数理解析研究所講究録1086 巻 1999 年 37-56

37

PRECIOUS SIEGEL MODULAR FORMS OF GENUS TWO
V. GRITSENKOABSTRACT. We give a review of the recent results concerning Siegel modular forms withrespect to the paramodular groups of genus 2 and their applications to Algebraic Geometry
and Physics. Some facts mentioned bellow have not been published before.

\S 0. GENERAL SET UP
be an even integral lattice of signature $(2, n)$ . Let us consider the correspondingLet”
denotes the subgroup of elements with the real spin
orthogonal group $O^{+}(L_{2,n})$ (“norm 1) and the homogeneous domain of type IV$L_{2,n}$$\mathrm{p}\mathrm{l}\mathrm{u}\mathrm{s}$$H_{L_{2,n}}^{+}\cong O^{+}(L_{2},n\otimes \mathbb{R})/Kmax\{\cong \mathbb{C}\omega\subset \mathrm{P}(L_{2,n}\otimes \mathbb{C})|\omega\cdot\omega=0, \omega\cdot\overline{\omega}>0\}^{+}$

where $K_{\max}$ is the maximal compact subgroup of the orthogonal group. The pairis a symmetric pair of $BD$ type. The corresponding bounded symmetric domain having complex dimension is a complex domain of type IV in the Cartan
as a tube
we denote a standard realization ofclassification. By(see . [B2], [Od], [G3]).
the tube of future) indomain (this isof finite index we may consider the d-graded
For an arbitrary subgroupof weight $dk$ with respect toof modular forms
ring$(O^{+}(L_{2,n}\otimes \mathbb{R}), K_{\max})$$n$$O^{+}(L_{2,n}\otimes \mathbb{R})/K_{\max}$$\mathcal{H}_{L_{2n}}^{+}$$\mathrm{S}\mathrm{O}^{-}\mathrm{C}\mathrm{a}\mathrm{l}\mathrm{i}\mathrm{e}\mathrm{d}$$\mathbb{C}^{n}$$\mathrm{f}.\mathrm{e}$$\Gamma\subset O^{+}(L_{2,n})$$(d\in \mathrm{N})$$\Gamma$$\mathfrak{M}_{dk}(\Gamma)$$m_{*}^{(d)}(\Gamma)=\oplus^{\infty}k=0\mathfrak{M}_{dk(\Gamma)}$.

It is well known that this ring defines the Satake compactification of the modular variety$A\Gamma=\Gamma\backslash \mathcal{H}_{L_{2,n}}+$.

More exactly
Proj$( \bigoplus_{k=0}^{\infty}\mathfrak{M}dk(\Gamma))=\overline{A}_{\Gamma}(s_{at}ake)$.

are the moduli spaces of the polarized
Classical examples of modular varieties of type
$K3$ , Enriques, Abelian and Kummer surfaces. In the case of polarized $K3$ surfaces we havethe moduli space of dimension 19 related with lattice$A_{\Gamma}$$L_{2,n}=L_{2,19}\cong 2U\oplus 2E_{8}(-1)\oplus<-2t>$

38
where

$U=$

is the unimodular hyperbolic plane, $E_{8}(-1)$ is the negative definite

unimodular lattice of rank 8 andone dimensional -lattice with quadratic form$(-2t)$ (the number
defines a polarization of $K3$ surface). For Enriques surfaces$<-2t>\mathrm{i}\mathrm{s}$$\mathbb{Z}$$2t$$L_{2,n+2}=L_{2,10}\cong U\oplus U(2)\oplus E_{8}(-2)$(see . [BPV]). The moduli spaces of $(1, t)$ -polarized Abelian and corresponding Kummer
surfaces have dimension 3. They are related with the lattice$\mathrm{f}.\mathrm{e}$$L_{2,\mathrm{s}}=2U\oplus<-2t>$(see$\mathrm{f}.\mathrm{e}$.

$[\mathrm{v}\mathrm{d}\mathrm{G}],$$[\mathrm{G}\mathrm{H}1]$). We may formulate the following problem:

Problem Al. To describe lattices
and correspondingthat theringis a polynomial ring over and to find genera tors of thisring.
$L_{2,n}$$gr\mathrm{a}ded$$gro\mathrm{u}_{\mathrm{P}^{S}}\Gamma\subset O^{+}(L_{2,n})s\mathrm{u}ch$$\mathfrak{M}_{*}^{(d)}(\Gamma)$$\mathbb{C}$More generally we may put
Problem A2. For a given lattice$eq$ uations of the modvariety$\mathrm{u}l\mathrm{a}r$

up
and a
to write a syst $em$ ofin terms of generators of the ring.

$L_{2,n}$$A_{\Gamma}$$\Gamma\subset O^{+}(L_{2,n})$$g\mathrm{r}o$$\mathfrak{M}_{*}^{(d)}(\Gamma)$

We would like to mention that a solution of these problems related with description of
the graded rings of typegives us not only good models of some classical moduli
spaces but defines the so-called arithmetic mirror symmetry for $K3$ surfaces (see)
and provides us with important invariants in the theory of root systems of different types(of elliptic root systems, of hyperbolic roots systems and of root systems of type IV in thesense of K. Saito).
Problems Al and A2 are closely related with the following$\mathfrak{M}_{*}^{(d)}(\Gamma)$$[\mathrm{G}\mathrm{N}3|$

Problem B. To constr
automorphic forms with resp $ect$ to the $gro$ up
a small weight with a “simple” divisor.
$\mathrm{u}ct$$\Gamma\subset O^{+}(L_{2,n})$of

One can reformulate this problem as a construction of first generators of the corresponding graded rings. By a simple divisor we mean a union of divisors of Humbert type.
A Humbert divisor is a projection of a rational quadratic divisor defined by some vectorof negative norm ( is the lattice dual to )$l\in L_{2,n}^{*}$$L^{*}$$L$$H_{l}=\pi \mathrm{r}(\{_{3}\in \mathcal{H}_{L_{2,n}}+|(\mathrm{g}, l)=0\})$

,

where
:
is the natural projection. The Humbert divisor
is amodular variety of type (2) of dimension $n-1$ .
We call a meromorphic automorphic form
Humbert modular form if its
divisor is a sum of Humbert divisors with some multiplicities$\pi_{\Gamma}$$H_{l}$$\mathcal{H}_{L_{2,n}}^{+}arrow\Gamma\backslash \mathcal{H}_{L_{2,n}}^{+}$$F\in \mathfrak{M}_{k}(\Gamma)$$\mathrm{d}\mathrm{i}\mathrm{v}_{A_{\Gamma}}(F)=\sum_{i}aiHl_{i}$.

39Such form $F$ is called reflective (resp. 2-reflective) if the reflection defined by belongs$l_{i})=-2$ ) for all . The reflective
to the integral orthogonal group $O(L_{2,n})^{+}$ (resp.
For example, they provide us
manyin
importantconsiderations.
very
modular forms arewith automorphic corrections of hyperbolic Kac-Moody algebras (see[G6] .
A particular answer on Problem formulated above is given by the lifting construction(see also [G4] where the case of the paramodular
proposed by the author ingroups, i.e. $n=3$ , was considered). This construction gives us all modular forms of verysmall weights, more exactly for the weights for which modular forms are defined by the
first Fourier-Jacobi coefficient. .
The information about divisors of modular forms is given by the Borcherds constructionof the exponential lifting. In the present talk we shall concentrate ourselves mainly on thecase of the homogeneous domain of dimension 3, i.e. on the lattices on signature $(2, 3)$ .
we construct a theory of reflective and 2-reflective
In our RIMS-preprints
forms in the case of the lattices of signature $(2, 3)$ . As it was mentioned above this caseis connected with the theory of the moduli spaces of Abelian and Kummer surfaces (see. The application of some reflective modular forms constructed in[G3] and),[GN1] and [GN6] to the heterotic string theory were found by T. Kawai (see
H.
E.
and
Verlinde
Dijkgraaf,[Ca]),[CCL],R.
(see. Cardoso, G. Curio and D. Luest.
Verlinde (see [DVV]). See also the papers of J. Harvey and G. Moore
The most part of the results presented bellow were obtained by the author together with
V. Nikulin and with K. Hulek during his stay at RIMS (09.1996-02.1997) and were puband [GH2]. Using this opportunity we express
lished in the RIMS-preprints
our gratitude to Research Mathematical Institute of Kyoto University for hospitality. I
give my special thanks to all members of Algebraic Geometry Seminars at RIMS for usefuland stimulating discussions.
$l_{i}$$(l_{i},$$\dot{i}$$[\mathrm{B}1]-[\mathrm{B}3],$$[\mathrm{G}\mathrm{N}1]-$$)$$\mathrm{B}$$[\mathrm{G}1]-[\mathrm{G}3]$$([\mathrm{G}\mathrm{N}5]-[\mathrm{G}\mathrm{N}6])$$[\mathrm{G}\mathrm{H}1]-[\mathrm{G}\mathrm{H}2])$$[\mathrm{K}\mathrm{a}\mathrm{l}]-[\mathrm{K}\mathrm{a}2]$$\mathrm{G}.\mathrm{L}$$[\mathrm{H}\mathrm{M}1]-[\mathrm{H}\mathrm{M}2]$$[\mathrm{G}\mathrm{N}5]-[\mathrm{G}\mathrm{N}7]$\S 1. ARITHMETIC LIFTING
In case of the lattice $L_{2,3}=2U\oplus<-2t>\mathrm{o}\mathrm{f}$ signature $(2, 3)$ the corresponding homois isomorphic to the Siegel upper-half plane of genus 2
geneous domain$\mathcal{H}_{L_{2,3}}$$\mathbb{H}_{2}=\{Z={}^{t}Z\in M_{2}(\mathbb{C}), Z=X+\dot{i}Y, Y>0\}$.

The integral orthogonal group $PO^{+}(L_{2,3})$ is isomorphic to a normal extension of the socalled paramodular group . The last group is conjugated to the integral symplectic groupof a skew-symmetric form with elementary divisors $(1, t)$ . It can be realized as the following
subgroup of$\Gamma_{t}$$Sp_{4}(\mathbb{Q})$$\Gamma_{t}:=\{$$(_{t*}^{*}**$$t*t*t**$$t****$$t^{-1_{*}}***)\in Sp_{4}(\mathbb{Q})|\mathrm{a}\mathrm{l}\mathrm{l}*\mathrm{a}\mathrm{r}\mathrm{e}\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{l}\}$The quotient$A_{t}=\Gamma_{t}\backslash \mathbb{H}_{2}$.

40
is isomorphic to the coarse moduli space of abelian surfaces with a polarization of type$(1, t)$ .
:the standard slash operator on the space of functions on
We denote by$|_{k}(k\in \mathbb{Z}/2)$$\mathbb{H}_{2}$$(F|_{k}M)(z):=\det(CZ+D)^{-k}F(M<Z>)$

where$M=\in Sp_{n}(\mathbb{R})$For a half-integral$\sqrt{\det(Z}/i)>0$ for

$k$

and

$M<Z>=(AZ+B)(Cz+D)^{-1}$ .

we choose one of the holomorphic square roots by. the condition$Z=\dot{i}Y\in \mathbb{H}_{n}$.

with a character
Definition. A modular form of weight $k(k\in \mathbb{Z}/2)$ with respect toholomorphic
half-integer)isaa:function on
ifissystemmultiplier(or asatisfies the functional equation
which for arbitrary$\Gamma_{t}$$k$$\chi$$\Gamma_{t}arrow \mathbb{C}^{*}$$\mathbb{H}_{2}$$M\in\Gamma_{t}$$F|_{k}M=\chi(M)F$$\forall M\in\Gamma_{t}$.

We denote the space of such modular (resp. cusp) forms bytively).

$\mathfrak{M}_{k}(\Gamma_{t}, \chi)(\mathfrak{R}_{k}(\Gamma_{t}, x)$

respec-in order to construct roots ofHere we admit a character of the paramodular groupwith trivial character. The classicalcertain orders of modular forms with respect toexample is the Dedekind eta-function$\Gamma_{t}$$\Gamma_{t}$$\eta(\tau)=q^{\frac{1}{24}}\prod_{n\geq 1}(1-q^{n})=\sum_{\mathrm{N}n\in}(\frac{12}{n})q^{n^{2}}\in \mathfrak{R}1/2(sL_{2}(\mathbb{Z})/24, v_{\eta})$

where$\tau\in \mathbb{H}_{1},$$q=\exp(2\pi\dot{i}\tau)$,

1 if $n\equiv\pm 1$ mod 12$( \frac{12}{n})=\{$ $-1$
if $n\equiv\pm 5$ mod 12
if $(n, 12)\neq 1$
$0$

and

$v_{\eta}$:$SL_{2}(\mathbb{Z})arrow\{^{2}\sqrt[4]{1}\}$

is -multiplier system. Then$\eta$$\Delta(\tau)=\eta(\tau)^{24}=q-24q+252q^{\mathrm{s}_{-}}1472q^{4}+\cdots\in \mathfrak{R}_{12}(SL_{2}(\mathbb{Z}))$

with trivial character.
is the first cusp form with respect to
All possible characters of the paramodular group are described in [GH2]. In particular
we have$SL_{2}(\mathbb{Z})$$\Gamma_{t}$41, Theorem 2.1].) For any integer $t\geq 1$ let
Theorem about Commutator. (seeis the comm tator $su$ bgroup of the aramodular $gro$ up$t_{1}=(t, 12)$ and $t_{2}=(2t, 12)$ . If
, then$[\mathrm{G}\mathrm{H}2$$\Gamma_{t}’$$p$$\mathrm{u}$$\Gamma_{t}$$\Gamma_{t}/\Gamma_{t}’\cong \mathbb{Z}/t_{1}\mathrm{x}\mathbb{Z}/t_{2}$.

of the Picard group ofis isomorphic to the torsion Tor Pic
Remark. The group$(1,t)$-polarized abelian surfaces.
ofthe moduli stack
can be considered as a root of some order of a modular
A modular form
where is the order of . From Theorem above follows that $d|12$ ifform fromis integer.
isThe classical example of modular forms with a non-trivial character ofthe product of all even Siegel theta-constants$(\underline{A}_{t})$$\Gamma_{t}/\Gamma_{t}’$$\underline{A}_{t}$$d$$F\in m_{k}(\mathrm{r}_{t,\chi})$

$k$

$d$

$\mathfrak{M}_{dk}(\Gamma_{t})$$\chi$$\Gamma_{1}=s_{p_{4}}(\mathbb{Z})$$(Z[l]|={}^{t}lZl)$$\theta_{a,b}(Z)=\sum_{\mathbb{Z}^{2}l\in}\exp(\pi\dot{i}(Z[l+\frac{1}{2}a]+{}^{t}bl))$

,

(1.1)

i.e.

(1.2)

$\triangle_{5}(Z)=2^{-6}.\prod_{={}^{t}ab0(2)}\theta a,b(z)\in \mathfrak{R}_{5}(\Gamma_{1}, \chi_{2})$:where) The graded ringsof. In particular$s_{p_{4}}(\mathbb{Z})arrow\{\pm 1\}([\Gamma_{1} :$\chi_{2}$, thusand\Gamma_{1}’]=\mathbb{Z}_{2}$$\mathfrak{M}_{*}^{(2)}(sp_{4}(\mathbb{Z}))$$\Gamma_{1}.$$\chi_{2}$is the unique non-trivial character
where defined by Igusa (see$\mathfrak{M}_{*}^{(1)}(Sp_{4}(\mathbb{Z}))$$[\mathrm{I}\mathrm{g}\mathrm{l}]-[\mathrm{I}\mathrm{g}2])$$\mathfrak{M}_{*}^{(2)}(Sp4(\mathbb{Z}))=\mathbb{C}[E4, E6, F10, F12]$-Eisenstein series
andare theis the polynomial ring in four variables, where
cuspform of weightis the unique, up to a constant,
andof weight 4 and 6,
12. Moreover$E_{4}$$F_{10}=\Delta_{5}^{2}$$Sp_{4}(\mathbb{Z})$$E_{6}$$F_{12}$$\mathfrak{M}_{*}((1)Sp4(\mathbb{Z}))=\mathfrak{M}(2)*(Sp_{4}(\mathbb{Z}))[F35]$

is the unique, up to a constant, Siegel cusp form of weight 35.
where
The modular forms of very small weights automatically belong to the set of generators
andare the. In the above example the Eisenstein seriesof the graded ring-modular forms. Similar Eisenstein series can be defined for arbitrary .
first two
-cusp forms.
are the first two
andThe modular forms
The cusp forms of the smallest possible weight can be constructed using a special proceof integral
dure, Arithmetic Lifting. The datum for this lifting is a Jacobi form
weight and index $R(R\in \mathrm{N}/2)$ together with a character of the full Jacobi group. We
define Jacobi forms as modular forms with respect to the maximal parabolic subgroupwhich fixes a line, i.e.
of$F_{35}$$E_{4}$$\mathfrak{M}_{*}^{(1)}(\mathrm{r}_{t})$$E_{6}$$\Gamma_{t}$$Sp_{4}(\mathbb{Z})$$F_{10}=\triangle_{5}^{2}$$F_{12}$$Sp_{4}(\mathbb{Z})$$\phi_{k,R}(\tau, z)$$k$$\Gamma_{\infty}$$s_{p_{4}}(\mathbb{Z})=\Gamma_{1}$$\mathrm{r}_{\infty}=\{\in Sp_{4}(\mathbb{Z})\}$

Let

$k$

and

$R$

be integral or half-integral.

.

42
on
Definition. We call a holomorphic function
and index with a multiplier system (or a character)$\emptyset(\mathcal{T}, Z)$$R$$\mathbb{H}_{1}\cross \mathbb{C}$a Jacobi form of weightif the function$k$$v:\Gamma_{\infty}arrow \mathbb{C}^{*}$$\overline{\phi}(Z):=\phi(\tau, z)\exp(2\pi\dot{i}R\omega)$,$Z=\in \mathbb{H}_{2}$ ,satisfies the functional equation
for any $M=\in\Gamma_{\infty}$$\overline{\phi}(M<Z>)=v(M)\det(cZ+D)^{k}\overline{\phi}(Z)$and$\phi$

has a Fourier expansion of type$\phi(\tau, z)=\sum_{n,l,-}f(n, l)\mathrm{e}\mathrm{x}4Rnl^{2}\geq 0\mathrm{p}(2\pi\dot{i}(n\mathcal{T}+lZ))$

where the summation is taken over and from some free -modules depending on .
The condition $f(n, l)=0$ unless $4Rn-l^{2}\geq 0$ is equivalent to the holomorphicity of at
is called a Jacobi cusp form if $f(n, l)=0$ unless $4Rn-l^{2}>0$ .
infinity. The form
We call the number 4 $Rn$ –l2 the norm of the index of the Fourier coefficient $f(n, l)$ . We
denote the finite dimensional space of such Jacobi forms (resp. cusp forms) by $J_{k,R}(v)$). If the function has a Fourier expansion of type(resp.
$l$$n$$\mathbb{Z}$$v$$\phi$$\emptyset(\mathcal{T}, Z)$$\phi$$J_{k,R}^{c}(v)$$\phi(\mathcal{T}, Z)=\sum_{n\geq 0,l}f(n, l)\exp(2\pi\dot{i}(n\mathcal{T}+lZ))$

then we call it a weak Jacobi form. The spacedimensional.

$J_{k,R}^{w}(v)$,of all such forms is again finite

Here we admit Jacobi forms of half-integral indices. This is the only difference between
the definition of the Jacobi forms given above and the definition of [EZ].
Many. examples of Jacobi forms of half-integral index see in [GN6]. The oldest example
of such forms is the Jacobi triple product. The Jacobi theta-series is defined as,$\theta(\tau, z)=\sum_{mn\equiv 1od2}(-1)\frac{n-1}{2}\exp(\frac{\pi\dot{i}n^{2}}{4}\tau+\pi inz)=\sum m\in \mathbb{Z}(\frac{-4}{m})q^{m^{2}/}8m/r2$

where $q=\exp(2\pi\dot{i}\tau),$ $r=\exp(2\pi\dot{i}z)$ and$\pm 1$$( \frac{-4}{m})=\{$$0$

if $m\equiv\pm 1$ mod 4
if
mod 2.
$m\equiv$$0$This is a Jacobi form of weight 1/2 and index 1/2, i.e. an element of-multiplier system
multiplier system
is induced by the-function and the following character of the integral Heisenberg group$J_{\frac{1}{2}\frac{1}{2}},(v_{\eta}^{3}\cross v_{H})$$SL_{2}(\mathbb{Z})$$v_{\eta}^{3}\cross v_{H}$. Theof the Dedekind$v_{\eta}$$\eta$$v_{H}([\lambda, \mu;\kappa]):=(-1)\lambda+\mu+\lambda\mu+\kappa$

.

(1.3)

We recall the famous Jacobi triple product formula$\theta(\tau, z)=-q^{1}r-81/2\square /(1-q-1r)n(1-qr-1)n\geq 1n(1-q^{n})$The next theorem is proved in [in [G4]$)$$\mathrm{G}\mathrm{N}6$

.

(1.4)

, Theorem 1.12] (compare with the main theorem

43
Arithmetic Lifting Theorem. Let$\phi(\tau, z)=$$f(ND, l) \exp(2\pi\dot{i}(\frac{N}{Q}\tau+\frac{l}{2}z))\in J_{k,R}^{cuS}\mathrm{P}(v\eta 24/Q\mathrm{x}v_{H}^{\epsilon})$$\sum_{N\equiv 1modQ}$

,

mod 2
$N>0,4NDt>l2$$l\equiv\epsilon$

where $Q=1,2,3,4,6$ or 12,
simplicity th at $t=QR\in \mathrm{N}$ and fix$\epsilon\equiv 2R$

mod 2, the weight is an integer. We assume for. Then the function$k$$\mu\in(\mathbb{Z}/Q\mathbb{Z})^{*}$$\mathrm{L}\mathrm{i}\mathrm{f}\mathrm{t}_{\mu}(\oint)(Z)$,$=$$(_{a|(,L,M)} \sum_{N}a-1D(k\frac{NMD}{a^{2}}v_{\eta}(\sigma a)f, \frac{L}{a}))\exp(2\pi\dot{i}(\frac{N}{Q}\tau+\frac{L}{2}Z+Mt\omega))$$\sum_{N,M>0}$$N,M\equiv\mu$

mod Q$L\equiv\epsilon mod2$

such that $\sigma_{a}\equiv$ mod
wherewith a chara cterto the paramodular group$\sigma_{a}\in SL_{2}(\mathbb{Z})$$Q$, is a cusp form of weight kwith respect$\Gamma_{t}$X$Qarrow\Gamma_{t}arrow\{Q\sqrt{1}If\mu=1,thenLift(\}_{o\mathrm{f}o}r_{\emptyset)(}der_{Z}QinduCedbyv)=Li\mathrm{f}t1(\emptyset)(z)\not\equiv\eta H24/Q\mathrm{o}for\phi \mathrm{x}v^{\epsilon}\not\equiv 0$

space$J_{k,t}^{cusp}(v^{D}\eta\cross v_{H}^{\epsilon})$, i.e. we have an embedding of theinto the space of Siegel modular forms.

Remarks. 1. If $QR$ is half-integral see Theorem 1.12 in [GN6].
2. For $\mu\neq 1$ the lifting could be zero for non-vanishing Jacobi forms. An examples of
non-zero -lifting see (1.11) bellow.
3. The construction of the arithmetic liftin.g in the case of orthogonal groups of signature$(2, n)$ see in [G3]. Its variant with a “commutator” character is similar to the theorem
above. See also [G3] for the case of non-cusp Jacobi forms.
$\mu$

.

-series of modular forms. We de.fine below the Siegel cusp
(the index denotes the weight). Let$\triangle$$\phi_{5},1/2(_{\mathcal{T},z})=\eta(\tau)^{9}\theta(\mathcal{T}, Z)$,$\phi_{2,1/2}(\tau, z)=\eta(\mathcal{T})^{3}\theta(_{\mathcal{T},Z})$

,

.rms

$\mathrm{f}\mathrm{o}$$\Delta_{1},$$\triangle_{2},$$\Delta_{11}$$\Delta_{5},$$\phi_{1,1/2}(_{\mathcal{T},z})=\eta(\mathcal{T})\theta(\tau, Z)$.

Then using the arithmetic lifting with $\mu=1$ we can define the following modular forms$\Delta_{5}=\mathrm{L}\mathrm{i}\mathrm{f}\mathrm{t}(\phi 5,1/2)\in \mathfrak{R}_{5}(\Gamma_{1}, \chi 2)$$\triangle_{2}=$Lift$(\phi_{2,1/2})\in \mathfrak{R}_{2}(\Gamma_{2,x4})$$\Delta_{1}=\mathrm{L}\mathrm{i}\mathrm{f}\mathrm{t}(\phi_{1},1/2)\in \mathfrak{R}_{1}(\Gamma_{1}, x6)$,,.

Moreover the lifting construction gives us the following Fourier expansions$\triangle_{1}(Z)=\sum$$M \geq 1\sum_{n,m>0,l\in \mathbb{Z}}(\frac{-4}{l})(\frac{12}{M})_{a}\sum_{|(n,\iota,m)}(\frac{6}{a})q^{n//2m}r^{l}S\in/2\mathfrak{R}_{1}6(\Gamma 3, x6)$

$n,m\equiv 1$

mod 6$4nm-3l^{2}=M^{2}$

(1.5)

44
and
$\triangle_{2}(Z)=\sum$$l \mathbb{Z}\sum_{\in}N(\frac{-4}{Nl})\sum_{a|(n,l,m)}(\frac{-4}{a})q^{n/4l}rs/2m/2\in \mathfrak{R}_{2}(\Gamma_{2}, x_{4})$(1.6)$N\geq 1n,m>0,$$n,m\equiv 1$

mod 4

.

$2nm-\downarrow^{2}=N2$

where$Z=\in \mathbb{H}_{2},$ $e(z)=\exp(2\pi\dot{i}z),$ $q=e(\tau),$$r=e(z),$ $s=e(\omega)$ .

(1.7)

Hence all primitive (i.e. with primitive matrices $N$ ) Fourier coefficients $a(N)$ of the cusp
formof weight one are equal toor (compare with the Fourier expansion of)
the Dedekind
According Igusa
is the first cusp form with respect to. One can prove (see(resp.
) is the first cusp form with respect to[G5] and [GH2] that
(resp. F3)
with trivial character. The first -cusp form of an odd weight (with trivial character) is
the modular form of weight 11$\pm 1$$\triangle_{1}(Z)$

$0$

$!$

$\eta- \mathrm{f}\mathrm{u}\dot{\mathrm{n}}\dot{\mathrm{C}}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}$$s_{p_{4}}(\mathbb{Z})$$\triangle_{5}^{2}$$\triangle_{2}^{4}$$)$$\triangle_{1}^{6}$$\Gamma_{2}$$\Gamma_{2}$

.

(1.8)

$\Delta_{11}(Z)=\mathrm{L}\mathrm{i}\mathrm{f}\mathrm{t}(\eta(\mathcal{T})^{2}.1\theta(\tau, 2Z))\in \mathfrak{R}_{11}(\mathrm{r}_{2})$$D$-series. Let us consider the Jacobi form of weight 1/2 and index 3/2 with the multipliersystem$v_{\eta}\cross v_{H}$(see (1.3))$\theta_{3/2}(\mathcal{T}, z)=\sum(n\in \mathbb{Z}\frac{12}{n})q^{n^{2}}/24rn/2\in J_{\frac{1}{2},\frac{3}{2}(v_{\eta}}\mathrm{x}v_{H})$.

This is the so-called quintiple product$\dot{\theta}_{3/2}(\mathcal{T}, Z)=q^{\frac{1}{24}}r-\frac{1}{2}\square (1+q^{n-}r)1(1+qr^{-1})n(1-n\geq 1)q^{21}-r^{2}(n1-q2n-1r-2)(1-q^{n})$. $(1.9)$

Then we can define two cusp forms of weight one and two with elementary Fourier coefficients$D_{1}(Z)=\mathrm{L}\mathrm{i}\mathrm{f}\mathrm{t}(\eta\theta_{3/}2)=$$\sum_{M\geq 1}\sum_{n,m>0,l\in \mathbb{Z}}(\frac{12}{Ml}\mathrm{I}\sum_{a|(n,l,m)}(\frac{-4}{a})q^{n/12}rsl/23m/2\in \mathfrak{R}_{1}(\Gamma_{18}, v_{\eta}^{2}\mathrm{x}v_{H})$$n,$ $m\equiv 1$,

mod 12$2nm-l^{2}=M2$(compare with the quintiple product) and$D_{2}(Z)=\mathrm{L}\mathrm{i}\mathrm{f}\mathrm{t}(\eta\theta_{3/}3)2=$.

$\sum_{N\geq 1}\sum_{n,m>0,\iota\in \mathbb{Z}}N(\frac{-4}{N})(\frac{12}{l})_{a|(}\sum_{)n,l,m}(\frac{6}{a})qrS/n/6l/23m2\in \mathfrak{R}_{2}(\Gamma_{9}, v_{\eta}4\mathrm{x}v_{H})$$n,$ $m\equiv 1$

mod 6$4nm-l^{2}=N^{2}$

45.

The first cusp forms for the paramodular groups:
that$\Gamma_{5},$$\Gamma_{6}$, F7,$\Gamma_{10}$. One can prove!

$\theta(_{\mathcal{T},aZ})\theta(\mathcal{T}, bz)\in J_{1}^{Cus},\frac{1}{2}(p+a2b^{2})(v\cross vH)\eta 6.\cdot$

is a cusp Jacobi form of weight one. It gives us an infinite series of Siegel cusp forms ofweight one
Lift$(\theta(\tau, az)\theta(\mathcal{T}, b\mathcal{Z}))\in \mathfrak{R}1(\mathrm{r}2(a+b^{2}), v_{\eta^{\mathrm{X}}}^{6}v2H)$

with a character of order 4. One can prove that
$F_{1}^{(10)}(Z)^{4}=\mathrm{L}\mathrm{i}\mathrm{f}\mathrm{t}(\theta(\mathcal{T}, Z)\theta(\mathcal{T}, 2Z))4\in \mathfrak{R}4(\mathrm{r}\mathrm{l}\mathrm{o})$with trivial character.
-cusp ‘form of the‘ minim:alis theFor the paramodular group F5 the first cusp form with trivial character has weight 5.
This is the lifting$\mathrm{w}\mathrm{e}\mathrm{i}\mathrm{g}\acute{\mathrm{h}}\mathrm{t}$$\Gamma_{10}$(.1.10)$F_{5}(5)(z)=\mathrm{L}\mathrm{i}\mathrm{f}\mathrm{t}(\eta(_{\mathcal{T})}3\theta(_{\mathcal{T},Z})6\theta(\tau, 2z))\in \mathfrak{R}_{5}(\mathrm{r}_{5}).$For$\Gamma_{6}$the first cusp form is the product of$\mathrm{t}\mathrm{w}\mathrm{o}$liftings(1.11)$F_{6}^{(6)}(z)=\mathrm{L}\mathrm{i}\mathrm{f}\mathrm{t}(\eta(\mathcal{T})5\theta(\mathcal{T}, 2z))\cdot \mathrm{L}\mathrm{i}\mathrm{f}\mathrm{t}_{2}(\eta(\tau)^{5}\theta(\tau, 2Z))\in \mathfrak{R}_{6}(\Gamma_{6})$is 2-lifting of the theorem above.
where
the first cusp form is the square of the lifting
For the group$\mathrm{L}\mathrm{i}\mathrm{f}\mathrm{t}_{2}$$\Gamma_{7}$$F_{2}^{(7)}(Z)^{2}=\mathrm{L}\mathrm{i}\mathrm{f}\mathrm{t}(\theta(\tau, z)3\theta(\tau, 2_{Z))^{2}}\in \mathfrak{R}_{4}(\Gamma_{7})$.

(1.12)In the next section we show that all these modular forms have an infinite product.

$\mathrm{e}\mathrm{x}\mathrm{p}\mathrm{a}\mathrm{n}\mathrm{s}\mathrm{i}_{0}\dot{\mathrm{n}}$Some other interesting examples of th.e arithmetic liftings with non-trivial commutator:
character see in [GN6] and [GH2].
the geoto. At this point \‘it isApplications to Algebraic
relatedtocanonical
closely
cusp
are
forms
metric implications of our results. Weight 3
modular variety. If is a cusp
corresponding
theofmodelssmoothondifferential forms$\omega_{F}=F(Z)dZ$
is a holomorphic 3-form
form of weight 3 with respect to a group , thenthreefold outside thetheconsider
we
that
means
, whereon the space. A very useful extension theorem
tobranch locus of the natural projection from.
due to E. Freitag implies that such a form can be extended to any smooth model of, which contains a principal
To be more precise, let be an arbitrary subgroup ofof some level . We then have the following
congruence subgroup$\mathrm{n}\mathrm{a}\mathrm{t}\mathrm{u}\mathrm{r}\mathrm{a}\dot{\mathrm{l}}$$\mathrm{G}\mathrm{e}\mathrm{o}\mathrm{m}’ \mathrm{e}\mathrm{t}\mathrm{r}\mathrm{y}$$\mathrm{d}\mathrm{i}_{\mathrm{S}}\mathrm{c}\mathrm{u}\mathrm{s}\mathrm{S}$$F$$\Gamma$$o$$A_{\Gamma}^{O}=(\Gamma\backslash \mathbb{H}_{2})^{o}$$\mathbb{H}_{2}$$A_{\Gamma}$$A\mathrm{r}$$s_{p_{4}}(\mathbb{R})$$\Gamma$$\Gamma_{1}(q)\subset Sp_{4}(\mathbb{Z})$$q$

Criterion. (Freit ) An elem ena anonic differential form on a non-singular model
is square integra $ble$ .
and only if the differenti form$t\omega_{F}=F(Z)dZ\in H^{0}\underline{(}A_{\Gamma}^{o},$$\mathrm{a}g$$c$$A\mathrm{r}$$\mathrm{a}l$$\mathrm{a}l$

Proof. See [F], Hilfsatz 3.2.1.

$\omega_{F}$an be extended toifof a compactifica tion of$\Omega_{3}(A_{\Gamma}^{o}))c$$A\mathrm{r}$46
It is well known that a -invariant differential form $\omega_{F}=F(Z)dZ$ is square-integrable
if and only if $F$ is a cusp form of weight 3 with respect to the group F. Thus we have thefollowing identity for the geometric genus of the variety:$\Gamma$$\tilde{A}\mathrm{r}$$p_{\mathit{9}}(\overline{A}\mathrm{r})=h^{3}’ 0(\overline{A}\Gamma)=\dim_{\mathbb{C}}\mathfrak{R}_{3}(\Gamma)$.

(1.14)In [G1] cusp forms of weight 3 with respect to the paramodular groupcharacter were constructed for all except$\Gamma_{t}$with trivial$t$$t=1,2,$$12,14,15,16,18,20,24,30,36$ .

$\ldots,$(1.15)

We call these polarizations exceptional. Thus we have

Non-Unirationality Theorem.
tions.

$A_{t}$

is not unira tional for all non-exceptional polariza-

Remark. A similar (but a little weaker) result is true in the case of moduli space of polarized$K3$ surfaces (see [G3]).
We also remark at this point that -cusp forms of weight 2 can be very useful when
one wants to prove that some modular threefolds are of general type (see [GS]). All thesefacts explain our interest in Siegel cusp forms of small weight $k(k\leq 3)$ .
One can formulate a standard conjecture:$\Gamma_{t}$$A_{t}$

is unirational for all exceptional polarizations.

In [GH2] it was proved that the geometrical genus ofis zero for $t\leq 8$ (see also [GP]).
Nevertheless as it was found in [GH2] the threefolds
have very natural modular coveringsof degree 2 or 3 with positive geometrical genus.
$A_{t}$$A_{t}$

Definition. Let be an arithmetic subgroup of
we define the threefold$\Gamma$$s_{p_{4}}(\mathbb{R})$$A(\chi \mathrm{r})=\mathrm{k}\mathrm{e}\mathrm{r}(\chi \mathrm{r})\backslash \mathbb{H}2$

The covering$A(\chi_{\Gamma})arrow A_{\Gamma}=\Gamma\backslash \mathbb{H}_{2}$. For any character$\chi \mathrm{r}$:$\Gammaarrow \mathbb{C}^{*}$.

is Galois with a finite abelian Galois group.

For example,
is the maximal abelian covering of the moduli space( is the commutator subgroup of ).
The next theorem is a direct corollary of the Arithmetic Lifting Theorem.
Theorem on Abelian Modular Coverings. (see, Corollary 1.6].) Let be oneof the exception polarizations.
(1) If $t\neq 1,2,4,5,8,16$ , then the modular double covering$A_{t}^{com}=\Gamma_{t}’\backslash \mathbb{H}_{2}$$\Gamma_{t}’$$A_{t}$$\Gamma_{t}$$[\mathrm{G}\mathrm{H}2$$t$$\mathrm{a}l$$A(\chi_{2})arrow.A_{t}2.1$

of the moduli spaceof abelian urfaces with polariza tion of type $(1, t)$ has
positive geometric genus, and in articular the $Kod$ aira dimension ofis notegative. Moreover, for $t=12,15,18,20,30$ and 36 the Kodaira dimension of thecorresponding
is positive.
$A_{t}$$s$$p$$A(\chi_{2})$$\mathrm{n}$$A_{t}(\chi)$(2) If $t=6,12,15,18,24,30,36$ , then the threefold $A_{t}(x_{3})arrow 31$ At

metric genus.

$h$as positive geo-(3) If $t=8$ or 16, then the covering $A_{t}(\chi_{4})arrow A_{t}41h$ as positive geometric genus.

47
In particular all these mod$\mathrm{u}l\mathrm{a}r$varieties are not unirational.

Moreover we have the following corollary about maximal abelian covering ofabove).
pare with Theorem on Non-Unirationality of$A_{t}$(com-$A_{t}$

Theorem on Maximal Abelian Coverings. Letof the $m$axim abelian
has geometric genus
of . Then$\overline{A}_{t}^{com}$$\mathrm{a}l$$A_{t}^{\mathrm{c}om}=\Gamma_{t}’\backslash \mathbb{H}_{2}$be a smooth projective model$co\mathrm{v}e\mathrm{r}in\underline{g}$$A_{t}^{com}$$A_{t}$$0$

if an only if $t=1,2,4$ or 5.
$d$

Let us consider the particular case of polarization of type $(1, 3)$ . The cube of the cusp
(see example above) defines a canonical differential form on the double coverform
thus $h^{3,0}(A3(x^{3}6))\geq 1$ . In fact we have equality!
whereofwith geometrical genus one. One can prove theIn [GH2] we found six coverings offollowing result$\Delta_{1}$$A_{3}(\chi_{6}^{3})$$\Delta_{1}^{3}\in \mathfrak{R}_{3}(\Gamma_{3}, x^{2}6)$$A_{3}$$A_{3}$

Threefolds with the Geometrical Genus One. Let
be two maximal abelian coverings. Then$h^{3,0}(A_{\mathrm{s}^{\circ m}}C)=1$$Rema\dot{r}k$,. One can prove that the threefolds$A_{3}^{Com}arrow A_{3}18\cdot 1$$h^{3,0}(A_{7}^{C}om)=1$$\overline{A_{3}}$and$\overline{A_{7}}$and$A_{7}^{com}arrow A_{7}21$.

are rational (see [G6])

We would like to put the following question
Question. How far are the threefolds$A_{3}^{com}$and$A_{7}^{com}$from$C\mathrm{a}l$a?

$bi-Y\mathrm{a}u.$\S 2. EXPONENTIAL OR BORCHERDS LIFTING
of modular forms, exponential or Borcherds
In this section we give anotherthe Humlifling, which gives us modular forms with Humbert divisors. In the case of
bert divisors are classical Humbert surfaces. For our purpose it is more convenient to consider Humbert surfaces as divisors of a double cover of the moduli space of $(1, t)$ -polarized
Abelian surfaces . To define this covering we consider a double normal extension of theparamodular group$\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{S}\mathrm{t}\mathrm{r}\mathrm{u}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}.\mathrm{n}$$Sp_{4}$$A_{t}$$\Gamma_{t}$$\Gamma_{t}^{+}=\Gamma_{t}\cup\Gamma_{t}Vt$,$V_{t}= \frac{1}{\sqrt{t}}$.

The double quotient$A_{t}arrow.A_{t}^{+}=2.1\Gamma t+\backslash \mathbb{H}_{2}$can be interpreted as a moduli space of lattice-polarized $K3$ surfaces for arbitraryor as the moduli space of Kummer surfaces of $(1,p)$ -polarized abelian surfaces for a prime, Theorem 1.5]).
$t=p$ (seeof discriminant can be represented in the form
Any Humbert surface in

of

$t$

$A_{t}$$[\mathrm{G}\mathrm{H}1$$A_{t}^{+}$$D$$H_{D}^{+}(b)= \pi^{+}t(g\in\bigcup_{\Gamma_{t}^{+}}g^{*}(\{Z\in \mathbb{H}_{2}|a\mathcal{T}+bZ+t\omega=0\}))$

48$D=b^{2}-4ta,$
mod and
where
(see
:
). Remarkdepends only
mod .
that
The datum for the exponential lifting is a nearly-holomorphic Jacobi form
$2t$

$b$

$a,$ $b\in \mathbb{Z},$$H_{D}^{+}(b)$$\pi_{t}^{+}$$\mathbb{H}_{2}arrow A_{t}^{+}$$[\mathrm{v}\mathrm{d}\mathrm{G}],$$[\mathrm{G}\mathrm{H}1]$$2t$$\mathrm{o}\mathrm{n}\pm b$(2.1)$\phi_{0,t}(\tau, z)=\sum_{n,l\in \mathbb{Z}}f(n, l)q^{nl}r\in J_{0_{)}t}^{nh}$and index . Nearly holomorphic means that there exist a number $m$ such thatis a weak Jacobi form. If we chose the minimal non-negative $m$ with thisproperty then $n\geq-m$ in the Fourier expansion (2.1). Bellow we use the notation (1.7).

of weight

$0$

$t$

$\triangle^{m}(\tau)\phi(\tau, Z)$

Exponential Lifting Theorem. (see [GN6]) Assume that the Fourier coefficients of
form
from (2.1) are in tegral. Then the product$\phi_{0,t}$$J\mathrm{a}cob,i$

,

(2.2)

$\mathrm{E}\mathrm{x}\mathrm{p}- \mathrm{L}\mathrm{i}\mathrm{f}\mathrm{t}(\phi_{0},t)(z)=B\phi(Z)=qrsABC(nn,’ ll,’ m\prod_{0m)>}\in \mathbb{Z}(1-q^{nl}rstm)f(nm,l)$

where$A= \frac{1}{24}\sum_{l}f(\mathrm{o}, l)$,$B= \frac{1}{2}\sum_{l>0}lf(0, l)$,$C= \frac{1}{4}\sum_{l}l^{2}f(0, l)$,

and $(n, l, m)>0$ means that if $m>0$ , then and are arbitrary integers, if $m=0$ ,or $l<0$ if $n=m=0$ , defin es a meromorphic modu 1 form ofthen $n>0$ andwith acter (or a multiplier system if the weightwith respect toweight. All divisors of-integral) induced byison
are theHumbert modular surfaces $H_{D}(b)$ of discrimin ant $D=b^{2}-4ta$ with multiplicities
$l$

$n$

$l\in \mathbb{Z}$$\frac{f(0,0)}{2}$$\mathrm{a}r$$ch\mathrm{a}r\mathrm{a}$$\Gamma_{t}^{+}$$h\mathrm{a}lf$$v_{\eta}^{24A}\cross v_{H}^{2B}$$\mathrm{E}\mathrm{x}\mathrm{p}- \mathrm{L}\mathrm{i}\mathrm{f}\mathrm{t}(\phi 0,t)(Z)$$m_{D,b}= \sum_{n>0}f(na, nb2)$$A_{t}^{+}$.

Moreover$B_{\phi}(V_{t}(z))=(-1)^{D}B\phi(Z)$with$D= \sum_{n<0,l\in \mathbb{Z}}\sigma_{1}..(-n)f.(n, l)$where $\sigma_{1}(n)=\sum_{d|n}d$ .
The infinite product expansion of the modular form
. We recall that there
exists the unique, up to a constant, weak Jacobi form of weight zero and index one$\triangle_{5}(Z)$$\phi_{0,1}(\tau, Z)=\frac{\phi_{12,1}(\mathcal{T},Z)}{\eta(_{\mathcal{T})^{24}}}=(r+10+r^{-1})+q(10r^{-2}-64r^{-1}+108-64r+10r^{2})+q^{2}($

...

$)$

where
is the unique Jacobi cusp form of weight 12 and index 1. There are several
formulae for Fourier coefficients of this Jacobi form. In [EZ] one can find a formula for
it in terms of Cohen’s numbers (values of special L-functions
Fourier coefficients ofat integral points). For a very convenient formula in terms of Hecke operators see,(3.34) . Using the theorem above for the function
we get the following result from
[GN1]:$\phi_{12,1}$$\phi_{12,1}$$[\mathrm{G}\mathrm{N}6$$])$$\phi_{0,1}$$\triangle_{5}(Z)=(qrs)^{1}/2\prod_{n,l,m\in \mathbb{Z}}(1-q^{n}rls^{m})^{f_{1(n}}m,l)\in \mathfrak{R}_{5}(\Gamma_{1}, x2)$$(n,l,m)>0$

where $f_{1}(n, l)$ are the Fourier coefficients of$\phi_{0,1}(\tau, Z)$.

49
The products expression for$\Delta_{1}$and$\triangle_{2}$. Let us define a weak Jacobi form$\emptyset 0,\mathrm{s}(\tau, z)=(\frac{\theta_{3/2}(_{\mathcal{T},z)}}{\eta(\tau)})^{2}=(\frac{\theta(\tau,2_{Z})}{\theta(\tau,z)})^{2}=\sum_{n\geq 0,l}f3(n, l)qnr^{\iota}$$=r^{-1}(_{n\geq 1}\square (1+q-1)n(r1+qr^{-1})n(1-q-1r^{2})(1-q-1r^{-})2n2n2)2$$=(r+2+r-1)+q(-4r\pm 3-4r+2\pm 2r+4\pm 1)+q^{2}($ .. .

$)$Then we have$\triangle_{1}(Z)=\sum$$\sum_{m>0,l\in \mathbb{Z}}(\frac{-4}{l})(\frac{12}{M})_{a|(m}\sum_{n,l,)}(\frac{6}{a})q^{n/6}r^{l/}sm2/2$$M\geq 1$$n,m\equiv 1$

mod 6$4nm-3l^{2}=M^{2}$$=q^{\frac{1}{6}}r^{\frac{1}{2}\frac{1}{2}}s \prod_{n,l,m\in \mathbb{Z}}(1-qrn\iota_{s^{3})}mf3(nm,l)\in \mathfrak{R}_{1}(\Gamma_{3}^{+}, \chi 6)$.

$(n,l,m)>0$

To get a similar formula forweight 2$\triangle_{2}$we consider the following holomorphic Jacobi form of$\phi_{2,2}(_{\mathcal{T},Z})=\frac{1}{2\pi\dot{i}}(3\frac{\partial\theta(\tau,z)}{\partial z}\theta 3/2(\tau, z)-\frac{\partial\theta_{3/2}(_{\mathcal{T},z)}}{\partial z}\theta(\mathcal{T}, Z))$.

$= \frac{1}{2}\sum_{m,n\in \mathbb{Z}}(3m-n)(\frac{-4}{m})(\frac{12}{n})q^{\frac{3m^{2}+n^{2}}{24}}r^{\frac{m+n}{2}}\in J_{2,2}(v_{\eta}^{4}\cross \mathrm{i}\mathrm{d}_{H})$Moreover the divisor of these modular forms is the irreducible Humbert surface
corresponding moduli space$\mathrm{D}\mathrm{i}\mathrm{v}_{A_{1}}(\Delta 5(Z))=H1$,$\mathrm{D}\mathrm{i}_{\mathrm{V}_{A_{2}^{+}}}(\Delta_{2}(Z))=H1$,$H_{1}$$\mathrm{D}\mathrm{i}\mathrm{v}A_{3}^{+(}\triangle_{1}(Z))=H1$

in the.

One can show that these -functions are discriminant of some moduli space of $K3$ surfaces
with a condition on the Picard lattice (see [GN3]).
$\Delta$. Now we define two special singularandThe singular modular forms. The first form isandmodular forms with respect to the full paramodular grouptheta-constanteven
one of the classical Siegel theta-constants. The “most odd”$\triangle_{1/2}$$D_{1/2}$$\Gamma_{4}$$\Gamma_{36}$,$\ominus_{1,1}(Z)=\frac{1}{2}\sum_{\mathbb{Z}l_{1},,l_{2}\in}\exp(\pi\dot{i}(Z [ l+\frac{1}{\frac{21}{2}}l_{2}+]1+l_{1}+l_{2}))=\frac{1}{2}\sum_{n,m\in \mathbb{Z}}(\frac{-4}{n})(\frac{-4}{m})q^{n^{2}//}r^{n}s^{m}8m4/28$

where $Z[M]={}^{t}MZM$ , is a modular form with respect to the subgroup(see [GN6]). Our second example is a modular
conjugated to the paramodular group$\Gamma_{2,1}\subset s_{p_{4}}(\mathbb{Z})$$\Gamma_{4}$50form of weight 1/2 with respect to the full paramodular group
Jacobi form$\Gamma_{36}$. Firstly we define a$\emptyset 0,4(\tau, Z)=\frac{\theta(\tau,3_{Z})}{\theta(\tau,z)}=\sum n\geq 0,$$\iota\in \mathbb{Z}f4(n, l)qrnl$$=r^{-1} \prod_{m\geq 1}(1+q^{m-1}r+q-2r)2m2(1+q^{m}r^{-}+q^{2}r-2)1m\prod_{n\geq 1}(1-q^{n3}r)(1-qr)n\equiv 1,2mod3n-3$$=(r+1+r^{-1})-q(r^{4}+r^{3}-r+2-r^{-1}+r^{-3}+r^{-4})+q^{2}(\ldots)$(2.10)

where all Fourier coefficients $f_{4}(n, l)$ of the weak Jacobi form are integral (in fact they are). Moreover we get the following
Fourier coefficients of automorphic forms ofidentity for the exponential lifting of$\mathrm{w}\mathrm{e}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t}-1/2$$\phi_{0,4}$$\Delta_{1/2}(z)=\frac{1}{2}\sum_{n,m\in \mathbb{Z}}(\frac{-4}{n})(\frac{-4}{m})q^{n^{2}/}rs=8nm/2m^{2}/2$$q^{1/8}rs \prod_{>}1/21/2(1-q^{n}r^{\iota}s^{4})(n,l,m)0mf_{4(}nm,l)\in \mathfrak{M})1/2(\Gamma_{4}, x8)$of order 8 induced by the multiplier system
is a multiplier system ofwhereof the Jacobi group. The divisor of this theta-series is again$\Gamma_{4}$$\chi_{8}$$v_{\eta}^{3}\cross v_{H}$$H_{1}$$\mathrm{D}\mathrm{i}\mathrm{v}_{A_{4}}(\triangle 1/2(Z))=H1$

To define the form

$D_{1/2}$.

we determine a Jacobi form$\phi_{0,3}6(\mathcal{T}, Z)=\frac{\theta_{3/2}(\mathcal{T},5_{Z})}{\theta_{3/2}(_{\mathcal{T},z)}}=\frac{\theta(_{\mathcal{T},10}z)\theta(_{\mathcal{T}},Z)}{\theta(\mathcal{T},5z)\theta(\tau,2_{Z})}=\sum_{n\geq 0,l\in \mathbb{Z}}f_{36(n},$

$l)$

qrnl

$=r^{-2} \prod_{n\geq 1}\frac{(1+q^{n}-1r)5(1+qr^{-5})n(1-q-110)2n(r1-q-r-10)2n1}{(1+q^{n-1}r)(1+qr^{-1})n(1-qr2n-12)(1-q^{2}r^{-2})n-1}$$=(r^{2}-r^{1}+1-r^{-1}+r^{-2})+q^{2}(-r^{17}+\ldots)+q^{5}(r^{27}+\ldots)$$+q^{7}(r^{32}+\ldots)+q^{8}(r+\ldots)34+\ldots$,with the negative norm $144n-l^{2}$ :where we include in the last formula only summands
144 $\cdot 2-17^{2}=-1,144\cdot 5-27^{2}=-9,144\cdot 7-32^{2}=-16,144\cdot 8-34^{2}=-4$. Then one
can prove the following identity$q^{n}r^{l}$$\frac{1}{2}\sum_{m,n\in \mathbb{Z}}(\frac{12}{n})(\frac{12}{m}\mathrm{I}^{q^{n^{2}/nm}s^{3/2}=}24/2m1/24/23/2\prod_{0(n,\iota,m)>}(rqr^{1}s1-2qrsnl36m)f36(nm,l)$.

The divisor of this modular form is more complicated, but this is still a reflective modular
form:$\mathrm{D}\mathrm{i}\mathrm{v}_{A_{36}^{+}}(D_{1/2}))=H_{4}(2)+H_{4}(34)+H_{9}(27)+H_{16}(32)$.

The example of modular forms given above give us some evidence for the following
conj ecture

51
of modular form
Conjecture C. Let $t>1$ be $sq$ uare free. Then the spaceof the paramodularof the minimal weight with respect to the commutator subgroupwords,
we can choose a Humbertcont ains a Humbert modular form. In other
group
$tor$
.
of the minimal weight of the graded ring
modular form as a genera$\mathfrak{M}_{\min}(\Gamma_{t}’)$$\Gamma_{t}’$$\Gamma_{t}$$\mathfrak{M}_{*}^{(1)}(\Gamma_{t}’)$We can formulate the same conjecture for arbitrary , but then we should admit modular
forms of half-integral weight.
Using considerations from the last section of [GH2] we can prove rather important
property of modular forms of type .
$t$$\Delta$

Uniqueness Theorem. The modular forms
with the divisor
ular forms with respect toone).

$\Delta_{5},$$\Gamma_{t}^{+}$$\Delta_{2},$$equ\mathrm{a}l$$\Delta_{1},$$\Delta_{1/2}$

are the only Siegel mod-exactly to$H_{1}^{+}$(with multiplicity

The result about the divisor of modular forms of -type gives us a particular answer
on Problems A and formulated in \S 1. As corollary we get
Rationality Theorem K. The moduli space of Kummer surfaces associa $ted$ with $(1, 2)-$ ,$(1, 3)- or(1,4)$ -polarized Abeli an surfaces is ration al.
$\Delta$$\mathrm{B}$Remark. For $t=2$ this result was obtained by Freitag (see [F1]) and Ibukiyama (see [Ib])by other methods.
Using Theorem on Exponential Lifting one can find divisor of the first cusp forms forfrom the example above. It gives us rationality of the corresponding, F7 andF5,
moduli spaces (see [G6]).
$\Gamma_{6}$$\Gamma_{10}$All modular forms constructed above are symmetric, i.e. invariant with respect to theexterior involution . To prove rationality of the moduli space of Abelian surfaces weshould construct modular forms anti-invariant with respect to .
Anti-symmetric modular forms. Using the exponential lifting, one can construct antiinvariant modular forms, i.e. forms satisfying $F(V_{t}(z))=-F(Z)$ . For example for $t=1$
we have the anti-invariant form $F(Z)=\triangle_{35}.(z)$ . This is so called Igusa modular forms.
.
In [GN4] we proposed a new construction of this form as a Hecke product of. Using the systeminwhich has the divisor
Let us take the modular form
$T(p)$
get,we thenof representatives$V_{t}$$V_{t}$$\triangle_{5}(Z)$$H_{1}$$\Delta_{5}(Z)$$[\triangle_{5}(Z)]\tau(2)=$$\prod\Delta_{5}(\frac{z_{1}+a}{2},$$\frac{z_{2}+b}{2},$$\frac{z\mathrm{s}+c}{2})$$\prod$$\Delta_{5}(\frac{z_{1}+a}{2},$$A_{1}$$z2,2_{Z)}3\Delta_{5}(2_{Z_{1}},$ $z_{2},$$\frac{z_{3}+a}{2})$amod 2$a,b,c$ mod 2$\cross\Delta_{5}(2_{Z_{1}},2z_{2},2z3)bmod\prod_{2}\Delta 5(2_{Z_{1}}, -z_{1}+z2, \frac{z-2z+z+b}{2})$One can check that$\mathrm{d}\mathrm{i}\mathrm{v}_{A_{1}}([\triangle_{5}(Z)]_{T(2}))=9H_{1}+H_{4}$$\triangle_{35}(Z)=\frac{[\Delta_{5}(Z)]_{\tau}(2)}{\triangle_{5}(Z)^{8}}=\mathrm{E}\mathrm{x}\mathrm{p}$-Lift. Thus$(\phi_{0,1}|(T\mathrm{o}(2)-2))\in \mathfrak{R}_{35}(\mathrm{r}_{1})$and$\Delta_{35}(z)=q^{2}rs^{2}(q-s)\prod_{n,l_{)}m\in \mathbb{Z}}(1-q^{n}r^{\iota}S^{m})f_{1}^{()}(24nm-\iota^{2})$$(n,l,m)>0$.

52
where $f_{1}(4n-l^{2})=f_{1}(n, l)$ are the Fourier coefficients of$\phi_{0,1}(\tau, Z)$and$f_{1}^{(2)}(N)=8f_{1}(4N)+2(( \frac{-N}{2})-1)f_{1}(N)+f_{1}(\frac{N}{4})$Remark that we cannot constructform. Nevertheless (3.29) gives us$\triangle_{35}(z)$$\triangle_{5}(Z)$.

as an arithmetic lifting of a holomorphic Jacobi
as a finite Hecke product of the lifted form$\Delta_{35}(Z)$.

A similar construction gives us the unique anti-symmetric modular forms for , F3 andof weight 12. The uniqueness will follow from the fact that these modular forms have
only the Humbert surface $H_{4t}(0)=\pi_{t}\{\tau-t\omega=0\}$ as their divisor.
where
Let us consider the function$\Gamma_{2}$$\Gamma_{4}$$\psi_{0,t}(\mathcal{T}, Z)=\triangle(\tau)-1E12,t(\mathcal{T}, z)$$\triangle(\tau)=q\prod_{\geq n\mathrm{i}}(1-q)n24-=q24q^{2}+253q.+\backslash \mathrm{s}\ldots$

is a Jacobi-Eisenstein series of weight 12 and index .
andin terms of H.
There exists a formula for Fourier coefficients ofnumbers(seeand\S 2]). One can find the table of the values of Fourier coefficients ofand. Using the basic Jacobi forms
we definein$t$$E_{12,t}(\mathcal{T}, Z)$$\mathrm{C}\mathrm{o}\mathrm{h}\mathrm{e}\mathrm{n}’\dot{\mathrm{s}}$$E_{k,1}$$E_{4,1}(\mathcal{T}, Z)$$[\mathrm{E}\mathrm{Z},$$E_{6,1}(\mathcal{T}, Z)$$\phi_{0,1},$$[\mathrm{E}\mathrm{Z}, \S 1]$$\phi_{0,3}(\tau, Z)$$\phi_{0,2}$$\psi_{0,2}(_{\mathcal{T}}, z)=\triangle(\mathcal{T})-1E_{6,1}(\mathcal{T}, z)2-2\phi_{0,1}2+216\phi 0,2(\tau, z)$$= \sum_{\in n\geq 0,l\mathbb{Z}}c_{2}(n, l)qrnl=q^{-1}+24+q(\ldots)$,$\psi_{0},\mathrm{s}(_{\mathcal{T},Z})=\triangle(\mathcal{T})-1E4,1(_{\mathcal{T},Z})3-3\phi 0,\mathrm{s}|\tau_{0}(2)(\tau, Z)-180\phi_{0,3}(_{\mathcal{T},Z})$$= \sum_{n\geq 0,l\in \mathbb{Z}}C_{3}(n, l)qr=q-1+nl24+q(\ldots)$.

.

$\psi_{0_{\mathrm{h}}},p(p=2)_{\mathrm{C}}\mathrm{o}\mathrm{n}\mathrm{t}\mathrm{a}\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{o}_{\mathrm{T}}\mathrm{n}1\mathrm{y}\mathrm{t}\mathrm{y}\mathrm{p}\mathrm{e}\mathrm{o}\mathrm{f}\mathrm{F}\mathrm{o}\mathrm{u}\mathrm{r}\mathrm{i}\mathrm{e}\mathrm{r}_{\mathrm{O}}\mathrm{C}\mathrm{o}\mathrm{e}\mathrm{f}\mathrm{f}\mathrm{i}\mathrm{c}\mathrm{i}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{S}\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}\mathrm{i}\mathrm{n}\mathrm{o}\mathrm{f}\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}4p.\mathrm{h}\mathrm{u}\mathrm{s}\mathrm{W}\mathrm{e}\mathrm{c}\mathrm{a}\mathrm{n}\mathrm{u}\mathrm{S}\mathrm{e}\mathrm{b}\mathrm{t}\mathrm{h}\mathrm{f}\mathrm{u}\mathrm{n}\mathrm{C}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{o}\mathrm{P}^{\mathrm{r}}\mathrm{o}\mathrm{d}\mathrm{d}\mathrm{u}\mathrm{i}_{\mathrm{C}\mathrm{e}\mathrm{s}}\mathrm{C}\mathrm{e}$$\mathrm{o}\mathrm{f}\mathrm{n}\mathrm{e}\mathrm{T}\mathrm{h}\mathrm{e}\mathrm{J}\mathrm{a}\mathrm{c}\mathrm{o}\mathrm{b}\mathrm{g}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{e}\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{T}\mathrm{i}_{\mathrm{S}\mathrm{i}}\mathrm{S}q-\mathrm{i}^{3}-\mathrm{i}\mathrm{f}_{0}\mathrm{r}\mathrm{m}\mathrm{S}$the exponential liftings,$\Psi_{12}^{(2)}(z)=\mathrm{E}\mathrm{X}\mathrm{p}- \mathrm{L}\mathrm{i}\mathrm{f}\mathrm{t}(\psi_{0,2})=q\prod_{n,l,m\in \mathbb{Z}}(1-qrsnl2m)C_{2(nm,l)}\in \mathfrak{M}_{12}(\Gamma_{2})$$(n,i,m)>0$.
$\Psi_{12}^{(3)}(Z)=\mathrm{E}_{\mathrm{X}}\mathrm{P}- \mathrm{L}\mathrm{i}\mathrm{f}\mathrm{t}(\psi_{0,3})=q,\square (1-q^{n}r^{l}s^{3m})\mathrm{C}3(nm,l)(nn,l,m)>0l,m\in \mathbb{Z}\in \mathfrak{M}_{12}(\Gamma_{3})$According to Theorem on Exponential lifting$\Psi_{12}^{(p)}(V_{p}<Z>)=-\Psi_{12}^{()}p(z)$$(p=2,3)$

and$H_{8}$$\mathrm{D}\mathrm{i}\mathrm{v}_{A_{\mathrm{p}}}(\Psi_{12}^{(p)})=\{$$H_{12}$The Fourier-Jacobi expansion of$\Psi_{12}^{(p)}$starts with coefficients$\Psi_{12}^{(p)}(z)=\triangle 12(\tau)-\triangle 12(\tau)\psi 0,p(\tau, Z)\exp(2\pi\dot{i}p\omega)+\ldots$.

for $p=2$
for $p=3$ .

53Therefore the constructed modular forms$\Psi_{12}^{(p)}(z)(p=2,3)$

are not cusp forms.

in terms of theta-constants was found r\‘ecently in [IO].
Remark. An expression for. Let usIf we do the same for $t=4$ , we get a Jacobi form we used to constructtake the Jacobi form$\Phi_{12}^{(2)}$$\triangle_{35}(Z)$$\phi_{0,1}|(\tau 0(2)-2)(\tau, 2z)=q-1+(r+704+r-4)+q(\ldots)$.

16. To delete
Its exponential lifting is zero along two Humbert surfaces with discriminant
has thewhich
seriesthe second component, we consider the additional Jacobi-Eisensteinsquare). For
constant term equals zero (such a series exists if the index contains a perfect.
-Eisenstein series is the eight power of the Jacobi theta-series$t=4$ this, we define
Using$\theta(\tau, z)$$\mathrm{J}\mathrm{a}\mathrm{c}\mathrm{o}\mathrm{b}\mathrm{i}$$\theta(\tau, z)^{8}$$\psi_{0,4}(\tau, Z)=(\phi_{0,1}|(\tau 0(2)+26))(\mathcal{T}, 2_{Z)}-\triangle(\tau)-1E4(_{\mathcal{T})\theta}(\tau, Z)8-8(\phi_{0},4|(\tau_{0}(3)+4))(\tau, Z)$$= \sum_{\in n\geq 0,l\mathbb{Z}}C_{4}(n, l)q^{n}r=q.+24+q(\ldots)l-.1$.

contains only the Fourier coefficients of typethe Jacobi form
andSimilarly tothewith index of negative norm. Taking its exponential lifting we obtainform of weight 12$\psi_{0,2}$$\psi_{0,4}$$\psi_{0,3}$$\Gamma_{4^{-\mathrm{m}\mathrm{o}}}\mathrm{d}\mathrm{u}\mathrm{l}\mathrm{a}\mathrm{r}$$q^{-1}$$\Psi_{12}^{(4)}(z)=\mathrm{E}\mathrm{X}\mathrm{p}-\mathrm{L}\mathrm{i}\mathrm{f}\mathrm{t}(\psi 0,4)(z)=qn,l,m\in \mathbb{Z}\square (1-qr)n\iota_{s}4mc_{4}(nm,l)\in \mathfrak{M}_{12}(\Gamma_{4})$.

$(n,l,m)>0$$\Psi_{12}^{(4)}(Z)$is anti-invariant and$\mathrm{D}\mathrm{i}\mathrm{v}_{A_{4}}(\Psi)(4)H8(120=)$

The construction of the modular forms
visor gives us$\Psi_{12}^{(t)}$.

together with information about their di-

Rationality Theorem A. The moduli space of $(1, 2)-,$ $(1,3)-or(1,4)- pol\mathrm{a}riZed$ Abeliansurfaces is tional.
ratioIn March, 1997, a new preprint of Ibukiyama and Onodera [IO] where the$r\mathrm{a}$Remark.

nality of$A_{2}$

was also proved by another method.

were used in [Ka2] and in [CCL],
andThe pair of modular forms[C] to define mirror symmetry for heterotic and IIA strings.
between
Applications to Lorentzian Kac-Moody algebras. Almost all identities
for Lorentarithmetic and exponential liftings mentioned above give denominator formulae
Thesedetails).
for morezian Kac-Moody algebras (see [GN1] and speciallycorrespondence bealgebras are generalized Kac-Moody superalgebras. Bellow we give a
hyperbolic type which
tween constructed modular forms and generalized Cartan matrices ofdescribe the set of real simple roots of the algebras under construction.
$(\Delta_{5}, \Delta_{35})$$(\Delta_{2}, \Psi_{12}^{(2)})$$[\mathrm{G}\mathrm{N}5]^{-}[\mathrm{G}\mathrm{N}6]$

54
We denote by
and with the Gram matrix$M_{t}=U(4t)\oplus\langle 2\rangle(t\in \mathrm{N}/4)$

a hyperbolic lattice with the bases

We denote an element $nf_{2}+lf_{3}\wedge+mf_{-2}$ of this lattice (or of$(n, l, m)$$M_{t}\otimes \mathbb{Q}$.

$f_{2},$$f_{-2}\wedge$$f3,$) by its coordinates

All our formulae have the form:$\Phi(Z)=(n,\iota_{m)},\sum_{\in M_{t}}N(n, l, m)q^{\rho_{1}+n}r^{\rho 2}S+lt(\rho 3+m)=$$= \sum_{w\in W}\epsilon(w)\sum_{M_{t}a\in \mathbb{R}_{++}\mathcal{M}\mathrm{n}}(e+w(\rho)N(a)e^{w(\rho})+a)=q\rho 1\rho_{2}rSt\rho 3(n,l,m\prod_{)>0}(1-q^{nl}rS)^{f}tm(n,l,m)$

for some $W,$
which we define below. The “exponent” $e^{()}n,l,m$ $:=$. The Weyl group $W$ is a reflection subgroup $W\subset W(M_{t})\subset O(M_{t})$ , andisa fundamental polyhedron of $W$ in the hyperbolic space
defined by the hyperbolic.
latticethe automorphic point of view the Weyl group $W$ is defined by themodular property of the modular form .
The elements together with coefficients $N(a)$ at the left hand side of the denominator
formula form a multi-set of imaginary simple rootsof the constructed generalized
Kac-Moody algebra (one should take every exactly $|N(a)|$ -times in this multi-set). The-graduation of the superalgebra is defined by the sign of the Fourier coefficients $N(a)$ .
Then the Siegel modular forms are related with the following generalized Cartan matrices, equivalently, hyperbolic root systems
1. The product of all even theta-constant–the right triangle with vertices at infinity(i.e the fundamental polygon of the Weyl group is a right triangle).
$\mathcal{M},$$\rho=(\rho_{1}, \rho_{2}, \rho_{3})$$q^{n}r^{\iota_{s}t}m$$\mathcal{M}_{t}$$\mathcal{L}(M_{t})$$M_{t}$$i^{\mathrm{F}\mathrm{r}\mathrm{o}\mathrm{m}}$$\Phi$$a$$s\Delta^{im}$$a$$\mathbb{Z}_{2}$$\Delta_{5}$$A_{2,II}=$
2. The Igusa modular form$\Delta_{35}$–a right triangle with two vertices at infinity.

$A_{1,0}=$ .
This is the “simplest” hyperbolic Cartan matrix (see [FF]).
3. The form
–the right square with all vertices at infinity.
$\Delta_{2}$$A_{2,II}=$ .

55
4. The form$\Delta_{1}$–the right hexagon with all vertices at infinity.

$-16-18–8028$$A_{2,III}=(_{-14}^{-2}$See$[\mathrm{G}\mathrm{N}5]-[\mathrm{G}\mathrm{N}6]$$-16-1-1-2-8-80_{4}28$$-18-16-14-8-2-028$$-16-1-18-2-8-80_{4}2$$-14-18-1-8-2-0826$$-18-16-14-8-2-802$$-1-16-14-8-8-0228$$-16-18-1-8-2-80_{4}2$).

for more information.
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ST. PETERSBURG DEPARTMENT OF STEKLOV MATHEMATICAL INSTITUTE,
FONTANKA 27, 191011 ST. PETERSBURG, RUSSIA$E$-mail address: gritsenk@mpim-bonn.mpg.de