MathOverflow will be down for maintenance for approximately 3 hours, starting Monday evening (06/24/2013) at approximately 9:00 PM Eastern time (UTC-4).

## Return to Answer

13 added 1168 characters in body
• The $g=4$, $n=2$ Mumford form is$$\mu_{4,2}=\pm{1\over c\widehat{\omega_i\omega_j}\wedge\cdots\wedge \omega_4\omega_4\over(\omega_1\wedge\cdots\wedge\omega_4)^{13}} \ ,with c a constant. • trasforms with affine terms proportional to F_4, so that it is a vector-valued modular form only when F_4=0, that is in the Jacobian. This motivates the name vector-valued Teichmueller modular forms. Note that also the square root of \chi_{68}(\tau) exists only in {\cal I}_4. It follows that \det S_4 and \chi_{68}(\tau)^{1/2} are modular forms (of weight 34) only when restricted to {\cal I}_4. • The This clearly shows that the vector-valued Teichmueller modular formsg=4, [i_{N_n+1},\ldots,i_{M_n}|\tau]are deeply related to the geometry of the Jacobian and to the Schottky's problem. The structure of the vector-valued Teichmueller modular at any n=2 Mumford form g, generated by Mumford's forms, and their properties, such as the one of generating canonical curves, that is$$\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, $$with c strongly support Mumford's suggestion that Petri's relations arefundamental and should have basic applications: see pg.241, D. Mumford, The Red Book of Varieties and Schemes, Springer Lecture Notes in Math. 1358, 1999. Actually, it seems Mumford was right, one has just to use his forms. A suggestion for the literature: the papers by John Fay are excellent and not very well-known as they should. The one in the Memoirs of the AMS is a constantmasterpiece. • 12 added 4 characters in body$$\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)}\ . $$For g>3 one has g(g+1)/2-(3g-3)>0, so apparently there is no a natural extension. Nevertheless one may continue to take 3g-3 elements of {\rm Sym}^2 H^0(K_C), or, more generally N_n:=(2n-1)(g-1) elements of {\rm Sym}^n H^0(K_C). Doing this leads to some surprise. 1. To simplify notation, denote here by \omega^{(n)} the basis {\omega^{(n)}_k} with k=i_1,\ldots,i_{N_n}\in{1,\ldots,M_{n}}.$$[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} \ . $$Note that the vector-valued nature is just a consequence of the inequality M_n-N_n>0 for some g. For example, for n=2 one has g(g+1)/2-(3g-3)>0 satisfied for g>3. This implies that there are free indeces: the i_{N_n+1},\ldots,i_{M_n}, a nice hint that the theory of vector-valued Teichmueller modular forms is a key tool to investigate the Schottky problem, see below for the case g=4 of genus 4 (presumably here should also appear some interesting Number Theoretical structures). 2. For each integer n\geq 2 and for all i_{2},\ldots,i_{K_n}\in{1,\ldots,M_n} one has$$\sum_{i=1}^{M_n}[i,i_{2},\ldots,i_{K_n}|\tau]\omega^{(n)}_{i}(x) =0\ .$$In particular, for n=2 these are all the quadrics characterizing the canonical curve in projective space. 3. Remarkably, one finds that at g=4, [(ij)|\tau]\equiv [i|\tau] (see the paper for the indexing) is proportional to$$S_{4ij}(Z):={1+\delta_{ij}\over 2}{\partial F_4(Z)\over \partial Z_{ij}} \ .$$4. For g=4 the discriminant of the quadrics is proportional to the square root of \chi_{68}, the g=4 Thetanullwerte$$ \det S_4(\tau)=d\chi_{68}(\tau)^{1/2}\ , $$with d a constant. A key step here is the following lemma. Let C be either a non-hyperelliptic Riemann surface of genus g=4 or a non-trigonal surface of g=5. Then, the canonical model of C is contained in a quadric of rank 3 if and only if \prod_{\delta\hbox{ even}}\theta[\delta]=0. Note that \det S_4 and \chi_{68}(\tau)^{1/2} are modular forms (of weight 34) only when restricted to {\cal I}_4. 5. The g=4, n=2 Mumford form is$$\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}} \ , $$with c a constant. 11 deleted 123 characters in body$$\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)}\ . $$For g>3 one has g(g+1)/2-(3g-3)>0, so apparently there is no a natural extension. Nevertheless one may continue to take 3g-3 elements of {\rm Sym}^2 H^0(K_C), or, more generally N_n:=(2n-1)(g-1) elements of {\rm Sym}^n H^0(K_C). Doing this leads to some surprise. 1. To simplify notation, denote here by \omega^{(n)} the basis {\omega^{(n)}_k} with k=i_1,\ldots,i_{N_n}\in{1,\ldots,M_{n}}.$$[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} \ . $$Note that the vector-valued nature is just a consequence of the inequality M_n-N_n>0 for some g. For example, for n=2 one has g(g+1)/2-(3g-3)>0 satisfied for g>3. This implies that there are free indeces: the i_{N_n+1},\ldots,i_{M_n}. This may be seen as i_{N_n+1},\ldots,i_{M_n}, a nice hint that the theory of vector-valued Teichmueller modular forms is a key tool to investigate the Schottky problem, see below for the case g=4 (presumably here should also appear some interesting Number Theoretical structures). 2. For each integer n\geq 2 and for all i_{2},\ldots,i_{K_n}\in{1,\ldots,M_n} one has$$\sum_{i=1}^{M_n}[i,i_{2},\ldots,i_{K_n}|\tau]\omega^{(n)}_{i}(x) =0\ .$$In particular, for n=2 these are all the quadrics characterizing the canonical curve in projective space. 3. Remarkably, one finds that at g=4, [(ij)|\tau]\equiv [i|\tau] (see the paper for the indexing) is proportional to S_{4ij}(\tau), where$$S_{4ij}(Z):={1+\delta_{ij}\over 2}{\partial F_4(Z)\over \partial Z_{ij}} \ .$$4. For g=4 the discriminant of the quadrics is proportional to the square root of \chi_{68}, the g=4 Thetanullwerte$$ \det S_4(\tau)=d\chi_{68}(\tau)^{1/2}\ , $$with d a constant. A key step here is the following lemma. Let C be either a non-hyperelliptic Riemann surface of genus g=4 or a non-trigonal surface of g=5. Then, the canonical model of C is contained in a quadric of rank 3 if and only if \prod_{\delta\hbox{ even}}\theta[\delta]=0. Note that \det S_4 and \chi_{68}(\tau)^{1/2} are modular forms (of weight 34) only when restricted to {\cal I}_4. 5. The g=4, n=2 Mumford form is$$\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 deleted 247 characters in body
9 added 105 characters in body; [made Community Wiki]
8 added 348 characters in body
7 added 323 characters in body
6 added 1276 characters in body
5 deleted 7 characters in body
4 added 1566 characters in body
3 deleted 43 characters in body
2 added 703 characters in body
1