|
13
|
|
edited Mar 20 2011 at 23:47 |
The $g=4$, $n=2$ Mumford form is$$\mu_{4,2}=\pm{1\over c\widehat{\omega_i\omega_j}\wedge\cdots\wedge \omega_4\omega_4\over(\omega_1\wedge\cdots\wedge\omega_4)^{13}} \ ,with $c$ a constant. trasforms with affine terms proportional to $F_4$, so that it is a vector-valued modular form only when $F_4=0$, that is in the Jacobian. This motivates the name vector-valued Teichmueller modular forms. Note that also the square root of $\chi_{68}(\tau)$ exists only in ${\cal I}_4$. It follows that $\det S_4$ and $\chi_{68}(\tau)^{1/2}$ are modular forms (of weight $34$) only when restricted to ${\cal I}_4$. The This clearly shows that the vector-valued Teichmueller modular forms$g=4$, [i_{N_n+1},\ldots,i_{M_n}|\tau]$are deeply related to the geometry of the Jacobian and to the Schottky's problem. The structure of the vector-valued Teichmueller modular at any $n=2$ Mumford form g$, generated by Mumford's forms, and their properties, such as the one of generating canonical curves, that is $$\mu_{4,2}=\pm{1\over c\widehat{\omega_i\omega_j}\wedge\cdots\wedge \omega_4\omega_4\over(\omega_1\wedge\cdots\wedge\omega_4)^{13}} \ Petri's relations, $$with $c$ strongly support Mumford's suggestion that Petri's relations arefundamental and should have basic applications: see pg.241, D. Mumford, The Red Book of Varieties and Schemes, Springer Lecture Notes in Math. 1358, 1999. Actually, it seems Mumford was right, one has just to use his forms. A suggestion for the literature: the papers by John Fay are excellent and not very well-known as they should. The one in the Memoirs of the AMS is a constantmasterpiece.
|
|
|
|
|
12
|
|
edited Mar 20 2011 at 13:41 |
$$\hbox{Vector-valued Teichmueller Modular forms}$$
Vector-valued Siegel
modular forms are the natural generalization of the classical elliptic
modular forms as seen by studying the cohomology of the universal abelian variety.
In spite of their relevance they have been studied essentially for genus $g=2$, where
correspond to suitable commutators of Siegel modular forms.
In the case $g=2$ and $g=3$ Ichikawa introduced the concept of Teichmueller modular forms.
It turns out that the Mumford forms for $g>3$ lead to the concept of vector-valued Teichmueller modular forms.
The main steps are the following.
For each fixed positive integers
$g,n$, define $$M_n(g)=M_n:={g+n-1\choose n}\ ,\;
N_n(g)=N_n:=(2n-1)(g-1)\ ,\quad K_n:=M_n-N_n\ , $$
so that, for a curve $C$ of genus $g\ge
2$, $M_n$ and $N_n$ are the dimensions of ${\rm Sym}^n H^0(K_C)$ and
$H^0(K_C^n)$, respectively.
Let ${\frak H}_g:={Z\in M_g({\Bbb C})\mid {}^tZ=Z,\mathop{\rm Im} Z>0}$, be the Siegel
upper half-space. Let
${\alpha_1,\ldots,\alpha_g,\beta_1,\ldots,\beta_g}$ be a
symplectic basis of $H_1(C,{\Bbb Z})$. Denote by
$\omega_1,\ldots,\omega_g$ the basis of $H^0(K_C)$ satisfying the
standard normalization condition
$\oint_{\alpha_i}\omega_j=\delta_{ij}$, and by
$\tau_{ij}:=\oint_{\beta_i}\omega_j$ the Riemann period matrix,
$i,j=1,\ldots,g$.
Denote by ${\cal I}_g$ the closure of the locus of Riemann
period matrices in ${\frak H}_g$ and by ${\cal M}_g$ the moduli space of curves of genus $g$.
Consider the case $g\ge 2$ and a given symplectic basis for $H_1(C,{\Bbb Z})$. For each positive integer $n$, consider the basis $\tilde\omega_1^{(n)},\ldots,\tilde\omega_{M_n}^{(n)}$ of ${\rm Sym}^n H^0(K_C)$ whose elements are symmetrized tensor products of $n$-tuples of vectors of the basis $\omega_1,\ldots,\omega_g$, taken with respect to an arbitrary ordering chosen once and for all. Denote by $\omega_i^{(n)}$, $i=1,\ldots, M_n$, the image of $\tilde\omega_i^{(n)}$ under the natural map $\psi:{\rm Sym}^n H^0(K_C)\to H^0(K_C^n)$. It is well known that such a map is surjective if and only if $g=2$ or $C$ is non-hyperelliptic of genus $g>2$. For $n=2$, $g=2$ and $g=3$ non-hyperelliptic, this map is an isomorphism.
Consider the Thetanullwerte
$\chi_k(Z):=\prod_{\delta\hbox{ even}} \theta[\delta](0,Z)$,
$Z\in{\frak H}_g$, with $k=2^{g-2}(2^g+1)$.
Set
$$F_g:=2^g
\sum_{\delta\hbox{
even}}\theta^{16}[\delta](0,Z)-\bigl(\sum_{\delta\hbox{
even}}\theta^{8}[\delta](0,Z)\bigr)^2 \ .
$$
It turns out that $F_4$, the Schottky-Igusa form, vanishes only on the Jacobian. Furthermore, there is a nice relation
between $F_g$ and the theta series $\Theta_\Lambda$
corresponding to the even unimodular lattices $\Lambda=E_8$ and
$\Lambda=D_{16}^+$:
$$
F_g=2^{-2g}(\Theta_{D_{16}^+}-\Theta_{E_8}^2) \ .
$$
Let ${\phi^n_i}_{1\le i\le N_n}$ be a basis of
$H^0(K_C^n)$, $n\geq2$.
The Mumford form is, up to a universal constant
$$
\mu_{g,n}={\kappa[\omega]^{(2n-1)^2}\over
\kappa[\phi^n]}{\phi^n_1\wedge\cdots\wedge\phi^n_{N_n}\over
(\omega_1\wedge\cdots\wedge\omega_g)^{c_n}} \ ,
$$
where $\kappa[\omega]$ is a constant that depends only on the choice of the homological basis whereas $\kappa[\phi^n]$ also depends on the choice of the basis $\phi^n$ (see Prop.1.2). In the case $n=2$ and $g<4$ one may choose the natural basis
${\rm Sym}^2 H^0(K_C)$ for $H^0(K_C^2)$, and for $g=2$ gets
$${\kappa[\omega]^{9}\over
\kappa[\omega^{(2)}]} ={1\over \pi^{12}\chi_{5}^2(\tau)}\ ,$$
whereas for $g=3$
$${\kappa[\omega]^{9}\over
\kappa[\omega^{(2)}]} ={1\over 2^6\pi^{18}\chi_{18}^{1/2}(\tau)}\ . $$
For $g>3$ one has $g(g+1)/2-(3g-3)>0$, so apparently there is no a natural extension. Nevertheless one may continue to take $3g-3$ elements of ${\rm Sym}^2 H^0(K_C)$, or, more generally $N_n:=(2n-1)(g-1)$ elements of ${\rm Sym}^n H^0(K_C)$. Doing this leads to some surprise.
To simplify notation, denote here by $\omega^{(n)}$ the basis ${\omega^{(n)}_k}$ with
$k=i_1,\ldots,i_{N_n}\in{1,\ldots,M_{n}}$.
$$[i_{N_n+1},\ldots,i_{M_n}|\tau]=\epsilon_{i_1,\ldots,i_{M_n}}
{\kappa[\omega^{(n)}]\over\kappa[\omega]^{(2n-1)^2}}\ ,$$
are vector-valued Teichmuller modular forms without poles on ${\cal M}_g$ and vanishing on the hyperelliptic locus, of weight
$$
d_n:=6n^2-6n+1-{g+n-1\choose n-1} \ .
$$
Note that the vector-valued nature is just a consequence of the inequality $M_n-N_n>0$ for some $g$. For example, for $n=2$ one has $g(g+1)/2-(3g-3)>0$ satisfied for $g>3$. This implies that there are free indeces: the $i_{N_n+1},\ldots,i_{M_n}$,
a nice hint that the theory of vector-valued Teichmueller modular forms is a key tool to investigate the Schottky problem, see below for the case $g=4$ of genus 4 (presumably here should also appear some interesting Number Theoretical structures). For each integer $n\geq 2$ and for all $i_{2},\ldots,i_{K_n}\in{1,\ldots,M_n}$ one has
$$\sum_{i=1}^{M_n}[i,i_{2},\ldots,i_{K_n}|\tau]\omega^{(n)}_{i}(x)
=0\ .$$ In particular, for $n=2$ these are all the quadrics characterizing the canonical curve in projective space. Remarkably, one finds that at $g=4$, $[(ij)|\tau]\equiv [i|\tau]$ (see the paper for the indexing) is proportional to
$$S_{4ij}(Z):={1+\delta_{ij}\over 2}{\partial F_4(Z)\over \partial Z_{ij}} \ .$$ For $g=4$ the discriminant of the quadrics is proportional to the square root of
$\chi_{68}$, the $g=4$ Thetanullwerte
$$ \det S_4(\tau)=d\chi_{68}(\tau)^{1/2}\ , $$
with $d$ a constant.
A key step here is the following lemma.
Let $C$ be either a non-hyperelliptic Riemann surface of genus $g=4$
or a non-trigonal surface of $g=5$. Then, the canonical model of $C$
is contained in a quadric of rank $3$ if and only if
$\prod_{\delta\hbox{ even}}\theta[\delta]=0$.
Note that $\det S_4$ and $\chi_{68}(\tau)^{1/2}$ are modular forms (of weight
$34$) only when restricted to ${\cal I}_4$. The $g=4$, $n=2$ Mumford form is
$$\mu_{4,2}=\pm{1\over c
S_{4ij}}{\omega_1\omega_1\wedge\cdots\wedge
\widehat{\omega_i\omega_j}\wedge\cdots\wedge \omega_4\omega_4\over
(\omega_1\wedge\cdots\wedge\omega_4)^{13}} \ ,
$$
with $c$ a constant.
|
|
|
|
|
11
|
|
edited Mar 16 2011 at 5:41 |
$$\hbox{Vector-valued Teichmueller Modular forms}$$
Vector-valued Siegel
modular forms are the natural generalization of the classical elliptic
modular forms as seen by studying the cohomology of the universal abelian variety.
In spite of their relevance they have been studied essentially for genus $g=2$, where
correspond to suitable commutators of Siegel modular forms.
In the case $g=2$ and $g=3$ Ichikawa introduced the concept of Teichmueller modular forms.
It turns out that the Mumford forms for $g>3$ lead to the concept of vector-valued Teichmueller modular forms.
The main steps are the following.
For each fixed positive integers
$g,n$, define $$M_n(g)=M_n:={g+n-1\choose n}\ ,\;
N_n(g)=N_n:=(2n-1)(g-1)\ ,\quad K_n:=M_n-N_n\ , $$
so that, for a curve $C$ of genus $g\ge
2$, $M_n$ and $N_n$ are the dimensions of ${\rm Sym}^n H^0(K_C)$ and
$H^0(K_C^n)$, respectively.
Let ${\frak H}_g:={Z\in M_g({\Bbb C})\mid {}^tZ=Z,\mathop{\rm Im} Z>0}$, be the Siegel
upper half-space. Let
${\alpha_1,\ldots,\alpha_g,\beta_1,\ldots,\beta_g}$ be a
symplectic basis of $H_1(C,{\Bbb Z})$. Denote by
${\omega_i}_{1\le i\le g}$ \omega_1,\ldots,\omega_g$ the basis of $H^0(K_C)$ satisfying the
standard normalization condition
$\oint_{\alpha_i}\omega_j=\delta_{ij}$, and by
$\tau_{ij}:=\oint_{\beta_i}\omega_j$ the Riemann period matrix,
$i,j=1,\ldots,g$.
Denote by ${\cal I}_g$ the closure of the locus of Riemann
period matrices in ${\frak H}_g$ and by ${\cal M}_g$ the moduli space of curves of genus $g$.
Consider the case $g\ge 2$ and a given symplectic basis for $H_1(C,{\Bbb Z})$. For each positive integer $n$, consider the basis $\tilde\omega_1^{(n)},\ldots,\tilde\omega_{M_n}^{(n)}$ of ${\rm Sym}^n H^0(K_C)$ whose elements are symmetrized tensor products of $n$-tuples of vectors of the basis $\omega_1,\ldots,\omega_g$, taken with respect to an arbitrary ordering chosen once and for all. Denote by $\omega_i^{(n)}$, $i=1,\ldots, M_n$, the image of $\tilde\omega_i^{(n)}$ under the natural map $\psi:{\rm Sym}^n H^0(K_C)\to H^0(K_C^n)$. It is well known that such a map is surjective if and only if $g=2$ or $C$ is non-hyperelliptic of genus $g>2$. For $n=2$, $g=2$ and $g=3$ non-hyperelliptic, this map is an isomorphism.
Consider the Thetanullwerte
$\chi_k(Z):=\prod_{\delta\hbox{ even}} \theta[\delta](0,Z)$,
$Z\in{\frak H}_g$, with $k=2^{g-2}(2^g+1)$.
Set
$$F_g:=2^g
\sum_{\delta\hbox{
even}}\theta^{16}[\delta](0,Z)-\bigl(\sum_{\delta\hbox{
even}}\theta^{8}[\delta](0,Z)\bigr)^2 \ .
$$
It turns out that $F_4$, the Schottky-Igusa form, vanishes only on the Jacobian. Furthermore, there is a nice relation
between $F_g$ and the theta series $\Theta_\Lambda$
corresponding to the even unimodular lattices $\Lambda=E_8$ and
$\Lambda=D_{16}^+$:
$$
F_g=2^{-2g}(\Theta_{D_{16}^+}-\Theta_{E_8}^2) \ .
$$
Let ${\phi^n_i}_{1\le i\le N_n}$ be a basis of
$H^0(K_C^n)$, $n\geq2$.
The Mumford form is, up to a universal constant
$$
\mu_{g,n}={\kappa[\omega]^{(2n-1)^2}\over
\kappa[\phi^n]}{\phi^n_1\wedge\cdots\wedge\phi^n_{N_n}\over
(\omega_1\wedge\cdots\wedge\omega_g)^{c_n}} \ .
,
$$
where $\omega_1,\ldots,\omega_g$ is the standard (normalized) basis of $H^0(K_C)$. $\kappa[\omega]$ is a constant that depends only on the choice of the homological basis whereas $\kappa[\phi^n]$ also depends on the choice of the basis $\phi^n$ (see Prop.1.2). In the case $n=2$ and $g<4$ one may choose the natural basis
${\rm Sym}^2 H^0(K_C)$ for $H^0(K_C^2)$, and for $g=2$ gets
$${\kappa[\omega]^{9}\over
\kappa[\omega^{(2)}]} ={1\over \pi^{12}\chi_{5}^2(\tau)}\ ,$$
whereas for $g=3$
$${\kappa[\omega]^{9}\over
\kappa[\omega^{(2)}]} ={1\over 2^6\pi^{18}\chi_{18}^{1/2}(\tau)}\ . $$
For $g>3$ one has $g(g+1)/2-(3g-3)>0$, so apparently there is no a natural extension. Nevertheless one may continue to take $3g-3$ elements of ${\rm Sym}^2 H^0(K_C)$, or, more generally $N_n:=(2n-1)(g-1)$ elements of ${\rm Sym}^n H^0(K_C)$. Doing this leads to some surprise.
To simplify notation, denote here by $\omega^{(n)}$ the basis ${\omega^{(n)}_k}$ with
$k=i_1,\ldots,i_{N_n}\in{1,\ldots,M_{n}}$.
$$[i_{N_n+1},\ldots,i_{M_n}|\tau]=\epsilon_{i_1,\ldots,i_{M_n}}
{\kappa[\omega^{(n)}]\over\kappa[\omega]^{(2n-1)^2}}\ ,$$
are vector-valued Teichmuller modular forms without poles on ${\cal M}_g$ and vanishing on the hyperelliptic locus, of weight
$$
d_n:=6n^2-6n+1-{g+n-1\choose n-1} \ .
$$
Note that the vector-valued nature is just a consequence of the inequality $M_n-N_n>0$ for some $g$. For example, for $n=2$ one has $g(g+1)/2-(3g-3)>0$ satisfied for $g>3$. This implies that there are free indeces: the $i_{N_n+1},\ldots,i_{M_n}$.
This may be seen as i_{N_n+1},\ldots,i_{M_n}$,
a nice hint that the theory of vector-valued Teichmueller modular forms is a key tool to investigate the Schottky problem, see below for the case $g=4$ (presumably here should also appear some interesting Number Theoretical structures). For each integer $n\geq 2$ and for all $i_{2},\ldots,i_{K_n}\in{1,\ldots,M_n}$ one has
$$\sum_{i=1}^{M_n}[i,i_{2},\ldots,i_{K_n}|\tau]\omega^{(n)}_{i}(x)
=0\ .$$ In particular, for $n=2$ these are all the quadrics characterizing the canonical curve in projective space. Remarkably, one finds that at $g=4$, $[(ij)|\tau]\equiv [i|\tau]$ (see the paper for the indexing) is proportional to
$S_{4ij}(\tau)$, where
$$S_{4ij}(Z):={1+\delta_{ij}\over 2}{\partial F_4(Z)\over \partial Z_{ij}} \ .$$ For $g=4$ the discriminant of the quadrics is proportional to the square root of
$\chi_{68}$, the $g=4$ Thetanullwerte
$$ \det S_4(\tau)=d\chi_{68}(\tau)^{1/2}\ , $$
with $d$ a constant.
A key step here is the following lemma.
Let $C$ be either a non-hyperelliptic Riemann surface of genus $g=4$
or a non-trigonal surface of $g=5$. Then, the canonical model of $C$
is contained in a quadric of rank $3$ if and only if
$\prod_{\delta\hbox{ even}}\theta[\delta]=0$.
Note that $\det S_4$ and $\chi_{68}(\tau)^{1/2}$ are modular forms (of weight
$34$) only when restricted to ${\cal I}_4$. The $g=4$, $n=2$ Mumford form is
$$\mu_{4,2}=\pm{1\over c
S_{4ij}}{\omega_1\omega_1\wedge\cdots\wedge
\widehat{\omega_i\omega_j}\wedge\cdots\wedge \omega_4\omega_4\over
(\omega_1\wedge\cdots\wedge\omega_4)^{13}} \ ,
$$
with $c$ a constant.
|
|
|
|
10
|
|
edited Feb 26 2011 at 15:22 |
$$\hbox{Vector-valued Teichmueller Modular forms}$$
Vector-valued Siegel
modular forms are the natural generalization of the classical elliptic
modular forms as seen by studying the cohomology of the universal abelian variety.
In spite of their relevance they have been studied essentially for genus $g=2$, where
correspond to suitable commutators of Siegel modular forms.
In the case $g=2$ and $g=3$ Ichikawa introduced the concept of Teichmueller modular forms.
It turns out that the Mumford forms for $g>3$ lead to the concept of vector-valued Teichmueller modular forms.
The main steps are the following.
For each fixed positive integers
$g,n$, define $$M_n(g)=M_n:={g+n-1\choose n}\ ,\;
N_n(g)=N_n:=(2n-1)(g-1)\ ,\quad K_n:=M_n-N_n\ , $$
so that, for a curve $C$ of genus $g\ge
2$, $M_n$ and $N_n$ are the dimensions of ${\rm Sym}^n H^0(K_C)$ and
$H^0(K_C^n)$, respectively.
Let ${\frak H}_g:={Z\in M_g({\Bbb C})\mid {}^tZ=Z,\mathop{\rm Im} Z>0}$, be the Siegel
upper half-space.
Let
${\alpha_1,\ldots,\alpha_g,\beta_1,\ldots,\beta_g}$ be a
symplectic basis of $H_1(C,{\Bbb Z})$. Denote by
${\omega_i}_{1\le i\le g}$ the basis of $H^0(K_C)$ satisfying the
standard normalization condition
$\oint_{\alpha_i}\omega_j=\delta_{ij}$, and by
$\tau_{ij}:=\oint_{\beta_i}\omega_j$ the Riemann period matrix,
$i,j=1,\ldots,g$.
Denote by ${\cal I}_g$ the closure of the locus of Riemann
period matrices in ${\frak H}_g$ and by ${\cal M}_g$ the moduli space of curves of genus $g$.
Consider the case $g\ge 2$ and a given symplectic basis for $H_1(C,{\Bbb Z})$. For each positive integer $n$, consider the basis $\tilde\omega_1^{(n)},\ldots,\tilde\omega_{M_n}^{(n)}$ of ${\rm Sym}^n H^0(K_C)$ whose elements are symmetrized tensor products of $n$-tuples of vectors of the basis $\omega_1,\ldots,\omega_g$, taken with respect to an arbitrary ordering chosen once and for all. Denote by $\omega_i^{(n)}$, $i=1,\ldots, M_n$, the image of $\tilde\omega_i^{(n)}$ under the natural map $\psi:{\rm Sym}^n H^0(K_C)\to H^0(K_C^n)$. It is well known that such a map is surjective if and only if $g=2$ or $C$ is non-hyperelliptic of genus $g>2$. For $n=2$, $g=2$ and $g=3$ non-hyperelliptic, this map is an isomorphism.
Consider the Thetanullwerte
$\chi_k(Z):=\prod_{\delta\hbox{ even}} \theta[\delta](0,Z)$,
$Z\in{\frak H}_g$, with $k=2^{g-2}(2^g+1)$.
Set
$$F_g:=2^g
\sum_{\delta\hbox{
even}}\theta^{16}[\delta](0,Z)-\bigl(\sum_{\delta\hbox{
even}}\theta^{8}[\delta](0,Z)\bigr)^2 \ .
$$
It turns out that $F_4$, the Schottky-Igusa form, vanishes only on the Jacobian. Furthermore, there is a nice relation
between $F_g$ and the theta series $\Theta_\Lambda$
corresponding to the even unimodular lattices $\Lambda=E_8$ and
$\Lambda=D_{16}^+$:
$$
F_g=2^{-2g}(\Theta_{D_{16}^+}-\Theta_{E_8}^2) \ .
$$
Let ${\phi^n_i}_{1\le i\le N_n}$ be a basis of
$H^0(K_C^n)$, $n\geq2$.
The Mumford form is, up to a universal constant
$$
\mu_{g,n}={\kappa[\omega]^{(2n-1)^2}\over
\kappa[\phi^n]}{\phi^n_1\wedge\cdots\wedge\phi^n_{N_n}\over
(\omega_1\wedge\cdots\wedge\omega_g)^{c_n}} \ .
$$
where $\omega_1,\ldots,\omega_g$ is the standard (normalized) basis of $H^0(K_C)$. $\kappa[\omega]$ is a constant that depends only on the choice of the homological basis whereas $\kappa[\phi^n]$ also depends on the choice of the basis $\phi^n$ (see Prop.1.2). In the case $n=2$ and $g<4$ one may choose the natural basis
${\rm Sym}^2 H^0(K_C)$ for $H^0(K_C^2)$, and for $g=2$ gets
$${\kappa[\omega]^{9}\over
\kappa[\omega^{(2)}]} ={1\over \pi^{12}\chi_{5}^2(\tau)}\ ,$$
whereas for $g=3$
$${\kappa[\omega]^{9}\over
\kappa[\omega^{(2)}]} ={1\over 2^6\pi^{18}\chi_{18}^{1/2}(\tau)}\ . $$
For $g>3$ one has $g(g+1)/2-(3g-3)>0$, so apparently there is no a natural extension. Nevertheless one may continue to take $3g-3$ elements of ${\rm Sym}^2 H^0(K_C)$, or, more generally $N_n:=(2n-1)(g-1)$ elements of ${\rm Sym}^n H^0(K_C)$. Doing this leads to some surprise.
To simplify notation, denote here by $\omega^{(n)}$ the basis ${\omega^{(n)}_k}$ with
$k=i_1,\ldots,i_{N_n}\in{1,\ldots,M_{n}}$.
$$[i_{N_n+1},\ldots,i_{M_n}|\tau]=\epsilon_{i_1,\ldots,i_{M_n}}
{\kappa[\omega^{(n)}]\over\kappa[\omega]^{(2n-1)^2}}\ ,$$
are vector-valued Teichmuller modular forms without poles on ${\cal M}_g$ and vanishing on the hyperelliptic locus, of weight
$$
d_n:=6n^2-6n+1-{g+n-1\choose n-1} \ .
$$
Note that the vector-valued nature is just a consequence of the inequality $M_n-N_n>0$ for some $g$. For example, for $n=2$ one has $g(g+1)/2-(3g-3)>0$ satisfied for $g>3$. This implies that there are free indeces: the $i_{N_n+1},\ldots,i_{M_n}$.
This may be seen as a nice hint that the theory of vector-valued Teichmueller modular forms is a key tool to investigate the Schottky problem, see below for the case $g=4$ (presumably here should also appear some interesting Number Theoretical structures). For each integer $n\geq 2$ and for all $i_{2},\ldots,i_{K_n}\in{1,\ldots,M_n}$ one has
$$\sum_{i=1}^{M_n}[i,i_{2},\ldots,i_{K_n}|\tau]\omega^{(n)}_{i}(x)
=0\ .$$ In particular, for $n=2$ these are all the quadrics characterizing the canonical curve in projective space. Such vector-valued forms seem to be a key tool to characterize the Jacobian. Such a problem has been explicitly solved only for $g=4$: there is a weight 8 Siegel modular forms vanishing only on the Jacobian, this is the Schottky-Igusa form $F_4$. Remarkably, one finds that at $g=4$, $[(ij)|\tau]\equiv [i|\tau]$ (see the paper for the indexing) is proportional to $S_{4ij}(\tau)$, where
$$S_{4ij}(Z):={1+\delta_{ij}\over 2}{\partial F_4(Z)\over \partial Z_{ij}} \ .$$ For $g=4$ the discriminant of the quadrics is proportional to the square root of
$\chi_{68}$, the $g=4$ Thetanullwerte
$$ \det S_4(\tau)=d\chi_{68}(\tau)^{1/2}\ , $$
with $d$ a constant.
A key step here is the following lemma.
Let $C$ be either a non-hyperelliptic Riemann surface of genus $g=4$
or a non-trigonal surface of $g=5$. Then, the canonical model of $C$
is contained in a quadric of rank $3$ if and only if
$\prod_{\delta\hbox{ even}}\theta[\delta]=0$.
Note that $\det S_4$ and $\chi_{68}(\tau)^{1/2}$ are modular forms (of weight
$34$) only when restricted to ${\cal I}_4$. The $g=4$, $n=2$ Mumford form is
$$\mu_{4,2}=\pm{1\over c
S_{4ij}}{\omega_1\omega_1\wedge\cdots\wedge
\widehat{\omega_i\omega_j}\wedge\cdots\wedge \omega_4\omega_4\over
(\omega_1\wedge\cdots\wedge\omega_4)^{13}} \ ,
$$
with $c$ a constant.
|
|
|
|
|
9
|
|
edited Feb 26 2011 at 14:32 |
N_n(g)=N_n:=(2n-1)(g-1)\ ,\quad K_n:=M_n-N_n\ . , $ $so that, for a curve $C$ of genus $g\ge2$, $M_n$ and $N_n$ are the dimensions of ${\rm Sym}^n H^0(K_C)$ and$ H^0(K_C^n)$, respectively.$ C$ be a Riemann surface of genus $g$ and ${\alpha_1,\ldots,\alpha_g,\beta_1,\ldots,\beta_g}$ be aperiod matrices in ${\frak H}_g$ .Denote and by ${\cal M}_g$ the moduli space of Riemann surfaces.curves of genus $g$.Consider the case $g\ge 2$ and a given symplectic basis for $H_1(C,{\Bbb Z})$. For each positive integer $n$, consider the basis $\tilde\omega_1^{(n)},\ldots,\tilde\omega_{M_n}^{(n)}$ of ${\rm Sym}^n H^0(K_C)$ whose elements are symmetrized tensor products of $n$-tuples of vectors of the basis $\omega_1,\ldots,\omega_g$, taken with respect to an arbitrary ordering chosen once and for all. Denote by $\omega_i^{(n)}$, $i=1,\ldots, M_n$, the image of $\tilde\omega_i^{(n)}$ under the natural map $\psi:{\rm Sym}^n H^0(K_C)\to H^0(K_C^n)$. It is well known that such a map is surjective if and only if $g=2$ or $C$ is non-hyperelliptic of genus $g>2$. For $n=2$, $g=2$ and $g=3$ non-hyperelliptic, this map is an isomorphism.
|
|
|
|
|
8
|
|
edited Feb 26 2011 at 12:30 |
$$\hbox{Vector-valued Teichmueller Modular forms}$$
Vector-valued Siegel
modular forms are the natural generalization of the classical elliptic
modular forms as seen by studying the cohomology of the universal abelian variety.
In spite of their relevance they have been studied essentially for genus $g=2$, where
correspond to suitable commutators of Siegel modular forms.
In the case $g=2$ and $g=3$ Ichikawa introduced the concept of Teichmueller modular forms.
It turns out that the Mumford forms for $g>3$ lead to the concept of vector-valued Teichmueller modular forms.
The main steps are the following.
For each fixed positive integers
$g,n$, define $$M_n(g)=M_n:={g+n-1\choose n}\ ,\;
N_n(g)=N_n:=(2n-1)(g-1)\ ,\quad K_n:=M_n-N_n\ . $$
Let ${\frak H}_g:={Z\in M_g({\Bbb C})\mid {}^tZ=Z,\mathop{\rm Im} Z>0}$, be the Siegel
upper half-space.
Let $C$ be a Riemann surface of genus $g$ and ${\alpha_1,\ldots,\alpha_g,\beta_1,\ldots,\beta_g}$ be a
symplectic basis of $H_1(C,{\Bbb Z})$. Denote by
$\tau_{ij}$ {\omega_i}_{1\le i\le g}$ the basis of $H^0(K_C)$ satisfying the
standard normalization condition
$\oint_{\alpha_i}\omega_j=\delta_{ij}$, and by
$\tau_{ij}:=\oint_{\beta_i}\omega_j$ the Riemann period matrixand ,
$i,j=1,\ldots,g$.
Denote by ${\cal I}_g$ the closure of the locus of Riemann
period matrices in ${\frak H}_g$.
Denote by ${\cal M}_g$ the moduli space of Riemann surfaces.
Consider the case $g\ge 2$ and a given symplectic basis for $H_1(C,{\Bbb Z})$. For each positive integer $n$, consider the basis $\tilde\omega_1^{(n)},\ldots,\tilde\omega_{M_n}^{(n)}$ of ${\rm Sym}^n H^0(K_C)$ whose elements are symmetrized tensor products of $n$-tuples of vectors of the basis $\omega_1,\ldots,\omega_g$, taken with respect to an arbitrary ordering chosen once and for all. Denote by $\omega_i^{(n)}$, $i=1,\ldots, M_n$, the image of $\tilde\omega_i^{(n)}$ under the natural map $\psi:{\rm Sym}^n H^0(K_C)\to H^0(K_C^n)$. It is well known that such a map is surjective if and only if $g=2$ or $C$ is non-hyperelliptic of genus $g>2$. For $g=2$ and $g=3$ non-hyperelliptic, this map is an isomorphism.
Consider the Thetanullwerte
$\chi_k(Z):=\prod_{\delta\hbox{ even}} \theta[\delta](0,Z)$,
$Z\in{\frak H}_g$, with $k=2^{g-2}(2^g+1)$.
Set
$$F_g:=2^g
\sum_{\delta\hbox{
even}}\theta^{16}[\delta](0,Z)-\bigl(\sum_{\delta\hbox{
even}}\theta^{8}[\delta](0,Z)\bigr)^2 \ .
$$
It turns out that $F_4$, the Schottky-Igusa form, vanishes only on the Jacobian. Furthermore, there is a nice relation
between $F_g$ and the theta series $\Theta_\Lambda$
corresponding to the even unimodular lattices $\Lambda=E_8$ and
$\Lambda=D_{16}^+$:
$$
F_g=2^{-2g}(\Theta_{D_{16}^+}-\Theta_{E_8}^2) \ .
$$
Let ${\phi^n_i}_{1\le i\le N_n}$ be a basis of
$H^0(K_C^n)$, $n\geq2$.
The Mumford form is, up to a universal constant
$$
\mu_{g,n}={\kappa[\omega]^{(2n-1)^2}\over
\kappa[\phi^n]}{\phi^n_1\wedge\cdots\wedge\phi^n_{N_n}\over
(\omega_1\wedge\cdots\wedge\omega_g)^{c_n}} \ .
$$
where $\omega_1,\ldots,\omega_g$ is the standard (normalized) basis of $H^0(K_C)$. $\kappa[\omega]$ is a constant that depends only on the choice of the homological basis whereas $\kappa[\phi^n]$ also depends on the choice of the basis $\phi^n$ (see Prop.1.2). In the case $n=2$ and $g<4$ one may choose the natural basis
${\rm Sym}^2 H^0(K_C)$ for $H^0(K_C^2)$, and for $g=2$ gets
$${\kappa[\omega]^{9}\over
\kappa[\omega^{(2)}]} ={1\over \pi^{12}\chi_{5}^2(\tau)}\ ,$$
whereas for $g=3$
$${\kappa[\omega]^{9}\over
\kappa[\omega^{(2)}]} ={1\over 2^6\pi^{18}\chi_{18}^{1/2}(\tau)}\ . $$
For $g>3$ one has $g(g+1)/2-(3g-3)>0$, so apparently there is no a natural extension. Nevertheless one may continue to take $3g-3$ elements of ${\rm Sym}^2 H^0(K_C)$, or, more generally $N_n:=(2n-1)(g-1)$ elements of ${\rm Sym}^n H^0(K_C)$. Doing this leads to some surprise.
To simplify notation, denote here by $\omega^{(n)}$ the basis ${\omega^{(n)}_k}$ with
$k=i_1,\ldots,i_{N_n}\in{1,\ldots,M_{n}}$.
$$[i_{N_n+1},\ldots,i_{M_n}|\tau]=\epsilon_{i_1,\ldots,i_{M_n}}
{\kappa[\omega^{(n)}]\over\kappa[\omega]^{(2n-1)^2}}\ ,$$
are vector-valued Teichmuller modular forms without poles on ${\cal M}_g$ and vanishing on the hyperelliptic locus, of weight
$$
d_n:=6n^2-6n+1-{g+n-1\choose n-1} \ .
$$
Note that the vector-valued nature is just a consequence of the inequality $M_n-N_n>0$ for some $g$. For example, for $n=2$ one has $g(g+1)/2-(3g-3)>0$ satisfied for $g>3$. This implies that there are free indeces: the $i_{N_n+1},\ldots,i_{M_n}$.
This may be seen as a nice hint that the theory of vector-valued Teichmueller modular forms is a key tool to investigate the Schottky problem, see below for the case $g=4$ (presumably here should also appear some interesting Number Theoretical structures). For each integer $n\geq 2$ and for all $i_{2},\ldots,i_{K_n}\in{1,\ldots,M_n}$ one has
$$\sum_{i=1}^{M_n}[i,i_{2},\ldots,i_{K_n}|\tau]\omega^{(n)}_{i}(x)
=0\ .$$ In particular, for $n=2$ these are all the quadrics characterizing the canonical curve in projective space. Such vector-valued forms seem to be a key tool to characterize the Jacobian. Such a problem has been explicitly solved only for $g=4$: there is a weight 8 Siegel modular forms vanishing only on the Jacobian, this is the Schottky-Igusa form $F_4$. Remarkably, one finds that at $g=4$, $[(ij)|\tau]\equiv [i|\tau]$ (see the paper for the indexing) is proportional to $S_{4ij}(\tau)$, where
$$S_{4ij}(Z):={1+\delta_{ij}\over 2}{\partial F_4(Z)\over \partial Z_{ij}} \ .$$ For $g=4$ the discriminant of the quadrics is proportional to the square root of
$\chi_{68}$, the $g=4$ Thetanullwerte
$$ \det S_4(\tau)=d\chi_{68}(\tau)^{1/2}\ , $$
with $d$ a constant.
A key step here is the following lemma.
Let $C$ be either a non-hyperelliptic Riemann surface of genus $g=4$
or a non-trigonal surface of $g=5$. Then, the canonical model of $C$
is contained in a quadric of rank $3$ if and only if
$\prod_{\delta\hbox{ even}}\theta[\delta]=0$.
Note that $\det S_4$ and $\chi_{68}(\tau)^{1/2}$ are modular forms (of weight
$34$) only when restricted to ${\cal I}_4$. The $g=4$, $n=2$ Mumford form is
$$\mu_{4,2}=\pm{1\over c
S_{4ij}}{\omega_1\omega_1\wedge\cdots\wedge
\widehat{\omega_i\omega_j}\wedge\cdots\wedge \omega_4\omega_4\over
(\omega_1\wedge\cdots\wedge\omega_4)^{13}} \ ,
$$
with $c$ a constant.
|
|
|
|
|
7
|
|
edited Feb 26 2011 at 11:36 |
$$\hbox{Vector-valued Teichmueller Modular forms}$$
Vector-valued Siegel
modular forms are the natural generalization of the classical elliptic
modular forms as seen by studying the cohomology of the universal abelian variety.
In spite of their relevance they have been studied essentially for genus $g=2$, where
correspond to suitable commutators of Siegel modular forms.
In the case $g=2$ and $g=3$ Ichikawa introduced the concept of Teichmueller modular forms.
It turns out that the Mumford forms for $g>3$ lead to the concept of vector-valued Teichmueller modular forms.
The main steps are the following.
For each fixed positive integers
$g,n$, define $$M_n(g)=M_n:={g+n-1\choose n}\ ,\;
N_n(g)=N_n:=(2n-1)(g-1)\ ,\quad K_n:=M_n-N_n\ . $$
Let ${\frak H}_g:={Z\in M_g({\Bbb C})\mid {}^tZ=Z,\mathop{\rm Im} Z>0}$, be the Siegel
upper half-space. Denote by $\tau_{ij}$ the Riemann period matrix and by ${\cal I}_g$ the closure of the locus of Riemann
period matrices in ${\frak H}_g$.
Denote by ${\cal M}_g$ the moduli space of Riemann surfaces.
Consider the case $g\ge 2$ and a given symplectic basis for $H_1(C,{\Bbb Z})$. For each positive integer $n$, consider the basis $\tilde\omega_1^{(n)},\ldots,\tilde\omega_{M_n}^{(n)}$ of ${\rm Sym}^n H^0(K_C)$ whose elements are symmetrized tensor products of $n$-tuples of vectors of the basis $\omega_1,\ldots,\omega_g$, taken with respect to an arbitrary ordering chosen once and for all. Denote by $\omega_i^{(n)}$, $i=1,\ldots, M_n$, the image of $\tilde\omega_i^{(n)}$ under the natural map $\psi:{\rm Sym}^n H^0(K_C)\to H^0(K_C^n)$. It is well known that such a map is surjective if and only if $g=2$ or $C$ is non-hyperelliptic of genus $g>2$. For $g=2$ and $g=3$ non-hyperelliptic, this map is an isomorphism.
Consider the Thetanullwerte
$\chi_k(Z):=\prod_{\delta\hbox{ even}} \theta[\delta](0,Z)$,
$Z\in{\frak H}_g$, with $k=2^{g-2}(2^g+1)$.
Set
$$F_g:=2^g
\sum_{\delta\hbox{
even}}\theta^{16}[\delta](0,Z)-\bigl(\sum_{\delta\hbox{
even}}\theta^{8}[\delta](0,Z)\bigr)^2 \ .
$$
It turns out that $F_4$, the Schottky-Igusa form, vanishes only on the Jacobian. Furthermore, there is a nice relation
between $F_g$ and the theta series $\Theta_\Lambda$
corresponding to the even unimodular lattices $\Lambda=E_8$ and
$\Lambda=D_{16}^+$:
$$
F_g=2^{-2g}(\Theta_{D_{16}^+}-\Theta_{E_8}^2) \ .
$$
Let ${\phi^n_i}_{1\le i\le N_n}$ be a basis of
$H^0(K_C^n)$, $n\geq2$.
The Mumford form is, up to a universal constant
$$
\mu_{g,n}={\kappa[\omega]^{(2n-1)^2}\over
\kappa[\phi^n]}{\phi^n_1\wedge\cdots\wedge\phi^n_{N_n}\over
(\omega_1\wedge\cdots\wedge\omega_g)^{c_n}} \ .
$$
where $\omega_1,\ldots,\omega_g$ is the standard (normalized) basis of $H^0(K_C)$. $\kappa[\omega]$ is a constant that depends only on the choice of the homological basis whereas $\kappa[\phi^n]$ also depends on the choice of the basis $\phi^n$ (see Prop.1.2). In the case $n=2$ and $g<4$ one may choose the natural basis
${\rm Sym}^2 H^0(K_C)$ for $H^0(K_C^2)$, and for $g=2$ gets
$${\kappa[\omega]^{9}\over
\kappa[\omega^{(2)}]} ={1\over \pi^{12}\chi_{5}^2(\tau)}\ ,$$
whereas for $g=3$
$${\kappa[\omega]^{9}\over
\kappa[\omega^{(2)}]} ={1\over 2^6\pi^{18}\chi_{18}^{1/2}(\tau)}\ . $$
For $g>3$ one has $g(g+1)/2-(3g-3)>0$, so apparently there is no a natural extension. Nevertheless one may continue to take $3g-3$ elements of ${\rm Sym}^2 H^0(K_C)$, or, more generally $N_n:=(2n-1)(g-1)$ elements of ${\rm Sym}^n H^0(K_C)$. Doing this leads to some surprise
To simplify notation, denote here by $\omega^{(n)}$ the basis ${\omega^{(n)}_k}$ with
$k=i_1,\ldots,i_{N_n}\in{1,\ldots,M_{n}}$.
$$[i_{N_n+1},\ldots,i_{M_n}|\tau]=\epsilon_{i_1,\ldots,i_{M_n}}
{\kappa[\omega^{(n)}]\over\kappa[\omega]^{(2n-1)^2}}\ ,$$
are vector-valued Teichmuller modular forms without poles on ${\cal M}_g$ and vanishing on the hyperelliptic locus, of weight
$$
d_n:=6n^2-6n+1-{g+n-1\choose n-1} \ .
$$
Note that the vector-valued nature is just a consequence of the inequality $M_n-N_n>0$ for some $g$. For example, for $n=2$ one has $g(g+1)/2-(3g-3)>0$ satisfied for $g>3$. This implies that there are free indeces: the $i_{N_n+1},\ldots,i_{M_n}$.
This may be seen as a nice hint that the theory of vector-valued Teichmueller modular forms is a key tool to investigate the Schottky problem, see below for the case $g=4$ (presumably here should also appear some interesting Number Theoretical structures). For each integer $n\geq 2$ and for all $i_{2},\ldots,i_{K_n}\in{1,\ldots,M_n}$ one has
$$\sum_{i=1}^{M_n}[i,i_{2},\ldots,i_{K_n}|\tau]\omega^{(n)}_{i}(x)
=0\ .$$ In particular, for $n=2$ these are all the quadrics characterizing the canonical curve in projective space. Such vector-valued forms seem to be a key tool to characterize the Jacobian. Such a problem has been explicitly solved only for $g=4$: there is a weight 8 Siegel modular forms vanishing only on the Jacobian, this is the Schottky-Igusa form $F_4$. Remarkably, one finds that at $g=4$, $[(ij)|\tau]\equiv [i|\tau]$ (see the paper for the indexing) is proportional to $S_{4ij}(\tau)$, where
$$S_{4ij}(Z):={1+\delta_{ij}\over 2}{\partial F_4(Z)\over \partial Z_{ij}} \ .$$ For $g=4$ the discriminant of the quadrics is proportional to the square root of
$\chi_{68}$, the $g=4$ Thetanullwerte
$$ \det S_4(\tau)=d\chi_{68}(\tau)^{1/2}\ , $$
with $d$ a constant.
A key step here is the following lemma.
Let $C$ be either a non-hyperelliptic Riemann surface of genus $g=4$
or a non-trigonal surface of $g=5$. Then, the canonical model of $C$
is contained in a quadric of rank $3$ if and only if
$\prod_{\delta\hbox{ even}}\theta[\delta]=0$.
Note that $\det S_4$ and $\chi_{68}(\tau)^{1/2}$ are modular forms (of weight
$34$) only when restricted to ${\cal I}_4$. The $g=4$, $n=2$ Mumford form is
$$\mu_{4,2}=\pm{1\over c
S_{4ij}}{\omega_1\omega_1\wedge\cdots\wedge
\widehat{\omega_i\omega_j}\wedge\cdots\wedge \omega_4\omega_4\over
(\omega_1\wedge\cdots\wedge\omega_4)^{13}} \ .
,
$$
with $c$ a constant.
|
|
|
|
|
6
|
|
edited Feb 26 2011 at 11:28 |
The extension to $g>3$ $\hbox{Vector-valued Teichmueller Modular forms}$$ Vector-valued Siegelmodular forms are the natural generalization of the classical ellipticmodular forms as seen by studying the cohomology of the universal abelian variety.In spite of their relevance they have been studied essentially for genus $g=2$, wherecorrespond to suitable commutators of Siegel modular forms. In the case $g=2$ and $g=3$ Ichikawa work leads introduced the concept of Teichmueller modular forms.It turns out that the Mumford forms for $g>3$ lead to the concept of vector-valued Teichmueller modular forms. Denote by ${\cal M}_g$ the moduli space of Riemann surfaces.(\omega_1\wedge\cdots\wedge\omega_g)^{c_n}} \ .where $\omega_1,\ldots,\omega_g$ is the standard (normalized) basis of $H^0(K_C)$. $\kappa[\omega]$ is a constant that depends only on the choice of the homological basis whereas $\kappa[\phi^n]$ also depends on the choice of the basis $\phi^n$ (see Prop.1.2). In the case $n=2$ and $g<4$ one may choose a the natural basis for $H^0(K_C^2)$: ${\rm Sym}^2 H^0(K_C)$ , for $H^0(K_C^2)$, and for $g=2$ gets ${\kappa[\phi^n]\over\kappa[\omega^{(n)}]^{(2n-1)^2}}$, that essentially correspond to what we denoted To simplify notation, denote here by $$[i_{N_n+1},\ldots,i_{M_n}|\tau] \ , \omega^{(n)}$ the basis ${\omega^{(n)}_k}$ with{\kappa[\omega^{(n)}]\over\kappa[\omega]^{(2n-1)^2}}\ ,$$are vector-valued Teichmueller Teichmuller modular forms without poles on ${\cal M}_g$ and vanishing on the hyperelliptic locus, of weight$$d_n:=6n^2-6n+1-{g+n-1\choose $$$$$Note that the vector-valued nature is just a consequence of the inequality $M_n-N_n>0$ for some $g$. For example, for $n=2$ one has $g(g+1)/2-(3g-3)>0$ satisfied for $g>3$. This implies that there are free indeces: the $i_{N_n+1},\ldots,i_{M_n}$.This may be seen as a nice hint that the theory of vector-valued Teichmueller modular forms is a key tool to investigate the Schottky problem, see below for the case $g=4$ (presumably here should also appear some interesting Number Theoretical structures).
|
|
|
|
|
5
|
|
edited Feb 26 2011 at 0:38 |
The extension to $g>3$ of Ichikawa work leads to vector-valued modular forms. The main steps are the following.
For each fixed positive integers
$g,n$, define $$M_n(g)=M_n:={g+n-1\choose n}\ ,\;
N_n(g)=N_n:=(2n-1)(g-1)\ ,\quad K_n:=M_n-N_n\ . $$
Let ${\frak H}_g:={Z\in M_g({\Bbb C})\mid {}^tZ=Z,\mathop{\rm Im} Z>0}$, be the Siegel
upper half-space. Denote by $\tau_{ij}$ the Riemann period matrix and by ${\cal I}_g$ the closure of the locus of Riemann
period matrices in ${\frak H}_g$.
Consider the case $g\ge 2$ and a given symplectic basis for $H_1(C,{\Bbb Z})$. For each positive integer $n$, consider the basis $\tilde\omega_1^{(n)},\ldots,\tilde\omega_{M_n}^{(n)}$ of ${\rm Sym}^n H^0(K_C)$ whose elements are symmetrized tensor products of $n$-tuples of vectors of the basis $\omega_1,\ldots,\omega_g$, taken with respect to an arbitrary ordering chosen once and for all. Denote by $\omega_i^{(n)}$, $i=1,\ldots, M_n$, the image of $\tilde\omega_i^{(n)}$ under the natural map $\psi:{\rm Sym}^n H^0(K_C)\to H^0(K_C^n)$. It is well known that such a map is surjective if and only if $g=2$ or $C$ is non-hyperelliptic of genus $g>2$. For $g=2$ and $g=3$ non-hyperelliptic, this map is an isomorphism.
Consider the Thetanullwerte
$$
\chi_k(Z):=\prod_{\delta\hbox{ \chi_k(Z):=\prod_{\delta\hbox{ even}} \theta\delta \,
$$
theta[\delta](0,Z)$,
$Z\in{\frak H}_g$, with $k=2^{g-2}(2^g+1)$.
Set
$$F_g:=2^g
\sum_{\delta\hbox{
even}}\theta^{16}[\delta](0,Z)-\bigl(\sum_{\delta\hbox{
even}}\theta^{8}[\delta](0,Z)\bigr)^2 \ .
$$
It turns out that $F_4$, the Schottky-Igusa form, vanishes only on the Jacobian. Furthermore, there is a nice relation
between $F_g$ and the theta series $\Theta_\Lambda$
corresponding to the even unimodular lattices $\Lambda=E_8$ and
$\Lambda=D_{16}^+$:
$$
F_g=2^{-2g}(\Theta_{D_{16}^+}-\Theta_{E_8}^2) \ .
$$
Let ${\phi^n_i}_{1\le i\le N_n}$ be a basis of
$H^0(K_C^n)$, $n\geq2$.
The Mumford form is, up to a universal constant
$$
\mu_{g,n}={\kappa[\omega]^{(2n-1)^2}\over
\kappa[\phi^n]}{\phi^n_1\wedge\cdots\wedge\phi^n_{N_n}\over
(\omega_1\wedge\cdots\wedge\omega_g)^{c_n}}
$$
where $\omega_1,\ldots,\omega_g$ is the standard (normalized) basis of $H^0(K_C)$. $\kappa[\omega]$ is a constant that depends only on the choice of the homological basis whereas $\kappa[\phi^n]$ also depends on the choice of the basis $\phi^n$ (see Prop.1.2). In the case $n=2$ and $g<4$ one may choose a natural basis for $H^0(K_C^2)$: ${\rm Sym}^2 H^0(K_C)$, and for $g=2$ gets
$${\kappa[\omega]^{9}\over
\kappa[\omega^{(2)}]} ={1\over \pi^{12}\chi_{5}^2(\tau)}\ ,$$
whereas for $g=3$
$${\kappa[\omega]^{9}\over
\kappa[\omega^{(2)}]} ={1\over 2^6\pi^{18}\chi_{18}^{1/2}(\tau)}\ . $$
For $g>3$ one has $g(g+1)/2-(3g-3)>0$, so apparently there is no a natural extension. Nevertheless one may continue to take $3g-3$ elements of ${\rm Sym}^2 H^0(K_C)$, or, more generally $N_n:=(2n-1)(g-1)$ elements of ${\rm Sym}^n H^0(K_C)$. Doing this leads to some surprise
${\kappa[\phi^n]\over\kappa[\omega^{(n)}]^{(2n-1)^2}}$, that essentially correspond to what we denoted by $$[i_{N_n+1},\ldots,i_{M_n}|\tau] \ , $$ are vector-valued Teichmueller modular forms of weight
$$d_n:=6n^2-6n+1-{g+n-1\choose n-1} \ .$$ For each integer $n\geq 2$ and for all $i_{2},\ldots,i_{K_n}\in{1,\ldots,M_n}$ one has
$$\sum_{i=1}^{M_n}[i,i_{2},\ldots,i_{K_n}|\tau]\omega^{(n)}_{i}(x)
=0\ .$$ In particular, for $n=2$ these are all the quadrics characterizing the canonical curve in projective space. Such vector-valued forms seem to be a key tool to characterize the Jacobian. Such a problem has been explicitly solved only for $g=4$: there is a weight 8 Siegel modular forms vanishing only on the Jacobian, this is the Schottky-Igusa form $F_4$. Remarkably, one finds that at $g=4$, $[(ij)|\tau]\equiv [i|\tau]$ (see the paper for the indexing) is proportional to $S_{4ij}(\tau)$, where
$$S_{4ij}(Z):={1+\delta_{ij}\over 2}{\partial F_4(Z)\over \partial Z_{ij}} \ .$$ For $g=4$ the discriminant of the quadrics is proportional to the square root of
$\chi_{68}$, the $g=4$ Thetanullwerte
$$ \det S_4(\tau)=d\chi_{68}(\tau)^{1/2}\ . $$
Note that $\det S_4$ and $\chi_{68}(\tau)^{1/2}$ are modular forms (of weight
$34$) only when restricted to ${\cal I}_4$. The $g=4$, $n=2$ Mumford form is
$$\mu_{4,2}=\pm{1\over c
S_{4ij}}{\omega_1\omega_1\wedge\cdots\wedge
\widehat{\omega_i\omega_j}\wedge\cdots\wedge \omega_4\omega_4\over
(\omega_1\wedge\cdots\wedge\omega_4)^{13}} \ .
$$
|
|
|
|
|
4
|
|
edited Feb 25 2011 at 23:39 |
Consider the case $g\ge 2$ and a given symplectic basis for $H_1(C,{\Bbb Z})$. For each positive integer $n$, consider the basis $\tilde\omega_1^{(n)},\ldots,\tilde\omega_{M_n}^{(n)}$ of ${\rm Sym}^n H^0(K_C)$ whose elements are symmetrized tensor products of $n$-tuples of vectors of the basis $\omega_1,\ldots,\omega_g$, taken with respect to an arbitrary ordering chosen once and for all. Denote by $\omega_i^{(n)}$, $i=1,\ldots, M_n$, the image of $\tilde\omega_i^{(n)}$ under the natural map $\psi:{\rm Sym}^n H^0(K_C)\to H^0(K_C^n)$. It is well known that such a map is surjective if and only if $g=2$ or $C$ is non-hyperelliptic of genus $g>2$. For $g=2$ and $g=3$ non-hyperelliptic, this map is an isomorphism. Consider the Thetanullwerte\chi_k(Z):=\prod_{\delta\hbox{ even}} \theta\delta \ ,$Z\in{\frak H}_g$, with $k=2^{g-2}(2^g+1)$.even}}\theta^{8}[\delta](0,Z)\bigr)^2 \ .It turns out that $F_4$, the Schottky-Igusa form, vanishes only on the Jacobian. Furthermore, there is a nice relationbetween $F_g$ and the theta series $\Theta_\Lambda$corresponding to the even unimodular lattices $\Lambda=E_8$ andF_g=2^{-2g}(\Theta_{D_{16}^+}-\Theta_{E_8}^2) \ .$$ For $g>3$ one has $g(g+1)/2-(3g-3)>0$, so apparently there is no a natural extension. Nevertheless you one may continue to take $3g-3$ elements of ${\rm Sym}^2 H^0(K_C)$, or, more generally $N_n:=(2n-1)(g-1)$ elements of ${\rm Sym}^n H^0(K_C)$. Doing this leads to some surprise For each integer $n\geq 2$ and for all $i_{2},\ldots,i_{K_n}\in{1,\ldots,M_n}$ we have The $g=4$, $n=2$ Mumford form is$$\mu_{4,2}=\pm{1\over c\widehat{\omega_i\omega_j}\wedge\cdots\wedge \omega_4\omega_4\over(\omega_1\wedge\cdots\wedge\omega_4)^{13}} \ .$$
|
|
|
|
|
3
|
|
edited Feb 17 2011 at 20:03 |
The extension to $g>3$ of Ichikawa work leads to vector-valued modular forms. The main steps are the following.
For each fixed positive integers
$g,n$, define $$M_n(g)=M_n:={g+n-1\choose n}\ ,\;
N_n(g)=N_n:=(2n-1)(g-1)\ ,\quad K_n:=M_n-N_n\ . $$
Let ${\frak H}_g:={Z\in M_g({\Bbb C})\mid {}^tZ=Z,\mathop{\rm Im} Z>0}$, be the Siegel
upper half-space. Denote by $\tau_{ij}$ the Riemann period matrix and by ${\cal I}_g$ the closure of the locus of Riemann
period matrices in ${\frak H}_g$.
Let ${\phi^n_i}_{1\le i\le N_n}$ be a basis of
$H^0(K_C^n)$, $n\geq2$.
For any points $p,x_1,\ldots,x_{N_n}\in C$,
the The Mumford form is, up to a universal constant
$$
\mu_{g,n}={\kappa[\omega]^{(2n-1)^2}\over
\kappa[\phi^n]}{\phi^n_1\wedge\cdots\wedge\phi^n_{N_n}\over
(\omega_1\wedge\cdots\wedge\omega_g)^{c_n}}
$$
where $\omega_1,\ldots,\omega_g$ is the standard (normalized) basis of $H^0(K_C)$. $\kappa[\omega]$ is a constant that depends only on the choice of the homological basis whereas $\kappa[\phi^n]$ also depends on the choice of the basis $\phi^n$ (see Prop.1.2). In the case $n=2$ and $g<4$ one may choose a natural basis for $H^0(K_C^2)$: ${\rm Sym}^2 H^0(K_C)$, and for $g=2$ gets
$${\kappa[\omega]^{9}\over
\kappa[\omega^{(2)}]} ={1\over \pi^{12}\chi_{5}^2(\tau)}\ ,$$
whereas for $g=3$
$${\kappa[\omega]^{9}\over
\kappa[\omega^{(2)}]} ={1\over 2^6\pi^{18}\chi_{18}^{1/2}(\tau)}\ . $$
For $g>3$ one has $g(g+1)/2-(3g-3)>0$, so apparently there is no a natural extension. Nevertheless you may continue to take $3g-3$ elements of ${\rm Sym}^2 H^0(K_C)$, or, more generally $N_n:=(2n-1)(g-1)$ elements of ${\rm Sym}^n H^0(K_C)$. Doing this leads to some surprise
${\kappa[\phi^n]\over\kappa[\omega^{(n)}]^{(2n-1)^2}}$, that essentially correspond to what we denoted by $$[i_{N_n+1},\ldots,i_{M_n}|\tau] \ , $$ are vector-valued Teichmueller modular forms of weight
$$d_n:=6n^2-6n+1-{g+n-1\choose n-1} \ .$$ For each integer $n\geq 2$ and for all $i_{2},\ldots,i_{K_n}\in{1,\ldots,M_n}$ we have
$$\sum_{i=1}^{M_n}[i,i_{2},\ldots,i_{K_n}|\tau]\omega^{(n)}_{i}(x)
=0\ .$$ In particular, for $n=2$ these are all the quadrics characterizing the canonical curve in projective space. Such vector-valued forms seem to be a key tool to characterize the Jacobian. Such a problem has been explicitly solved only for $g=4$: there is a weight 8 Siegel modular forms vanishing only on the Jacobian, this is the Schottky-Igusa form $F_4$. Remarkably, one finds that at $g=4$, $[(ij)|\tau]\equiv [i|\tau]$ (see the paper for the indexing) is proportional to $S_{4ij}(\tau)$, where
$$S_{4ij}(Z):={1+\delta_{ij}\over 2}{\partial F_4(Z)\over \partial Z_{ij}} \ .$$ For $g=4$ the discriminant of the quadrics is proportional to the square root of
$\chi_{68}$, the $g=4$ Thetanullwerte
$$ \det S_4(\tau)=d\chi_{68}(\tau)^{1/2}\ . $$
Note that $\det S_4$ and $\chi_{68}(\tau)^{1/2}$ are modular forms (of weight
$34$) only when restricted to ${\cal I}_4$.
|
|
|
|
|
2
|
|
edited Feb 17 2011 at 19:34 |
$g,n$, we define $$M_n(g)=M_n:={g+n-1\choose n}\ , \quadN_n(g)=N_n:=(2n-1)(g-1)\ ,\quad K_n:=M_n-N_n$$K_n:=M_n-N_n\ . $$Let ${\frak H}_g:={Z\in M_g({\Bbb C})\mid {}^tZ=Z,\mathop{\rm Im} Z>0}$, be the Siegelupper half-space. Denote by $\tau_{ij}$ the Riemann period matrix and by ${\cal I}_g$ the closure of the locus of Riemannperiod matrices in ${\frak H}_g$.where $\omega_1,\ldots,\omega_g$ is the standard (normalized) basis of $H^0(K_C)$. $\kappa[\omega]$ is a constant that depends only on the choice of the homological basis whereas $\kappa[\phi^n]$ also depends on the choice of the basis $\phi^n$ (see Prop.1.2). In the case $n=2$ and $ ng<4$ you one may choose a natural basis for $H^0(K_C^2)$: $ \Sym^2 {\rm Sym}^2 H^0(K_C)$, and you getFor $g>3$ you have one has $g(g+1)/2-(3g-3)>0$, so apparently there is no a natural extension. Nevertheless you may continue to take $3g-3$ elements of $ \Sym^2 {\rm Sym}^2 H^0(K_C)$ (, or , more generally $N_n:=(2n-1)(g-1)$ elements of $ \Sym^n {\rm Sym}^n H^0(K_C)$. What you get isDoing this leads to some surprise${\kappa[\phi^n]\over\kappa[\omega^{(n)}]^{(2n-1)^2}}$, that essentially correspond to what we denoted by $$[i_{N_n+1},\ldots,i_{M_n}|\tau] \ , $$ are vector-valued Teichmueller modular forms of weight=0\ .$$ In particular, for $n=2$ these are all the quadrics characterizie characterizing the canonical curve in projective space. As a first step that such Such vector-valued are interesting objects forms seem to be a key tool to characterize the Jacobian. Such a problem has been explicitly solved only for $g=4$: there is a weight 8 Siegel modular forms vanishing only on the Jacobian, this is the Schottky-Igusa form $F_4$. Remarkably, one notes finds that at $g=4$g=4$, $$\det S_4(\tau)=d\chi_{68}(\tau)^{1/2}[(ij)|\tau]\equiv [i|\tau]$ (see the paper for the indexing) is proportional to $S_{4ij}(\tau)$, where$$S_{4ij}(Z):={1+\delta_{ij}\over 2}{\partial F_4(Z)\over \,partial Z_{ij}} \ .$$where For $S_4$ is g=4$ the derivative with respect to discriminant of the $Z_{ij}\in {\cal H_4}$ evaluated on quadrics is proportional to the Schottky locus square root ofthe Schottky-Igusa form and $\chi_{68}$ is \chi_{68}$, the $g=4$ Thetanullwerte $$ \det S_4(\tau)=d\chi_{68}(\tau)^{1/2}\ . It is nice $$Note that $\det S_4$ is proportional and $\chi_{68}(\tau)^{1/2}$ are modular forms (of weight$34$) only when restricted to $[i|\tau]$.{\cal I}_4$.
|
|
|
|
|
1
|
|
answered Feb 17 2011 at 17:53 |
The extension to $g>3$ of Ichikawa work leads to vector-valued modular forms. The main steps are the following.
For each fixed positive integers
$g,n$, we define $$M_n(g)=M_n:={g+n-1\choose n}\ ,\quad
N_n(g)=N_n:=(2n-1)(g-1)\ ,\quad K_n:=M_n-N_n$$
Let ${\phi^n_i}_{1\le i\le N_n}$ be a basis of
$H^0(K_C^n)$, $n\geq2$. For any points $p,x_1,\ldots,x_{N_n}\in C$,
the Mumford form is, up to a universal constant
$$
\mu_{g,n}={\kappa[\omega]^{(2n-1)^2}\over
\kappa[\phi^n]}{\phi^n_1\wedge\cdots\wedge\phi^n_{N_n}\over
(\omega_1\wedge\cdots\wedge\omega_g)^{c_n}}
$$
where $\omega_1,\ldots,\omega_g$ is the standard (normalized) basis of $H^0(K_C)$. $\kappa[\omega]$ is a constant that depends only on the choice of the homological basis whereas $\kappa[\phi^n]$ also depends on the choice of the basis (see Prop.1.2). In the case $n=2$ and $n<4$ you may choose a natural basis for $H^0(K_C^2)$: $\Sym^2 H^0(K_C)$, and you get
for $g=2$ $${\kappa[\omega]^{9}\over
\kappa[\omega^{(2)}]} ={1\over \pi^{12}\chi_{5}^2(\tau)}\ ,$$
whereas for $g=3$
$${\kappa[\omega]^{9}\over
\kappa[\omega^{(2)}]} ={1\over 2^6\pi^{18}\chi_{18}^{1/2}(\tau)}\ . $$
For $g>3$ you have $g(g+1)/2-(3g-3)>0$, so apparently there is no a natural extension. Nevertheless you may continue to take $3g-3$ elements of $\Sym^2 H^0(K_C)$ (or more generally $N_n:=(2n-1)(g-1)$ elements of $\Sym^n H^0(K_C)$. What you get is
${\kappa[\phi^n]\over\kappa[\omega^{(n)}]^{(2n-1)^2}}$, that essentially correspond to what we denoted by $$[i_{N_n+1},\ldots,i_{M_n}|\tau] \ , $$ are vector-valued modular forms of weight
$$d_n:=6n^2-6n+1-{g+n-1\choose n-1} \ .$$ For each integer $n\geq 2$ and for all $i_{2},\ldots,i_{K_n}\in{1,\ldots,M_n}$ we have
$$\sum_{i=1}^{M_n}[i,i_{2},\ldots,i_{K_n}|\tau]\omega^{(n)}_{i}(x)
=0\ .$$ In particular, for $n=2$ these are all the quadrics characterizie the canonical curve in projective space. As a first step that such vector-valued are interesting objects one notes that at $g=4$
$$\det S_4(\tau)=d\chi_{68}(\tau)^{1/2}\ ,$$
where $S_4$ is the derivative with respect to the $Z_{ij}\in {\cal H_4}$ evaluated on the Schottky locus of the Schottky-Igusa form and $\chi_{68}$ is the $g=4$ Thetanullwerte. It is nice that $S_4$ is proportional to $[i|\tau]$.
|
|
|
