined over a number field gives a number of the form $\frac{1}{2\pi\sqrt{-1}} \log f$ with $f\in \bar\Q$. For any $t$ for which $E_t$ has complex multiplication the class $\theta_t$ is a simple multiple of a class representable by an algebraic subvariety defined over a number field. This implies the statement.

The parts (i), (ii), (v) are already written and are contained in the two preprints I attach. Parts (iii) and (iv) are present only in drafts, so their writing is still in progress.
\end{enumerate}
\section{Future development}
It would be interesting to generalize the proof, or at least some parts of it, in the following directions:
\begin{enumerate}
\item The case of arbitrary group $\Gamma$ or arbitrary weight $k$. Then the conjecture can be formulated for certain linear combinations of values of the Green function.
\item Different choices of the complex multiplication point (which was $\sqrt{-1}$).
\end{enumerate}
The generalization of parts (i), (ii) and (v) is straightforward. 

For the part (iii) the problem is to obtain the formula for $\partial^k\langle\epsilon_t, \theta_t\rangle$ for higher weights (product of more that two elliptic curves). This involves studying cycle map to the de Rham cohomology and its generalization for higher Chow groups. The goal is to obtain some concrete formulae for the case of powers of families of elliptic curves. By concrete we mean some rational expressions involving coefficients of equations defining $x_t$.

For the part (iv) we need to construct a different family $x_t$ for each complex multiplication point. We have constructed one for $\sqrt{-1}$. We may try to construct one for $\frac{-1+\sqrt{-3}}2$, $\sqrt{-2}$ or $\frac{-1+\sqrt{-7}}2$ since in this cases the complex multiplication has small degree ($1$ or $2$).
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