数理解析研究所講究録1052 巻 1998 年 142-153

142

Theta liftings- a comparison between classical andrepresentation theoretic results
R. Schulze-Pillot

Introduction.
Theta liftings have been considered both from a classical and from a representation theoretic point of view. In the classical setting, one considers
holomorphic theta series attached to integral quadratic forms as Siegel (orHilbert-Siegel) modular forms. One is then interested in a description of

the linear relations between the members of a given set of such theta series
and in a characterization of the space of modular forms spanned by these
theta series. The theta series in such a set are usually quite restricted in
type, e.g., they belong to full lattices of some fixed level or they are theta

series with characteristic (thetanullwerte) attached to a single lattice butwith varying characteristic. The representation theoretic approach considers the more general theta correspondence between automorphic forms

on adelic orthogonal and symplectic (or metaplectic) groups defined using
the oscillator (or Weil-) representation of the metaplectic group. Here one
discusses for an irreducible representation space of automorphic forms on

one of the groups whether it is in the image under the theta correspondence of a representation space of automorphic forms on the other grouprespectively whether its image under the correspondence is zero or not.

Although both types of question appear to be extremely similar, they are
not quite the same. It is the purpose of this note to discuss some cases inwhich a transfer of results between the two settings is made possible byrecent results of Moeglin and to describe some of the difficulties that occur
in other cases.

143
1. The problems.
Let (V, ) be a non-degenerate quadratic space of even dimension $m=2k$
$q$

over

$\mathrm{Q}$, denote by $B(x, y)=q(x+y)-q(x)-q(y)$ the associated sym-

metric bilinear form, let$V$(i.e.,$q(L)\subseteq \mathrm{Z},$$L\subseteq V$$\mathrm{Q}\otimes L\cong V$

be an integral$\mathrm{Z}$) of level $N$ (i.e.,-lattice of full rank on$q(L\#)\mathrm{Z}=N^{-1}\mathrm{Z}$$L^{\neq}=\{y\in V|B(y, L)\subseteq \mathrm{Z}\}$is the dual lattice ofconsists of all lattices $K$ on

with

to

$K\otimes \mathrm{Z}_{p}$

$V$

$L$

, where). The genus ofisometric with respect to$L\otimes \mathrm{Z}_{p}$

$L$

$q$

for all primes . It consists of finitely many isometry classes$p$of lattices. We restrict attention to positive definite , let$q$Siegel upper half space of degree (or genus)$q( \mathrm{x})=(\frac{1}{2}B(x_{\dot{i}}, X_{j}))\in \mathrm{J}\text{ノ}l_{n}^{\mathrm{S}}\mathrm{y}\mathrm{m}(\mathrm{Q})$

$n$

for

$fi_{n}$denote the, put$\mathrm{x}=(x_{1)}\ldots , x_{n})\in V^{n}$and consider the theta series$\theta_{L}^{(n)}(z)=\sum_{\mathrm{x}\in L^{n}}\exp(2\pi i\mathrm{t}\mathrm{r}(q.(\mathrm{X})z))$

Inhomogeneous theta seriessummation over a coseta harmonic form

$P$

representation space:$\theta_{L,\mathrm{y}}^{(n)}(z)$$\mathrm{y}+L^{n}$withare defined in the same way, but with$\mathrm{y}\in V^{n}$$V^{n}\otimes \mathrm{R}arrow W_{\rho}$$W_{\rho}$of$\mathrm{G}\mathrm{L}_{n}(\mathrm{C})$.

. More generally we can consider

with values in some irreducible

and put$\theta_{L,\mathrm{y}}^{(n)}(P, z)=\sum_{Y\mathrm{x}\in+L^{n}}P(\mathrm{x})\exp(2\pi\dot{i}\mathrm{t}\mathrm{r}(q(\mathrm{x})z))$

[7].

For

$\mathrm{Y}=0,$character$\chi$

theta series$\theta_{L}^{()}n(P, Z)$

is a$W_{\rho}$-valued Siegel modular form for

depending on the discriminant of$L$$\Gamma_{0}^{(n)}(N)$with, for the inhomogenouss.ee [7]. We want to consider here the following questions:A) Find the linear relations between the$\theta_{L_{i}}^{(n)}$for$L_{\dot{i}}$running through a setof representatives of the isometry classes of lattices in the genus of thelattice

$L$

.

144
B) Characterize the space of modular forms generated by the$\theta_{L_{i}}^{(n)}$for$L_{i}$as above.
The question B) (the basis problem) has been considered in many variants,

for example$\mathrm{B}’)$

Let $N,$degree$\chi$$n$be given. Is the space, weight$k$$M_{k}^{(n)}(\Gamma_{0}(N), x)$

and character$\chi$

of modular forms of(or its subspace of cusp forms)

generated bya) theta series (maybe inhomogeneous) attached to (arbitray) quadratic

formsb) by theta series attached to full latticesc) by theta series attached to full lattices of level $N$d) bytheta- series with spherical harmonics as above

In each of these subproblems, determine an explicit representation ofa given$F\in M_{k}^{(n)}(\Gamma 0(N)_{\rangle}x)$.

For a survey of some results concerning B),$\mathrm{B}’$) see [2].

For the representation theoretic view of the problem let $(U^{(n)}, A)$ denote

a $2n$ -dimensional space over$\mathrm{Q}_{\mathrm{A}}/\mathrm{Q}arrow S^{1}$$\mathrm{Q}$with non-degenerate alternating form,:a nontrivial additive character. The symplectic group $G=$

$G_{n}=\mathrm{S}\mathrm{p}(U^{(n)}, A)=$

Sp(n) and the orthogonal group $H=O(V)$ form adual reductive pair in the sense of Howe. Let
representation of the adelic groupof Schwartz-Bruhat functions on$h\in H(\mathrm{A})$$\psi$$\omega=\omega_{\psi}$$G(\mathrm{A})\cross H(\mathrm{A})$$V(\mathrm{A})^{n}$denote the Weilon the space$S(V(\mathrm{A})^{n})$. For $f\in S(V(\mathrm{A})^{n}))g\in G(\mathrm{A})$ ,define the theta kernel$\theta(\mathit{9}_{\rangle}h,\cdot f):=\sum\omega(g)\varphi(\mathrm{x}\in V(L)^{n}h^{-1}\mathrm{X})$.

145
For a space$Y$of cuspidal automorphic forms on

we write$G(\mathrm{A})$$\Theta^{V}(Y)$forthe space of theta lifts$\Theta_{f}(\varphi’)(h):=\int_{G(\mathrm{Q})\backslash }G(\mathrm{A})\varphi’(g)\Theta(g, h;f)dg$

with

on

$\varphi’\in Y,$$f\in S(V(\mathrm{A})^{n}))$we write$H(\mathrm{A})$$\Theta^{U}(X)$similarly for a space $X$ of automorphic forms

for the space of theta lifts$\Theta_{f}(\varphi):=\int_{H(\mathrm{Q})\backslash }H(\mathrm{A})(\varphi h)\Theta(g, h,\cdot f)dh$of functions $f\in X$ with respect to $f\in S(V(\mathrm{A})^{n})$ to$\Theta_{f}(\varphi)$$\tilde{\mathrm{A}})$on$G(\mathrm{A})$tomorphic forms$\mathrm{a}_{\vee}\mathrm{u}$. We have thenGiven a (cuspidal) irreducible automorphic representation
decide whether$\Theta^{U^{(n)}}(\pi)$

is nonzero (find the first

$n$

$\pi$

of

$H(\mathrm{A})$,for which it isnonzero).
$\tilde{\mathrm{B}})$

Given a cuspidal irreducible automorphic representation
decide whether

With a lattice$L$$A(H(\mathrm{A})\rangle \mathit{0}_{\mathrm{A}(L)})$$\varphi(h)$foris a lift of some representation$\pi’$

on$V$

with adelic orthogonal groupbe the space of functions$\gamma\in H(\mathrm{Q}),$$u\in O_{\mathrm{A}}(L)$$\varphi$

:

$\pi$

of

$\pi’$$H(\mathrm{A})$of$G(\mathrm{A})$as above.

$O_{\mathrm{A}}(L)\subseteq H(\mathrm{A})$$H(\mathrm{A})arrow \mathrm{C}$

with

,

let

$\varphi(\gamma hu)=$

and consider a double coset decomposition$t$$H( \mathrm{A})=\bigcup_{\dot{i}}=1LH(\mathrm{Q})hio_{\mathrm{A}}()$

Let$f^{(n)}\in S(V(\mathrm{A})^{n})$be given as$f^{(n)}= \prod_{p}f_{p}^{(n)}$

.

with

$f_{p}^{(n)}=1_{l_{p}^{n}}$for finite

and$f_{\infty}^{(n)}(\mathrm{X})=\exp(-2.\pi(q(x_{1})+\cdots+q(xn)))$Then for$\varphi\in A(H(\mathrm{A}), O\mathrm{A}(L))$.

we have,$\mathrm{o}_{f^{()}}^{U^{(n}}-n)(\varphi)(g)=\sum^{t}j=1\frac{\varphi(h_{j})}{|O(h_{j}L)|}\omega(_{\mathit{9})}\sum_{\mathrm{x}\in(h_{j}L)^{n}}\exp(-2\pi\cdot \mathrm{t}\mathrm{r}(q(\mathrm{x})))$

$p$

146
and under the usual correspondence of automorphic forms onon the Siegel half space$\ovalbox{\tt\small REJECT}_{n}$of degree$n$$G(\mathrm{A})$andthis function corresponds to$\mathrm{O}_{L}^{(n)}-(\varphi)(Z):=0-f(n)(U^{(n})\varphi)(z)=\sum j=1h\frac{\varphi(h_{j})}{|O(h_{j}L)|}\theta^{(n)}(h_{j}L, z)$,where $L_{j}:=h_{j}L$ runs through a set of representatives of the isometry
classes in the genus of$L$

harmonic forms arise forright invariant under$\mathrm{i}\mathrm{r}\mathrm{r}\mathrm{e}\mathrm{d}\mathrm{u}\mathrm{c}\mathrm{i}\mathrm{b}\dot{\mathrm{l}}\mathrm{e}$$H(\mathrm{R})$. Similar expressions involving theta series with$\varphi$from a space of functions on$\prod_{p\neq\infty}O(L_{p})$representation of$H(\mathrm{R})$$O_{\mathrm{A}}(V)$that areand transform according to some fixedunder right translation by elements of, see [7, 4, 5].

On the other hand, given
cusp form $F$ of weight$\varphi’$on$k= \frac{m}{2}$$G(\mathrm{A})$that corresponds to a Siegel modular

we have$\theta^{(n}(L_{j})\rangle$$\mathrm{O}_{f^{()(\varphi’)(}}^{V}-nhj)=^{c\cdot\langle)}F,$

with some constant $C\neq 0$ depending on the normalization chosen andwithdenoting the Petersson inner product.

$\langle, \rangle$

Our questions concerning the relation between resu.l$\mathrm{t}\mathrm{s}$in the classical andin the representation theoretic setting are then$?\mathrm{A})$

Let

$\pi$

be an irreducible cuspidal representation of$A(H(\mathrm{A}), o\mathrm{A}(L))\neq\{0\}$some$\varphi\in\pi,$$f\in S(V_{\mathrm{A}}^{n})$and with$\Theta^{U^{(n)}}(\pi)\neq\{0\}$with$\pi\cap$. Then there aresuch that$\Theta_{f}^{U^{(n)}}(\varphi)\neq 0$,

but the lift may of course vanish for individual
conditions on$H(\mathrm{A})$$\varphi\in A(H(\mathrm{A}), O_{\mathrm{A}}(L))\cap\pi$that$\Theta_{L}^{(n)}(\varphi)\neq 0$?

and on$\varphi,$

$\pi$

$f$

. Under which

can we conclude

147$?\mathrm{B})$Let$\pi’$

be an irreducible cuspidal automorphic representation ofsuch that$\pi’=\Theta^{U^{(n)}}(\pi)$for some

of

$\pi$

modular form of level $N$ and weight$\varphi’\in\pi’$. Then there exist$\varphi\in\pi$

and

$F$

more specifically be written as$V$of level $N$ andcorresponding to some$f\in S(V_{\mathrm{A}}^{n})$$\varphi’=\Theta_{f}^{U^{(})}(\varphi n)$

Under which conditions onand let $F$ be a Siegel$H(\mathrm{A})$$k= \frac{m}{2}$and on$\pi’$

with.

can we conclude that

$F=\Theta_{L}^{(n)}(\varphi)(Z)$$\varphi\in A(H(\mathrm{A}), O\mathrm{A}(L))$$G_{n}(\mathrm{A})$$F$

for some lattice

can

$L$

on

?

We remark that question B) is important if we want to write down $F$ explicitly (e.g. by giving part of its Fourier expansion): It is for moderatedimensions and levels computationally feasible to tabulate the lattices ina given genus and to compute their theta series. On the other hand, theformulation$\pi’=\Theta^{U^{(n)}}(\pi)$gives only the existence of an expression of $F$

by possibly inhomogenous theta series attached to lattices of some levels(which are in no way restricted).
2. Results.

Our main tool will be the following theorem of Moeglin [15]:Theorem (Moeglin). Let be an irreducible subrepresentation of thespace of (cuspidal) automorphic forms on the adelic orthogonal group$\pi$$H(\mathrm{A})$and suppose that$\Theta^{U^{(n)}}(\pi)$contains a nonzero cusp form. Then$\Theta^{U^{(n)}}(\pi)$is an irreducible cuspidal representation, and one has. Conversely fora cuspidal irreducible automorphic representation$\Theta^{V}(\Theta^{U}(n)(\pi))=$$\pi’$$\pi$of$G(\mathrm{A}),$$\Theta^{(V)}(\pi’)$is irreducible (cuspida ) (if it contains a nonzero cusp$l$form) and$\Theta^{U^{(n)}V}\Theta(\pi’)=\pi’$.

148
Remark. For our present situation of positive definite
ity assumption is vacuous for forms on$V$the cuspidal-

and can hence be omitted.

$H(\mathrm{A})$

Moeglin’s result is valid for general (V, ).
$q$

Proposition 1. Let

automorphic forms on

form. Let
$(*)$

Then

such that$H(\mathrm{A})$contains a nonzero cusp$\Theta^{(n)}(\pi)$be a lattice on $V$ satisfying

$L$

For

the liftbe an irreducible subrepresentation of the space of$\pi$

and for all$r\in \mathrm{N}$$\Theta_{L}^{(r+1)}(\varphi)$$\Theta_{L}^{(n)}(\varphi)\neq 0$$\varphi\in A(H(\mathrm{A}), O_{\mathrm{A}}(L))$$\Theta_{L}^{r}(\varphi)=0$

is cuspidal.

for all$\varphi\neq 0$

in $A(H(\mathrm{A}), O\mathrm{A}(L))\cap\pi$ .

Proof. Ist is known [16] that there is$r$

the smallest suchis cuspidal,

$r$

with

the lifting$\Theta^{(r)}(\pi)$with$\Theta^{U^{(r)}}(\pi)\neq\{0\}$$\Theta^{(s)}(\pi)$and that for

non-cuspidal for$s>r$ . Also, the linear independence of theta series of degree bigger thanthe dimension implies that$\Theta_{L}^{(s)}(\varphi)\neq 0$

our assumption $(*)$ would imply that$s>n$ , hence$\Theta^{U^{(s)}}(\pi)$for some . If$s$$\Theta_{L}^{(s)}(\varphi)$$\Theta_{L}^{(n)}(\varphi)$

were ,$0$is nonzero cuspidal for someis cuspidal by Moeglin’s theorem. But then Rallis’

result quoted above contradicts$\Theta^{(n)}(\pi)\neq 0$.

Remark.
a) Condition $(*)$ is obviously true for$L$of level 1 (even unimodular$L$)and by [3] for square free $N$ . For general $N$ , little seems to be known.

Classically this condition says thatits image underb) If there is$r\in \mathrm{N}$)$\mathrm{S}\mathrm{i}\mathrm{e}\mathrm{g}\mathrm{e}1_{\mathrm{S}}$with$\Phi$$\Theta_{L}^{(s)}(\varphi)$

is cuspidal if and only if-operator is zero.

$\Theta_{L}^{(r)}(\varphi)=0,$$\theta_{L}^{(r+}(\varphi 1))$

is not cuspidal and hence by Rallis’ result

noncuspidal, then$\Theta^{(r)}(\pi)\neq 0$$\theta^{(r+1)}(\pi)$. Condition $(*)$

is therefore necessary for the validity of the conclusion of Proposition
1.

149c) Statement and proof of the proposition can be transferred to theta series
with spherical harmonics (liftings ofirreducible representationd) Let$(z_{\mathcal{T}}, \tau)$of$\varphi\in A(H(\mathrm{A}),$ $o_{\mathrm{A}}(L))\tau$) for an.

$H(\mathrm{R}))$be a definite quaternion algebra over Q) equipped with the (re-$V$

duced) norm form as quadratic form,$L$

an Eichler order infree level $N$ . Then in $[3, 5]$ examples have been given ofwith$\theta_{L}^{(2)}(\varphi)=0$. The proposition shows that then indeed

for the irreducible representation$\pi$

generated by$\varphi$$V$of squareon$\varphi$$H(\mathrm{A})$$\Theta^{U^{(2)}}(\pi)=0$, hence$\Theta^{U^{(3)}}(\pi)$iscuspidal. It has been shown by Roberts [18] that the representation(extended to the group of similitudes) has a nonzero theta lifting to to$\mathrm{G}\mathrm{S}\mathrm{p}_{2}$locally everywhere. This is in contrast to the situation in [11],where the nonvanishing of the theta lift tolocal conditions,

Proposition 2. Let$\mathrm{G}\mathrm{S}_{\mathrm{P}_{2}}$is decided b.y purely

sing- the result of [8].

$\mathrm{u}$$\pi’$

be an irreducible subrepresentation of the spaceof cuspidal automorphic forms on
sponding to the Siegel modular form

$G_{n}(\mathrm{A})$$F$

containing a functionof weight for$k$$\mathrm{r}_{0}^{(n)}(N)$$\varphi’$corre-. Assumethat $\pi:=\Theta^{V}(\pi’)\neq\{0\}$ and that$(**)$

there is a lattice$\pi\neq\{0\}$Then$\pi’$$L$onof level $N’|N$ such thatfrom Proposition 1 holds.

$V$and such that $(*)$

contains a Siegel cusp form

$A(H(\mathrm{A}), O_{\mathrm{A}(}L))\cap$$\tilde{F}$of weight for$k$$\Gamma_{0}^{(n}()N’)$such that$\overline{F}$is a linear combination of theta series attached to lattices in the genus of .
$L$

Proof. By Moeglin’s theorem$\pi$

by Proposition 1 we have$\tilde{F}\in\pi^{\prime_{\mathrm{h}\mathrm{o}}}1\mathrm{d}\mathrm{S}$

.

$\pi$

is irreducible. For$\tilde{F}:=\theta_{L}^{(n)}(\varphi)\neq 0$$0\neq\varphi\in A(H(\mathrm{A}), O\mathrm{A}(L))\cap$. Again by Moeglin’s theorem

150

Remark.
a) If the$\mathrm{Z}$-maximal lattices on$V$are even unimodular and $N$ is square

free and odd, the condition $(**)$ is satisfied: By the results of Aubert[1] for each $p|N$ the local representation$\pi_{p}$contains a vector invariant

under the orthogonal group of some lattice of level dividing

The resulting function$\varphi\in A(H(\mathrm{A}), O_{\mathrm{A}}(L))$

$p$

von

$V_{p}$.

satisfies $(*)$ as in theremark to Proposition 1. A modified version of Aubert’s result shouldbe true for the local theta correspondence with respect to arbitrary$V_{p}$(Aubert requires$V_{p}$

to carry a self-dual lattice and $p\neq 2$ ). This

would give the validity of $(**)$ for arbitrary $V$ and square free $N$ .

b) The condition $(**)$ is necessary for the validity of the conclusion of
the proposition: If

is as described, then$\tilde{F}$$\varphi\in\backslash A(H(\mathrm{A}), O_{\mathrm{A}}(L))$$\overline{F}=\Theta_{L}^{(n)}(\varphi)$, and $\varphi\in\Theta^{V}(\pi’)=\pi$ follows from Moeglin’s

theorem, hence $A(H(\mathrm{A}), o\mathrm{A}(L))\cap\pi\neq\{0\}$ . Thencuspidal by assumption, hence

satisfied for$\varphi$

Hence$\Theta_{L}^{(r)}(\varphi)=0$$\Theta_{L}^{(n)}(\varphi)=\tilde{F}$isfor all $r<n$ , and $(*)$ is

as well.

c) If $N=1$ , the cusp form$\mathrm{S}\mathrm{p}_{n}(\mathrm{Z}_{p})$for some$F$of level 1 in$\pi’$

is unique (for each , the$p$-fixed vectors in an irreducible local representation is unique).

$\tilde{F}$is proportional to $F$ , that is, $F$ itself is alinear combination oftheta series of even unimodular lattices. In this case we have therefore

proved that a representation theoretic solution of the basis problem
implies a classical solution.

d) If $n=1$ and $N$ is the level of$F$$\pi’$

then by [6]$\tilde{F}$is again proportional to. The condition $(**)$ can then be obtained from the local Jacquet-

Langlands correspondence [13], except for the additional requirementof $(*)$ . In particular, our result seems not be strong enough to show

151
that the correspondence of Jacquet-Langlands implies the solution of
the basis problem due to Hijikata, Pizer and Shemanske [12], except

for the case of square free level $N$ . In this case, a different way toderive the implications has been sketched in Section 9 of [12].

Proposition 3. Let$\pi$automorphic forms on$\Theta_{f}^{U^{(n)}}(\varphi)$tobe an irreducible subrepresentation of the space of$H(\mathrm{A})$and let$\varphi\in\pi,$$f\in S(V(\mathrm{A})^{n})$corresponds to a Siegel modular form$F$

. Assume that the -maximal lattices on$\mathrm{Z}$$\mathrm{S}\mathrm{p}_{n}(\mathrm{Z})$of weight$V$$k$be such thatwith respect

are even unimodular.

Then $F$ is a linear combination of theta series attached to even unimodular

lattices on$V$

Proof. By$\Theta_{V}(\pi’)=\pi$vector$\tilde{\varphi}$.

$\mathrm{M}\mathrm{o}\mathrm{e}\mathrm{g}\mathrm{l}\mathrm{i}\mathrm{n}^{)}\mathrm{S}$theorem$\Theta^{U^{(n)}}(\pi)=:\pi’$is cuspidal irreducible with

. By the local theta correspondence $[14, 17]$

invariant under the group$O_{\mathrm{A}}(L)$for a lattice

$\pi$

$L$

has to contain aof level 1. The

assertion follows from Proposition 2.
Remark. If$\Theta_{fn}^{U^{(n_{K}}}(()\varphi))$$\varphi\in A(H(\mathrm{A}), o\mathrm{A}(K))\cap\pi$

for some lattice $K$ on$V$

is such that

corresponds to $F$ with the special test function $f_{n}(K)$ associated

to $K$ the assertion can also be deduced using the Computation. of traces$\mathrm{S}$of theta series in [10].

References[1] A.-M. Aubert: Correspondance de Howe et sous-groupes parahoriques,J. f. d. reine u. angew. Math. 392, 176-186 (1988).
[2] S.B\"ocherer: Siegel modular forms and theta series, Proc. symp. Pure

Math. 49, vol. 2, 3-17 (1989)[3] S.B\"ocherer,R.Schulze-Pillot: Sie.g el modular forms and theta series at-tached to quaternion algebras. Nagoya Math.J. 121 (1991), 35-96

152[4] S.B\"ocherer, R. Schulze-Pillot: Mellin transforms of vector valued theta

series attached to quaternion algebras, Math. Nachr. 169 (1994), 31-57[5] S. B\"ocherer R. Schulze-Pillot: Siegel modular forms and theta series
attached to quaternion algebras II. Nagoya Math.J.147(1997), 71-106[6] W. Casselmann:$GL_{2}(k)$An assortment of results on representations of, p. 1-54 in Modular functions of one variable II, Lecture Notes

Math. 349,$\mathrm{B}\mathrm{e}\mathrm{r}\mathrm{l}\mathrm{i}\mathrm{n}rightarrow \mathrm{H}\mathrm{e}\mathrm{i}\mathrm{d}\mathrm{e}\mathrm{l}\mathrm{b}\mathrm{e}\mathrm{r}\mathrm{g}$-New York 1973

[7] E. Freitag: Singular modular forms and theta relations, LNM 1487,
Berlin, Heidelberg 1991
[8] S.Friedberg, J. Hoffstein: Nonvanishing theorems for automorphic L-

functions on$\mathrm{G}\mathrm{L}(2)$, Ann. of Math. (2) 142 (1995), no. 2, 385-423.

[9] J. Funke, Spuroperator und Thetareihen quadratischer Formen, Diplomarbeit, K\"oln (1994).
[10] S. B\"ocherer, J. Funke, R. S.chulze-Pillot: Trace Operator for Modular

Forms. In preparation.
[11] M. Harris, D. Soudry, R. Taylor:$\ell$-adic representations associatedto modular forms over imaginary quadratic fields, Invent. math. 112(1993), 377-411[12] H. Hijikata, A. Pizer, T. Shemanske: The basis problem for modular

forms on$\Gamma_{0}(N)$, Mem. Amer. Math. Soc. 82 (1989), no. 418[13] H. Jacquet, R. Langlands: Automorphic forms on $GL(2)$ , Lect. Notes
in Math. 114, Berlin-Heidelberg-New York 1970

[14] S. Kudla: On the local theta correspondence, Invent. math. 83 (1986),

229-255

153[15] C. Moeglin: Quelques propri\’etes de base des series theta, J. of Lie
Theory 7 (1997), 231-238[16] S. Rallis: Injectivity properties of liftings associated to Weil represen-tations. Comp. Math. 52 (1984), 139-169[17] S. Rallis: Langlands functoriality and the Weil representation, Amer.

J. of Math. 104 (1982), 469-515[18] B. Roberts: The non-Archimedean theta correspondence for GSp(2)and GO(4). Trans. Amer. Math. Soc., to appear.