# Siegel modular forms as sections of line bundles over the period domain

The transformation formula for a Siegel modular form can be interpreted as the statement that the modular form is a holomorphic section of a line bundle over the period domain (the quotient of the Siegel upper half-plane by a subgroup of finite index in the integral symplectic group). I've seen brief mentions of this fact in several references, but haven't been able to find one in which this geometric point of view is developped in detail. Does anyone know such a reference?

In particular, I would like to know if holomorphic line bundles over period domains have been classified.

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A small correction: the period domain is usually the thing you take the quotient of -- in this case the Siegel's upper half plane $H_g$. But otherwise yes, Siegel modular form should correspond to sections of line bundles on $A_g = H_g/Sp(2g, Z)$. I would image that $Pic(A_g)=Z$ for g large enough. I notice that Emerton has given some references. There may be something more classical also, although I don't have anything specific in mind. You could start with Birkenhake and Lange's book on complex abelian varieties for example. – Donu Arapura Aug 2 '10 at 16:24
Modulo torsion my guess about $Pic(A_g)$ ought to follow from Borel's "Stable real cohomology of arithmetic groups". If anyone knows anything more precise, please let me know. – Donu Arapura Aug 2 '10 at 16:37
@Donu : see my answer below. – Andy Putman Aug 2 '10 at 16:39
OK, thanks I'll take a look at your paper. – Donu Arapura Aug 2 '10 at 16:42
Thanks to all, and especially to Emerton for taking the time to explain in detail the construction of equivariant bundles. It turns out that the classification result I was after is contained in Andy's paper (with some mild restrictions on the genus and level structure). Thanks again! – Samuel Monnier Aug 2 '10 at 19:09

Modulo torsion, the Picard group of the quotient of the Siegel upper half plane by a finite-index subgroup of $\text{Sp}_{2g}(\mathbb{Z})$ is just $\mathbb{Z}$. This result should probably be attributed to Borel.

For a calculation of the torsion and explicit line bundles corresponding to the various pieces (plus references), see my paper "The Picard group of the moduli space of curves with level structures", available here. The results you want are discussed starting on page 3.

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Dear Samuel,

I don't know if the holomorphic line bundles on the quotients you ask about are classified, but one point of view that is adopted when thinking of (Siegel, or other) modular forms as sections of bundles is to consider $G$-equivariant holomorphic bundles on the symmetric space, so in your case, this would be $Sp_{2g}(\mathbb R)$-equivariant bundles on Siegel upper half-space. Such equivariant bundles can be used to induce a holomorphic bundle on the quotient of the upper half-space by any discrete group (in a way that is compatible with pull-back when one discrete group is contained in another), and so these are the bundles that are normally the most interesting in relation to automorphic forms.

The recipe for making these bundles (which works in some generality, say in an Shimura variety context; but I will stick to your Siegel situation) is as follows:

We first note that, just as the usual complex upper half-plane embeds into $\mathbb P^1(\mathbb C)$, the Siegel upper half-space embeds as an open subset of the complex points of a certain partial flag variety of $Sp_{2g},$ namely $Sp_{2g}/P$, where $P$ is the Siegel parabolic of $Sp_{2g}$ (the stablizer of a maximal isotropic subspace in the $2g$-dimensional symplectic space on which $Sp_{2g}$ acts). The Levi of $P$ is equal to $GL_g$, and so the irreducible $Sp_{2g}$-equivariant vector bundles on $Sp_{2g}/P$ correspond to irreducible reps. of $GL_g$. (If $V$ is a rep. of $GL_g$, we think of it as a rep. of $P$ through the quotient map $P \to GL_g$, and then make a bundle $\mathcal V$ on $Sp_{2g}/P$ via the formula $\mathcal V = (Sp_{2g} \times V)/P.$ In particular, the rank of $\mathcal V$ equals the dimension of $V$.

Restricting these bundles $\mathcal V$ to the Siegel upper half-space, one obtains a family of $Sp_{2g}(\mathbb R)$-equivariant holomorphic bundles on the Siegel upper half-space. As I already mentioned, because of their equivariance, they induce holomorphic bundles on the quotient by any discrete subgroup of the symplectic group.

The line bundles whose sections are classical Siegel modular forms correspond to the one-dimensional representations of $GL_g$ given by taking powers of the determinant.

Other (higher than one-dimensional) rep's of $GL_g$ give rise to bundles whose sections are vector-valued Siegel modular forms.

I found this kind of material a bit tricky to find in the literature, and pieced it together by reading Deligne's papers on Shimura varieties, as well as the early parts of various articles of Michael Harris and Jim Milne about automorphic vector bundles on Shimura varieties.

I wouldn't be surprised if there is a more classical literature that is complimentary to the Shimura-variety literature just mentioned, and which is perhaps more accessible. Unfortunately I don't know it!

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It seems to me that you are using $Sp_{2g}$ to denote both the real points as well as the algebraic group itself which is a little confusing; if I have understood correctly, in the first and fourth paragraph the groups should perhaps be $Sp_{2g}(\mathbb{R})$. – ulrich Sep 20 '11 at 5:37
Dear Ulrich, Thanks; now corrected. Regards, Matthew – Emerton Sep 20 '11 at 13:03

In the case $g=1$ modular forms are sections of a line bundle on a (geometrically meaningful) compactification of the stack obtained by quotienting the upper half plane by $SL_2(\mathbb Z)$ (usually denoted $\overline{M}_{1,1}$), whose Picard group is indeed isomorphic to $\mathbb Z$. By contrast, the Picard group before compactifying is torsion of order 12.

Since such compactifications exists AFAIK for every $g$, I suspect that this is what you have to consider. Bundles on the stack quotient of $X$ by $G$ are the same as $G$-equivariant bundles on $X$, so this answer has nontrivial overlap with Emerton's.

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