PUT A SHORT VAGUE DESCRIPTION OF SIEGEL MODULAR FORMS HERE.

Contents

  1. Remarks on notation
  2. Definitions
    1. Congruence subgroups
    2. Weights
    3. Definition of Siegel modular forms
  3. Fourier expansions and Fourier-Jacobi expansions
  4. Rings of modular forms
  5. Quadratic forms and theta series
  6. Cusp forms
  7. Dimension formulas
  8. Hecke Theory
  9. $L$-functions
  10. Liftings
  11. Well-known conjectures
  12. Applications
    1. Algebraic number theory
    2. Arithmetic geometry
    3. Cohomology of some arithmetic groups
    4. Coding theory
    5. Elliptic cohomology
    6. Physics
  13. Computational Aspects
    1. Databases for Degree 2 Level 1
    2. Software for Degree 2 Level 1
  14. References
    1. Books
    2. Survey Papers
    3. Survey Talks
  15. Links

Remarks on notation

We denote the transpose of a matrix A by A^t. We denote the symplectic group of 2g\times 2g matrices by Sp_{2g} (caveat: this is sometimes denoted Sp_g).

Definitions

Let g\geq 1 be an integer (referred to as the degree, or the genus).

Set

J:=\begin{pmatrix}0&I_g\\-I_g&0\end{pmatrix}.

The symplectic group over \mathbb{Z} is defined by

\Gamma_g:=Sp_{2g}(\mathbb{Z}):=\{\gamma\in GL_{2g}(\mathbb{Z}):\gamma^t J\gamma=J\}=\left\{\begin{pmatrix}A&B\\C&D\end{pmatrix}\in GL_{2g}(\mathbb{Z}):A^tC=C^tA,B^tD=D^tB,A^tD-C^tB=I_g\right\},
where A, B, C, D are g\times g blocks.

The Siegel upper half space is defined by

\mathcal{H}_g:=\{\text{symmetric $g\times g$ complex matrices with positive definite imaginary part}\}=\{\tau\in Mat_{g\times g}(\mathbb{C}):\tau^t=\tau,\Im(\tau)>0\}.

If \tau\in\mathcal{H}_g and \gamma=\begin{pmatrix}A&B\\C&D\end{pmatrix}\in\Gamma_g, then the g\times g matrix (C\tau+D) is invertible (see Proposition 1, Section 1 of \cite{Klingen1990}). One can therefore define an action of \Gamma_g on \mathcal{H}_g by "linear fractional transformations":

\begin{pmatrix}A&B\\C&D\end{pmatrix}\tau:=(A\tau+B)(C\tau+D)^{-1}.

In the classical (g=1) case, some mileage can be obtained from working explicitly with a fundamental domain for the action of \Gamma_1=SL_2(\mathbb{Z}) on \mathcal{H}_1. In the general case, such an approach would be very cumbersome: although fundamental domains were constructed by Siegel for all g, already for g=2 their boundary is made of 28 algebraic surfaces (see C.L. Siegel: Symplectic geometry, Amer. J. Math. 65 (1943), pp. 1-86 and E. Gottschling: Explizite Bestimmung der Randflächen des Fundamentalbereiches der Modulgruppe zweiten Grades, Math. Ann. 138, 1959, pp. 103-124).

Congruence subgroups

A congruence subgroup is a subgroup \Gamma of finite index of \Gamma_g=Sp_{2g}(\mathbb{Z}). Here are some examples:

One defines

\Gamma_g(N):=\{\gamma\in\Gamma_g:\gamma\equiv I_{2g}\pmod{N}\}.
By Serre's Lemma, if N\geq 3 then \Gamma_g(N) acts freely on \mathcal{H}_g (see \cite{Serre1961}).

The level of \Gamma is the smallest integer N such that \Gamma\subset\Gamma(N).

Weights

The weight of a classical modular form is an integer. The weight of a Siegel modular form will be a rational representation

\rho: GL_g(\mathbb{C})\to GL(V),
where V is a finite-dimensional complex vector space. It suffices to consider only irreducible representations.

* The case g=1

In the classical case, the weights are irreducible representations of \mathbb{C}^\times, which are one-dimensional (because \mathbb{C}^\times is abelian), and since they are by assumption rational the only possibilities are

z\mapsto z^k
for some integer k. Hence one recovers precisely the notion of weight of a classical modular form.

* The case g=2

Let std denote the standard representation of GL_2(\mathbb{C}), i.e. std is the identity map from GL_2(\mathbb{C}) to GL(\mathbb{C}^2). To any pair of integers (k_1,k_2) one associates the following representation of GL_2(\mathbb{C}):

\rho_{k_1,k_2}:=Sym^{k_1}(std)\otimes\det(std)^{k_2}.
These are actually all the irreducible representations of GL_2(\mathbb{C}).

* General g

Irreducible representations of GL_g(\mathbb{C}) are classified by g-tuples of integers, called highest weight vectors (see, for instance, \cite{Fulton1991}).

Definition of Siegel modular forms

To simplify the notation, one uses the slash operator: given \gamma\in\Gamma_g, \rho:GL_g(\mathbb{C})\to GL(V), and a function F:\mathcal{H}_g\to V set

(F|_\rho\gamma)(\tau):=\left(\rho(C\tau+D)\right)^{-1}F(\gamma\tau).
A Siegel modular form of degree g, weight \rho, on the group \Gamma is a holomorphic function F:\mathcal{H}_g\to V such that
F|_\rho\gamma=F\quad\text{for all }\gamma\in\Gamma_g.
If g=1, one must also impose the condition of holomorphicity at infinity: there exists \epsilon>0 such that
|F(x+iy)|\ll y^\epsilon\quad\text{for }y\geq 1,
where x+iy=\tau. This condition is automatically satisfied when g>1, by the Koecher principle (see Theorem 1, Section 4 of \cite{Klingen1990}).

One denotes the complex vector space of Siegel modular forms of weight \rho and level \Gamma by M_\rho(\Gamma). Note that a Siegel modular form of degree g is a holomorphic function in g(g+1)/2 complex variables.

If a weight \rho decomposes as a direct sum of representations

\rho=\rho_1\oplus\rho_2,
then it is easily seen that
M_\rho=M_{\rho_1}\oplus M_{\rho_2}.
Therefore it suffices to only work with weights that are irreducible representations.

Scalar-valued Siegel modular forms are obtained by restricting to weights of the form

\rho=\det(std)^k,
where k is an integer.

This gives rise to two graded rings:

M^{\text{scalar}}(\Gamma):=\bigoplus_{k\in\mathbb{Z}} M_k(\Gamma)\quad\text{and}\quad M(\Gamma):=\bigoplus_{\rho\text{ irrep}} M_\rho(\Gamma).

Fourier expansions and Fourier-Jacobi expansions

Consider the following 2g\times 2g matrix:

\gamma=\begin{pmatrix}I&S\\0&I\end{pmatrix},
where S is a symmetric g\times g matrix with integer entries. It is easily seen that \gamma\in\Gamma_g.

If F is a Siegel modular form, it satisfies

F(\tau+S)=F(\tau)\quad\text{for all }\tau\in\mathcal{H}_g,
so F is a periodic function in its g(g+1)/2 variables \tau_{ij}, where \tau=(\tau_{ij}).

Setting q_{ij}=e^{2\pi i\tau_{ij}}, one obtains a multivariate q-expansion of the form

F(q_{11},\ldots,q_{gg})=\sum_{n_{11},\ldots,n_{gg}\in\mathbb{Z}} a(n_{11},\ldots,n_{gg}) q_{11}^{n_{11}}\ldots q_{gg}^{n_{gg}}.
This notation obscures some of the features of the Fourier expansion, so the following alternate notation is often used.

Let N be a symmetric g\times g matrix; we say N is half-integral if 2N has integral entries with even integers on the diagonal. Given a half-integral matrix N, and \tau\in\mathcal{H}_g, one easily checks that

Tr(N\tau)=\sum_{i=1}^g N_{ii}\tau_{ii}+2\sum_{1\leq i < j\leq g}N_{ij}\tau_{ij}.
Running over all half-integral matrices N will yield all possible linear combinations of the \tau_{ij}'s with integer coefficients, so the Fourier expansion can be written as
F(\tau)=\sum_{N\geq 0\text{ h.-i.}}a(N)e^{2\pi i Tr(N\tau)},
which is equivalent to but more compact than~(\ref{eq:q}), and is reminiscent of classical q-expansions. The fact that only N\geq 0 are needed is due to the Koecher principle (or to the holomorphicity at infinity in the case g=1).

One can recover the Fourier coefficients using the formula

a(N)=\int_{x\text{ mod }1}F(\tau)e^{-2\pi i Tr(N\tau)}dx,\quad\tau=x+iy,
where dx is the standard measure on \mathbb{R}^g.

If u\in GL_g(\mathbb{Z}), then

a(u^tNu)=\rho(u^t)a(N).
This can be used to show that M_k=0 if kg\equiv 1\pmod{2}.

Rings of modular forms

generators, etc.

Quadratic forms and theta series

Cusp forms

FIX THIS SECTION SO IT MAKES SENSE FOR ARBITRARY GROUPS \Gamma AND VECTOR WEIGHTS.

We define the Siegel \Phi-operator

\Phi:M_k(\Gamma_g)\to M_k(\Gamma_{g-1})
by the rule
\Phi F(\tau'):=\lim_{t\to\infty} F\begin{pmatrix}\tau'&0\\0&it\end{pmatrix}\quad\text{for }\tau'\in\mathcal{H}_{g-1}.
This is a well-defined linear map (see Proposition 1, Section 5 in \cite{Klingen1990}).

In terms of the Fourier expansion, if F can be written as

F(\tau)=\sum_{N\geq 0\text{, h.-i.}} a(N)e^{2\pi i Tr(N\tau)},
then we have
\Phi F(\tau')=\sum_{N'\geq 0\text{, h.-i.}} a\!\begin{pmatrix}N'&0\\0&0\end{pmatrix} e^{2\pi i Tr(N'\tau')}.

An element F\in M_\rho(\Gamma_g) is called a Siegel cusp form if it belongs to the kernel of \Phi. Equivalently, F is a cusp form if its Fourier coefficients satisfy: a(N)=0 for all half-integral matrices N which are not positive definite. We denote the space of Siegel cusp forms of weight \rho by S_\rho(\Gamma_g).

PUT IN EXAMPLES OF CUSP FORMS

Dimension formulas

The problem of computing the dimensions of the spaces S_k(\Gamma_g) for general g is much more difficult and far from being settled. The state of the art in this area is contained in \cite{Poor2002}, \cite{Poor2001}, and \cite{Poor2000}; for example, they prove that

\dim S_{10}(\Gamma_4)=1\quad\text{and}\quad \dim S_{2k+1}(\Gamma_4)=0\text{ for }k=0,1,\ldots,6.

The cases g=2 and g=3 are completely solved thanks to \cite{Igusa1964} (see also \cite{Hashimoto1983}), resp. \cite{Tsuyumine1986}. There is an explicit yet rather complicated conjectural expression for the dimension in the general case; see~\cite{Ibukiyama1992}.

\dim S_k(\Gamma_3(N))=\left(2^{-16}3^{-6}5^{-2}7^{-1}N^{21}(2k-2)(2k-3)(2k-4)^2(2k-5)(2k-6)-2^{-10}3^{-2}5^{-1}N^{16}(2k-4)+2^{-8}3^{-3}N^{15}\right)\prod_{p\mid N}(1-p^{-2})(1-p^{-4})(1-p^{-6}),
where the product ranges over all prime divisors p of N.

For vector-valued forms of genus 2, a dimension formula is (apparently)

A related (and even more difficult) problem is finding a basis for the space M_k(\Gamma_g), or a set of generators for the graded ring M(\Gamma_g).

For g=2, \cite{Igusa1962} proved that

M(\Gamma_2)=\mathbb{C}[E_4,E_6,\chi_{10},\chi_{12},\chi_{35}]/\left(\chi_{35}^2=P(E_4,E_6,\chi_{10},\chi_{12})\right),
where E_{2k} is the Siegel-Eisenstein series of weight 2k, and \chi_{k} are Siegel cusp forms of weight k which can be expressed explicitly in terms of Siegel-Eisenstein series.

The case g=3 was solved by \cite{Tsuyumine1986}. He exhibits a set of 34 generators for M(\Gamma_3), some of which cannot be written in terms of Siegel-Eisenstein series. I think it is still unknown whether this set of generators is minimal.

Hecke Theory

We define G = \text{GSp}^+_{2g}(\mathbb{Q}) be the group of rational symplectic similitudes with positive scalar factor:

\{\gamma\in GL_{2g}(\mathbb{Z}):\gamma^t J\gamma=r(\gamma)J,\text{ for }\gamma\in\mathbb{Q}_+\}.

For degree g>1 and each prime p, there are g+1 linear operators called Hecke operators. We describe the algebra in which they are found.

Let L(\Gamma,G) be the free \mathbb{C}-module generated by the right cosets \Gamma\alpha where \alpha\in\Gamma \ G. Note \Gamma acts on L(\Gamma,G) be right multiplication and we set

\mathbb{H}_g(\Gamma,G)=L(\Gamma,G)^\Gamma.

Let T_1,T_2\in \mathbb{H}_g(\Gamma,G) and

T_i = \sum_{\alpha_i \in \Gamma \setminus G} c_i(\alpha) \Gamma\alpha.
Then
T_1 T_2 = \sum_{\alpha,\alpha'\in \Gamma \setminus G} c_1(\alpha)c_2(\alpha')\Gamma\alpha\alpha'.

As in the classical case, we pay most attention to the Hecke operators at a prime p. It is known that \mathbb{H}_g = \bigotimes_{p\text{ prime}} \mathbb{H}_{g,p} where the construction of the local Hecke algebra \mathbb{H}_{g,p} is the same as before but with G replaced with G_p = G\cap \text{GL}_{2g}(\mathbb{Z}[p^{-1}]). The generators of this local algrebra \mathbb{H}_{g,p} are the double cosets

T(p)=\Gamma\text{diag}(I_g;pI_g)\Gamma \text{ and }
and
T_i(p^2)=\Gamma \text{diag}(I_i,pI_{g-i};p^2I_i,pI_{g-i})\Gamma
for 1\leq i \leq g. Some authors also define T_0(p^2), too. The operator T(p^2)=\sum_{i=1}^g T_i(p^2).

\mathbb{H}_g acts on \mathcal{M}_k^g by

F|_k\left(\sum c_i\Gamma\alpha_i\right)=\sum c_i F|_k\alpha_i
where
\left(F|_k \alpha\right)(Z)=r(\alpha)^{gk-\frac{g(g+1)}{2}}\det(CZ+D)^{-k}F\left(\alpha\cdot Z \right).
Some authors use a different normalization in this definition. A Hecke eigenform is a form in M_k(\Gamma) is a simultaneous eigenform for all the operators T(p).

More naturally, perhaps, than Hecke eigenvalues, one can associate Satake parameters to a Hecke eigenform. In the 1960s Satake proved the following theorem (in much more generality):

\mathbb{H}_{g,p}\cong \mathbb{C}[x_0^{\pm 1},\dots,x_g^{\pm 1}]^{W_g}
where W_g is the Weyl group generated by the permutations of x_1,\dots,x_g and by the maps x_0\mapsto x_0x_j, x_j\mapsto x_j^{-1}, x_i\mapsto x_j (i\neq j,\, 1\leq i\leq g). The above can be rephrased as an isomorphism \Psi on:
\text{Hom}_\mathbb{C}\left ( \mathbb{H}_{g,p},\mathbb{C}\right ) = \left ( \mathbb{C}^\times\right )^{g+1}/W_g.

Let F be a Hecke eigenform and T\in\mathbb{H}_g write F|_k T = \lambda_F(T) F. Then

\Psi(T\mapsto \lambda_F(T)) = (\alpha_{0,p},\dots,\alpha_{g,p}).
The entries of the above (g+1)-tuple are the Satake parameters of F.

$L$-functions

There is a more complicated relationship between L-functions and Siegel modular forms of degree bigger than one than there is in the degree one case. For a Hecke eigenform F with p-Satake parameters (\alpha_{0,p},\dots,\alpha_{g,p}) we define two Langlands L-functions.

The spinor L-function of F is

L(F,s, \text{sp})=\prod_p [L_p(F,s,\text{sp})(p^{-s})]^{-1},
where each local factor is given by
L_p(F,X,\text{sp})=(1-\alpha_{0,p}X)\prod_{r=1}^g\prod_{1 \leq i_1 < \cdots < i_r \leq n}(1-\alpha_{0,p}\alpha_{i_1,p}\cdots\alpha_{i_r,p}X).

If g=2 and F is an eigenform of weight (j,k) with T(m)F=\lambda_F(m)F then

L(F,s,\text{sp})=\zeta(2s-j-2k+4)\sum_{m=1}^{\infty} \frac{\lambda_F(m)}{m^s}.

In order to discuss the functional equation (for g=2), we introduce the function

\Lambda(F,s,\text{sp})=(2\pi)^{-2s}\Gamma(s)\Gamma(s-k+2)L(F,s,\text{sp}).
Then
\Lambda(F,s,\text{sp})=(-1)^k\Lambda(F,2k+j-2-s,\text{sp}).
In order to get the Selberg data of the spinor L-function, we need to normalize \Lambda so that its symmetry is around 1/2. Set
\Phi(F,s)=\Lambda\left(F,s+k+\frac{j-3}{2}\right).
Then
\Phi(F,s)=(-1)^k\Phi(F,1-s).

The standard L-function is defined by

L(F,s,\text{st}) = \prod_p [L_p(F,s,\text{st})(p^{-s})]^{-1},
where
L_p(F,X,\text{st}) = (1 - X)\prod_{m=1}^g (1 - \alpha_{m,p} X)(1 - \alpha_{m,p}^{-1} X).

Define

\Lambda(F,s,\text{st})=(2\pi)^{-gs}\pi^{-s/2}\Gamma\left(\frac{s+\epsilon}{2}\right)\left(\prod_{m=1}^g\Gamma(s+k-m)\right) L(F,s,\text{st}),
where \epsilon=0 if g is even and \epsilon=1 if g is odd. Then \Lambda(F,s,\text{st}) can be extended to a meromorphic function on the whole complex plane. It satisfies the functional equation
\Lambda(F,s,\text{st})=\Lambda(F,1-s,\text{st}).

A third analytic object studied in the context of Siegel modular forms is the Koecher-Maass series given by...

Liftings

The most well-known lift is the Saito Kurokawa lift. Proved by Maass, Andrianov, and Zagier, the following describes this lift: For even weights k and (k+2)/2, each eigenform f \in S_1^k corresponds to an eigenform F \in S_2^{{k/2+1}} such that the standard L-function of F factors

L(F,s,\text{st})= \zeta(s)L(f,s + \frac{k}{2})L(f,s + \frac{k}{2}-1).

Ikeda recently generalized this lift to higher degree: For even k, g and (g+k)/2, each eigenform f \in S_1^k corresponds to an eigenform I_g(f) \in S_g^{(g+k)/{2}} such that the standard L-function factors

L(I_g(f),s,\text{st})= \zeta(s) \prod_{i=1}^g L(f, s+ \frac{g+k}{2} - i).

Ikeda also recently proved the existence of the following lift first conjectured by Miyawaki: Let k, g+r and {\hat k}=(g+r+k)/2 be even integers with r \le g. Each pair of eigenforms f\in S_1^{2k-4} and h\in S_r^{{\hat k}} corresponds to an element M_g(f,h)\in S_g^{{\hat k}} via M_g(f,h)(Z_g)= \left( I_{g+r}(f)(Z_g\oplus Z_r), g^c(Z_r)\right), where \left( \cdot,\cdot \right) is the Petersson inner product. If M_g(f,h) is nontrivial then it is an eigenform and

L(M_g(f,h),s,\text{st})= L(h,s,\text{st}) \prod_{i= r+1}^g L(f, s+{\hat k} -i).

Well-known conjectures

Resnikoff-Saldaña

Ramanujan-Petersson

Existence of associated Galois representations (see Paragraph 1.3 in A. Mokrane, J. Tilouine: Cohomology of Siegel varieties with p-adic integral coefficients and applications)

Applications

Algebraic number theory

\alpha:=\zeta+\zeta^3+\zeta^9,\quad\zeta:=e^{2\pi i/13}.
They consider various abelian extensions of K, and show how special values of the L-functions of these extensions can be computed explicitly using Siegel modular forms. They can also show that these special values are units.

Arithmetic geometry

Cohomology of some arithmetic groups

Coding theory

Elliptic cohomology

(\text{fudge factor})\cdot\prod_{m,n,\ell}\left(\frac{1}{1-q^my^\ell t^n}\right)^{c(mn,\ell)}
is a Siegel modular form.

Physics

Computational Aspects

Databases for Degree 2 Level 1

  1. Data accumulated by Ryan and Yuen (2007): Genus2DB.txt. The data is for cuspforms of degree 2 from weight 16 to 44 and is organized as follows. The weights are separated by an empty line and for each weight there are six lines of data. The first line is merely the weight of the space, the second is a basis (E4, E6, X10, Y12) are the generators identified by Igusa, the third through sixth are, respectively, the matrix representation of T(2), T_0(4), T_1(4), T_2(4).

  2. Interactive database by Skoruppa MODI. This site provides data related to degree 2 modular forms that are not Saito-Kurokawa lifts. It provides Fourier coefficients of a selected form, the Jacobi form associated to the selected form, and the T(p) for p < 1000 and T(p^2) for p^2<80.

  3. Here is how we envision a sample line for a table of spaces of Siegel modular forms of level 1: 

Data Column                  Known? Reference?
weight                       for free
dimension                    generating series for dim M_k(Gamma_2)
basis                        generators known, Igusa
basis of eigenforms          should be doable
characteristic poly of Hecke yes, formulas for Hecke action, basis known

We would want similar tables for certain interesting subspaces of M_k(Gamma_2): S_k(Gamma_2), subspace of lifts, subspace of nonlifts

  1. Here is how we envision a sample line for a table of Siegel modular eigenforms: 

Data Column                  Known? Reference?
level                        for free
weight                       for free
Fourier coefficients         product of generators, Igusa 
Hecke eigenvalues            formulas for Hecke action known
Central values of L-function Kohnen/Kuss

Software for Degree 2 Level 1

SAGE in the near future will include code to compute data associated to Siegel modular forms in degree 2.

References

  1. Hafner, James and Walling, Lynne "Explicit action of Hecke operators on Siegel modular forms." J. Number Theory 93 (2002), no. 1, 34--57.

  2. Igusa, Jun-ichi "On Siegel modular forms of genus two." Amer. J. Math. 84 1962 175--200.

  3. Kohnen, Winfried; Kuss, Michael "Some numerical computations concerning spinor zeta functions in genus 2 at the central point." (English summary) Math. Comp. 71 (2002), no. 240, 1597--1607

  4. Skoruppa, Nils-Peter "Computations of Siegel modular forms of genus two." Math. Comp. 58 (1992), no. 197, 381--398.

Books

Survey Papers

Survey Talks

Links

ModularForms/SiegelModularForms (last edited 2009-03-01 00:55:50 by localhost)