Teichmuller modular forms and number theory

Question

Do higher genus Teichmuller modular forms have, or are they expected to have, implications for number theory that generalize the sorts of results that flow from the study of classical modular forms?

Answer 1

@David Hansen: Teichmuller modular forms are basically the natural analogue of Siegel modular forms when one considers sections of line bundles on $M_g$ instead of $A_g$. Search for papers by Ichikawa.

I don't know very much about Teichmuller modular forms but my advisor knows a bit about them and from what I can tell from conversations with him, pretty much nothing at all is known about this question. I can think of some plausible reasons why this is so. (This answer is probably a bit naive, my apologies.)

One is that the difference between Siegel modular forms and Teichmuller modular forms does not become visible until genus three. For genus one and two the map from $M_g$ to $A_g$ is an isomorphism and an open immersion respectively, and all Teichmuller modular forms are just pullbacks of forms from $A_1$ and $A_2$. In genus three the Torelli map is an open immersion on coarse moduli spaces but on the level of stacks it is a ramified double cover, which makes it plausible that there should be more functions on $M_3$ than on $A_3$. And indeed the ring of Teichmuller modular forms in genus three is obtained from the ring of Siegel modular forms by adjoining a square root of $\chi_{18}$, which vanishes exactly at the hyperelliptic locus i.e. the branch locus of the Torelli map. Anyway, the point is that already Siegel modular forms in higher genera are not so well understood compared to the rich theory we have in low genus. In particular we know extremely few examples of genuine Teichmuller modular forms.

Another more serious reason is that it is not clear how any of the standard tools for studying ordinary modular forms can be applied to the Teichmuller case. As a very basic example, it is for instance completely open if or how one can define a notion of Hecke operators acting on Teichmuller modular forms: double cosets in the mapping class group are a scary prospect, and there seems to be no analogue of the standard Hecke correspondences in terms of cyclic subgroups of order p. What's worse is that none of the standard automorphic/Langlands etc tools seem to be applicable to the study of Teichmuller modular forms. Unlike Siegel's upper half space, Teichmuller space is not a symmetric space. If we believe in the Langlands philosophy then there should be a reductive group somewhere that the Teichmuller modular forms "come from", but there is no natural reductive group anywhere in sight.

Anyway, to actually answer your question: there probably is some connection between Teichmuller modular forms and number theory. In particular it seems that there are $\ell$-adic Galois representations naturally attached to Teichmuller modular forms. There are just no tools to study these representations.

Answer 2

$$\hbox{Vector-valued Teichmueller Modular forms}$$

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 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 $\prod_{\delta\hbox{ even}}\theta[\delta]=0$.

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.

Note that $S_{4ij}(Z)$ 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 ${\cal I}_4$.

This clearly shows that the vector-valued Teichmueller modular forms $[i_{N_n+1},\ldots,i_{M_n}|\tau]$ 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 $g$, generated by Mumford's forms, and their properties, such as the one of generating canonical curves, that is Petri's relations, strongly support Mumford's suggestion that Petri's relations are fundamental 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 masterpiece.

Answer 3

Marco Matone, Roberto Volpato, Vector-Valued Modular Forms from the Mumford Form, Schottky-Igusa Form, Product of Thetanullwerte and the Amazing Klein Formula, Proc. Amer. Math. Soc. 141 (2013), 2575-2587 doi:10.1090/S0002-9939-2012-11526-6, arXiv:1102.0006.

may be useful

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Abstract: 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. We show that for $ g\geq 4$, a new class of vector-valued modular forms, defined on the Teichmüller space, naturally appears from the Mumford forms, a question directly related to the Schottky problem. In this framework we show that the discriminant of the quadric associated to the complex curves of genus $ 4$ is proportional to the square root of the products of Thetanullwerte $ \chi _{68}$, which is a proof of the recently rediscovered Klein ``amazing formula''. Furthermore, it turns out that the coefficients of such a quadric are derivatives of the Schottky-Igusa form evaluated at the Jacobian locus, implying new theta relations involving the latter, $ \chi _{68}$ and the theta series corresponding to the even unimodular lattices $ E_8\oplus E_8$ and $ D_{16}^+$. We also find, for $ g=4$, a functional relation between the singular component of the theta divisor and the Riemann period matrix.