Osaka University Knowledge Archive

Chorro, Christophe

On an extension of the Hilbertian central limit theorem to Dirichlet forms

Murakami, Jun

Colored Alexander invariants and cone-manifolds

D'Ancona, Piero

Spagnolo, Sergio

Kinoshita, Tamotu

On the 2 by 2 weakly hyperbolic systems

Fujii, Satoshi

On the maximal pro-p extension unramified outside p of an imaginary quadratic field

Flapan, Erica

Naimi, Ramin

Osaka University
Title The Y-triangle move does not preserve intrinsic knottedness
Author(s)Flapan, Erica; Naimi, Ramin
CitationOsaka Journal of Mathematics. 45(1) P.107-P.111
Issue Date2008-03
Text Versionpublisher
URL http://hdl.handle.net/11094/11893
DOI
Rights
Flapan, E. and Naimi, R.
Osaka J. Math.
45 (2008), 107–111
THE Y-TRIANGLE MOVE
DOES NOT PRESERVE INTRINSIC KNOTTEDNESS
ERICA FLAPAN and RAMIN NAIMI
(Received March 7, 2006, revised January 31, 2007)
Abstract
We answer the question “Does the Y-triangle move preserve intrinsic
knottedness?” in the negative by giving an example of a graph that is obtained
from the intrinsically knotted graph K7 by triangle-Y and Y-triangle moves but is
not intrinsically knotted.
1. Introduction
A graph is said to be intrinsically knotted (IK) if every embedding of it in R3
contains a cycle that is a nontrivial knot. Similarly, a graph is said to be intrinsically
linked (IL) if every embedding of it in R3 contains a nontrivial link. Sachs [5] and
Conway and Gordon [1] showed that K6, the complete graph on six vertices, is IL.
Conway and Gordon [1] also showed that K7 is IK.
A ∇Y move on an abstract graph consists of removing the edges of a 3-cycle abc
in the graph, and then adding a new vertex v and connecting it to each of the vertices
a, b, and c, as shown in Fig. 1. The reverse of this move is called a Y∇ move. Note
that in a Y∇ move, the vertex v cannot have degree greater than three.
Sachs [5] noticed that additional IL graphs can be obtained from K6 by doing fi-
nite sequences of ∇Y and Y∇ moves on it. Motwani, Raghunathan, and Saran [3]
showed that performing a ∇Y move on any IK or IL graph produces a graph with
the same property. Robertson, Seymour, and Thomas [4] (Lemmas 1.2 and 5.1 (iii))
proved that a Y∇ move on any IL graph produces an IL graph again.
It has been an open question whether a Y∇ move on an IK graph always pro-
duces an IK graph again. We prove that the answer is negative, by giving a knotless
embedding of a graph G7 that is obtained from K7 by ∇Y and Y∇ moves.
A graph H is a minor of another graph G if H can be obtained from G by a
finite sequence of edge deletions and contractions and vertex deletions [2]. A graph is
said to be minor minimal with respect to a property if the graph has that property but
no minor of it has the property.
We work with connected, finite, simple graphs, i.e., graphs with no loops (an edge
whose endpoints are the same) and no double-edges (two edges with the same pair
2000 Mathematics Subject Classification. Primary 05C10; Secondary 57M25.
108 E. FLAPAN AND R. NAIMI
Fig. 1. ∇Y and Y∇ moves.
of endpoints). This is because loops and double-edges do not affect whether or not a
graph is IK or IL: they can always be embedded such that they bound small disks with
interiors disjoint from the rest of the graph. Thus, in edge contractions and Y∇ moves
on an abstract graph, whenever a double-edge is introduced, one of the two edges is
deleted.
2. Description of the graph G7
We label the seven vertices of the abstract graph K7 with the letters a through g.
We perform the following five ∇Y and two Y∇ moves on G0 = K7 to obtain the
graph G7.
1. G0 → G1 by ∇Y on abc, with new vertex h as center.
2. G1 → G2 by ∇Y on ade, with new vertex i as center.
3. G2 → G3 by ∇Y on afg, with new vertex j as center.
4. G3 → G4 by ∇Y on bdf, with new vertex k as center.
5. G4 → G5 by ∇Y on beg, with new vertex l as center.
6. G5 → G6 by Y∇ on hij, deleting vertex a.
7. G6 → G7 by Y∇ on hkl, deleting vertex b.
3. G7 is not IK
Theorem 1. The Y∇ move does not preserve intrinsic knottedness.
Proof. Recall that K7 is IK, ∇Y moves preserve IKness, and G7 is obtained from
K7 by ∇Y and Y∇ moves. Thus it suffices to prove that the embedding of G7 shown
in Fig. 2 has no nontrivial knots.
Fig. 2 contains seven crossings, numbered 1–7. Note that rotating this diagram
by 180◦ about a horizontal line through its center leaves the embedded graph invariant,
swaps crossing 1 with 2, and 6 with 7, and leaves crossings 3, 4, 5 fixed. And rotating
the diagram by 180◦ about a vertical line through the center also leaves the embedded
graph invariant, but swaps crossing 1 with 7, 2 with 6, and 3 with 5.
Suppose towards contradiction that this embedded graph contains a nontrivial knot
K . The proof consists of the following three steps.
Y-TRIANGLE MOVE DOESN’T PRESERVE INTRINSIC KNOTTEDNESS 109
Fig. 2. A knotless embedding of G7.
STEP 1. K must contain exactly one of the edges ef and ij.
Proof. We will show that if K contains neither or both edges, then it is a triv-
ial knot.
Suppose K contains neither ef nor ij. Then it does not contain any of the cross-
ings 1, 2, 6, and 7, and must therefore contain crossings 3, 4, and 5. Hence K
contains the edges ec, ih, dk, gl, fc, and jh. If K contains ei or fj, then at
least one of its crossings can be untwisted, making K trivial. So K must contain el,
fk, id, and jg. Then K is easily seen to be trivial.
Now suppose K contains both ef and ij. Then it cannot contain both 3 and 5,
since otherwise it would be a link. So, by symmetry, we can assume K does not
contain 5. Furthermore, if K contains fj, then it is trivial. It follows that K must
contain at least one of fk or jg. By symmetry, we can assume it contains fk. We
claim that K must contain dk, since otherwise it will contain at most three crossings,
3, 1, and 6; but 1 and 6 do not alternate, which makes K trivial. Now, dkfe can be
isotoped, with fixed endpoints, to eliminate 1, 4, and 7. So K must contain 3, 2, and
6. If K contains jh, 3 and 6 will not alternate, making K trivial. So K must contain
jg. But then 6 can be isotoped away, again making K trivial. This proves Step 1.
So, by symmetry, we can assume K contains ef and not i j . Hence K does not
contain crossings 2 or 6.
110 E. FLAPAN AND R. NAIMI
STEP 2. K must contain 1, 4, and 7.
Proof. Suppose, towards contradiction, that K does not contain gl. Then it con-
tains at most three crossings, 3, 5, and 7; but ec, cf, and fe form a cycle, and
therefore only links contain all three crossings 3, 5, and 7. Hence K contains gl. By
a symmetric argument, K contains dk. Thus K contains crossings 1, 4, and 7.
STEP 3. K contains exactly one of 3 and 5.
Proof. If it contains both, it will be a link. If it contains neither, it will be trivial,
since 1 and 4 do not alternate.
So, by symmetry, we can assume that K contains 1, 3, 4, and 7, and no other
crossings. As K does not contain ij, this implies that K contains di. But hidk is
isotopic, with fixed endpoints, to hk. Thus K is isotopic to a knot that contains only
crossing 1, and therefore is trivial.
ACKNOWLEDGMENTS. The second author thanks Caltech for its hospitality while
he worked on this paper during his sabbatical leave.
References
[1] J.H. Conway and C.McA. Gordon: Knots and links in spatial graphs, J. Graph Theory 7
(1983), 445–453.
[2] R. Diestel: Graph Theory, Graduate Texts in Mathematics 173, Springer-Verlag, New York,
1997.
[3] R. Motwani, A. Raghunathan and H. Saran: Constructive results from graph minors: Link-
less embeddings, 29th Annual Symposium on Foundations of Computer Science, IEEE, 1988,
398–409.
[4] N. Robertson, P. Seymour and R. Thomas: Sachs’ linkless embedding conjecture, J. Combin.
Theory Ser. B 64 (1995), 185–227.
[5] H. Sachs: On spatial representations of finite graphs; in Finite and Infinite sets, Vol. I, II (Eger,
1981), Colloq. Math. Soc. János Bolyai 37, North-Holland, Amsterdam, 1984, 649–662.
Y-TRIANGLE MOVE DOESN’T PRESERVE INTRINSIC KNOTTEDNESS 111
Erica Flapan
Mathematics Department
Pomona College
Claremont, CA 91711
USA
e-mail: eflapan@pomona.edu
Ramin Naimi
Mathematics Department
Occidental College
Los Angeles, CA 90041
USA
e-mail: rnaimi@oxy.edu


The Y-triangle move does not preserve intrinsic knottedness

We use results on the local theta correspondence to prove that for large degrees the Duke-Imamoglu-Ikeda lifting of an elliptic modular form is not a linear combination of theta series

Schulze-Pillot, Rainer

Osaka University
Title Local theta correspondence and the lifting of Duke, Imamogluand Ikeda
Author(s)Schulze-Pillot, Rainer
CitationOsaka Journal of Mathematics. 45(4) P.965-P.971
Issue Date2008-12
Text Versionpublisher
URL http://hdl.handle.net/11094/4222
DOI
Rights
Schulze-Pillot, R.
Osaka J. Math.
45 (2008), 965–971
LOCAL THETA CORRESPONDENCE
AND THE LIFTING OF DUKE, IMAMOGLU AND IKEDA
RAINER SCHULZE-PILLOT
(Received September 13, 2007)
Abstract
We use results on the local theta correspondence to prove that for large
degrees the Duke-Imamoglu-Ikeda lifting of an elliptic modular form is not a linear
combination of theta series.
1. Introduction
In the article [7] it is proved as a side remark that a Siegel modular form F of
degree 2n and weight n + k obtained by the lifting of Duke, Imamoglu and Ikeda [5]
(called the DII-lifting in the sequel) from an elliptic modular form of weight 2k is
not a linear combination of theta series of even unimodular positive definite quadratic
forms of rank m = 2(n + k) if n is bigger than k, whereas for n = k  0 mod 2 the DII
liftings lie in the space generated by theta series subject to a conjecture on L-functions
of elliptic cuspidal Hecke eigenforms. The proof uses Böcherer’s characterization of
the cuspidal Siegel eigenforms that lie in the space of theta series by special values of
their standard L-functions.
In this article we use results on the local theta correspondence to give a different
proof of the first result. In fact we prove a more general version including theta series
attached to arbitrary non degenerate quadratic forms and also theta series with spherical
harmonics. The underlying local fact has been noticed by several people (including the
author) immediately after the preprint version of [5] became available in 1999 but has
apparently never been published.
In the case n = k we show that the local representations attached to a DII-lift are
in the image of the local theta correspondence with the orthogonal group of a suitable
local quadratic space of dimension 4n = 2(n + k) for all places.
I thank Ralf Schmidt for discovering a mistake in an earlier version of this article.
2. The case n > k
Let f be an elliptic modular form of weight 2k for the full modular group SL2(Z).
It was conjectured by Duke and Imamoglu and proven by Ikeda in [5] that for any
2000 Mathematics Subject Classification. Primary 11F46; Secondary 11F27.
966 R. SCHULZE-PILLOT
n  k mod 2 there exists a nonzero Siegel cusp form F = F2n( f ) of weight n + k for
the group Sp2n(Z)  SL4n(Z) whose standard L-function is equal to
 (s)
2n
Y
i=1
L(s + k + n   i , f ),
where L(s, f ) is the usual Hecke L-function of f . We call F2n( f ) the DII-lift of de-
gree 2n of f .
In fact, it has been shown in [16, Lemma 1.3.1], that the Satake parameters ( j )p
of F at the prime p are given by
(2.1) (0)p =  np , ( j )p = p p n+ j 1=2 for 1  j  2n,
where we write the p-factor of L(s, f ) as (1   p p s+k 1=2) 1(1    1p p s+k 1=2) 1,
i.e., we have (by Ramanujan-Petersson) the normalization jpj = 1.
Moreover, Ikeda has recently announced a generalization of this result. In this
generalized version f is replaced by an irreducible cuspidal automorphic representation
 =
N

v
of GL2(AE ), where E is a totally real number field and v is assumed to be
a principal series representation attached to a character 
v
for all finite places v of E
and to be discrete series with minimal weight 
w
for the infinite places w of E .
Ikeda then proves the existence of an irreducible cuspidal automorphic representa-
tion  = (m,  ) of the metaplectic group eSpm(AE ) with local components v described
in terms of 
v
as follows: For a real place w the representation 
w
is the lowest weight
representation of lowest K -type detw+m=2, for a finite place v the representation 
v
is
a degenerate principal series representation which is induced from a character (m)
v
on
the maximal parabolic ˜Pm derived from v
If m = 2n is even one can obtain from  a representation of Sp2n(AE ), also de-
noted by  = (2n,  ); this representation is induced from the 2n-tuple of characters

v
j j
 n+ j 1=2 for 1  j  2n. If (for E = Q) in addition f is as above and  is the
automorphic representation of GL2(AQ) associated to f this gives the representation of
Sp2n(AQ) associated to F2n( f ).
We are interested in the question whether the Siegel modular form F2n( f ) can be
obtained as a linear combination of theta series, respectively in the more general situ-
ation whether the representation (2n,  ) is in the image of the theta correspondence
between Sp2n and a suitable orthogonal group.
We let V be a vector space over E of even dimension 2r with a nondegenerate
quadratic form q and associated symmetric bilinear form B(x , y) = q(x + y) q(x) q(y)
on it. If E = Q and q is positive definite we consider a homogenous pluriharmonic
form P 2 C[fX i j j 1  i  2r , 1  j  mg] in 2rm variables of weight  (see [3, III,
3.5]) as a function on (V 
R)m by identifying the latter space with R2rm using a basis
of V which is orthonormal with respect to the symmetric bilinear form B.
LOCAL THETA CORRESPONDENCE AND DII LIFTING 967
The theta series of degree m of L with respect to P is defined for any Z-lattice
L of full rank 2r on V by
#
(m)(Z , L , P) =
X
x2Lm
P(x) exp(2 i tr(q(x)Z )),
where Z 2 Hm is in the Siegel upper half space Hm  MSymm (C) and where q(x) =
(1=2)(B(xi , x j )) is half the Gram matrix of the m-tuple x = (x1, : : : , xm). It is a Siegel
modular form of weight r +  for some congruence subgroup of Spm(Z). One has a
similar definition for arbitrary totally real E and totally positive definite q.
If L is even unimodular the theta series is a Siegel modular form for the full mod-
ular group Spm(Z).
More generally one considers for arbitrary E and (non degenerate) q the theta cor-
respondence associating to a subset of the set of irreducible automorphic representations
of the adelic orthogonal group O(V ,q)(AE ) a subset of the set of irreducible automorphic
representations of the adelic group Spm(AE ). There is by now a vast literature (starting
with Weil’s [18] and Howe’s [4]) but no textbook reference on this correspondence, for
definitions and properties of it see [14, 11, 12, 8, 10].
Theorem 2.1. Let E , V , q,  be as above.
(1) If 2n > r   1 and there is a finite place v of E for which the completion V
v
of the quadratic space (V , q) is not split (i.e. is not an orthogonal sum of hyperbolic
planes) the representation (2n,  ) is not in the image of the theta correspondence with
O(V ,q)(AE ).
(2) If 2n > r the representation (2n,  ) is not in the image of the theta correspon-
dence with O(V ,q)(AE ).
Corollary 2.2. Let f be an elliptic modular form of weight 2k as above,  2 N0.
Then for n > k    the DII-lift F2n( f ) is not a linear combination of theta series of
positive definite quadratic forms with pluriharmonic forms of degrees  0  .
In particular for n > k the DII-lift F2n( f ) is not a linear combination of theta se-
ries attached to positive definite quadratic forms (with or without pluriharmonic forms).
Proof of the theorem. If (2n,  ) is in the image of the global theta correspon-
dence with O(V ,q)(AE ) its local components v(2n,  ) are in the image of the local
theta correspondence with O(V ,q)(Ev) for all places v of E ; we denote by  0
v
the cor-
responding representation of this orthogonal group.
The local theta correspondence has been described explicitly in terms of the Bernstein-
Zelevinsky data of the representations in [14, 8]. By construction the Bernstein-
Zelevinsky data of 
v
(2n,  ) are 
v
j j
 n+ j 1=2 for 1  j  2n. The characters

v
j j
 n+ j 1=2 are never of the type j ji for some integral i ; this is clear if 
v
is ram-
ified, and follows from j
v
(!
v
)j < q5=34
v
(see [6]) if 
v
is unramified (where !
v
is a
968 R. SCHULZE-PILLOT
prime element at the place v and q
v
is the norm of !
v
). Since by the results of [14, 8]
only characters of the type j ji for some integral i can be missing in the Bernstein-
Zelevinsky data of the representation  0 on the orthogonal side we see that all the

v
j j
 n+ j 1=2 for 1  j  2n have to appear in these Bernstein-Zelevinsky data. Obvi-
ously the rank of the orthogonal group O(V ,q)(Ev) has to be at least 2n for this to be
possible. If V
v
is split this requires r  2n, if V
v
is not split it requires r   1  2n
(and even r   2  2n if the anisotropic kernel of V
v
is of dimension 4).
Proof of the corollary. It is well known that one can associate to a cuspidal Siegel
modular form F for Sp2n(Z) which is an eigenfunction of all Hecke operators an ir-
reducible cuspidal automorphic representation (F) of Sp2n(AQ) with the same Satake
parameters, see e.g. [2]. Moreover, if F has weight ˜k and can be written as a lin-
ear combination of theta series of positive definite quadratic forms with pluriharmonic
forms of weights  0  , the fact that the space generated by theta series with pluri-
harmonic forms of weight  0 for lattices on a fixed quadratic space is Hecke invariant
guarantees that there is a positive definite quadratic space (V , q) of dimension 2r =
2(˜k   0) for some  0   such that the irreducible representation (F) is in the image
of the theta correspondence with O(V ,q)(AQ).
Considering this for the DII lift F2n( f ) and ,  0, k as in the assertion we have
˜k = k + n and therefore r = k + n    0  k + n    and k < n + , hence r < 2n.
By the theorem (F2n( f )) is not in the image of the theta correspondence with
O(V ,q)(AQ), so F2n( f ) can not be a linear combination of theta series as described
above.
REMARK. (1) If we omit the restriction that the quadratic form is positive def-
inite we still get the same result as long as we restrict attention to the holomorphic
theta series of weight r associated to indefinite quadratic forms of rank 2r that have
been constructed by Siegel and Maaß [17, 9]. Although there is no method known to
use suitable test functions in the oscillator representation in order to construct holo-
morphic theta series of weight k + n    0 associated (by the theta correspondence) to
indefinite quadratic forms of rank larger than 2(k + n    0), the results of Rallis [15]
seem to imply that such a construction is indeed possible. Such a construction would
then yield F2n( f ) without contradicting the theorem.
(2) The corollary could in principle be proved without using representation theoretic
tools by computing the possible Hecke eigenvalues of a linear combination of theta
series with the help of results of Andrianov [1] and Yoshida [19] and comparing with
the eigenvalues of a DII-lift; we expect this to be a rather tedious and unpleasant com-
putation.
LOCAL THETA CORRESPONDENCE AND DII LIFTING 969
3. The case n = k
With the notations of the previous section we assume now E = Q and n = k, so
that the weight k + n of a DII lift is equal to the rank 2n of the symplectic group
considered.
Theorem 3.1. For n = k the local components p of the representation (F2n( f ))
of the Duke-Imamoglu-Ikeda lift F2n( f ) are in the image of the local theta correspon-
dence with the split quadratic space over Qp of dimension 4n = 2(k + n) (which is the
orthogonal sum of 2n hyperbolic planes) for all (finite) primes p.
The component 
1
at the real place is in the image of the theta correspondence
with the orthogonal group of the positive definite quadratic space over R of dimension
4n and also in the image of the theta correspondence with the orthogonal group of the
quadratic space of dimension 4n and signature (4n   1, 1) over R.
Proof. The assertion for the finite primes is again an immediate consequence of
the results of [14, 8]. The assertion at the real place follows from Theorem 15 of [13]
since 
1
is a limit of discrete series.
REMARK. (1) At the real place it follows from Theorem 15 of [13] that 
1
oc-
curs also in the images of the theta correspondence with the orthogonal groups of the
quadratic spaces of dimension 4n + 2 and signatures (4n + 1, 1) and (4n, 2) over R, but
not for any other quadratic space of dimension 4n or 4n + 2.
(2) Since the split quadratic space of dimension 4n has square discriminant, there is
no quadratic space over Q which is split at all finite primes and of signature (4n 1, 1).
It is moreover well known that there is a positive definite quadratic space over Q of
even dimension 2r which is split at all finite primes if and only if r is divisible by 4
and that the same condition is necessary and sufficient for the existence of a quadratic
space over Q of even dimension 2r + 2 and signature (2r + 1, 1) which is split at all
finite primes
We see that (F2n( f )) can (in the case n = k) be in the image of the global theta
correspondence with some quadratic space of dimension 4n = 2(n + k) only if n is even
and the quadratic space is the unique positive definite space of that dimension which
is split at all finite places.
In a similar way for even n there is also a unique quadratic space (V , q) over Q
of dimension 4n + 2 and signature (4n + 1, 1) for which 
v
is in the image of the local
theta correspondence with the orthogonal group of the completion V
v
for all (finite or
infinite) places v of Q; this space is again split at all finite primes.
There are also spaces of signature (4n, 2) for which 
v
is in the image of the lo-
cal theta correspondence with the orthogonal group of the completion V
v
for all (finite
or infinite) places v of Q; such a space can be obtained as the orthogonal sum of an
imaginary quadratic field equipped with the norm form scaled by some negative num-
970 R. SCHULZE-PILLOT
ber and a positive definite space of dimension 4n which is split at all finite places. In
all cases our purely local methods allow no statement about occurrence in the global
theta correspondence for the respective space.
4. Iterated theta liftings
One might ask whether it is possible to construct the Duke-Imamoglu-Ikeda lift
F2n( f ) or the associated automorphic representation by a sequence of theta liftings be-
tween groups G i , where for each i the pair G i , G i+1 consists (in either order) of a
symplectic or metaplectic group and an orthogonal group, starting with the representa-
tion associated to f on SL2 or the representation on the metaplectic group eSL2 asso-
ciated to the form g which corresponds to f under the Shimura correspondence. We
have more generally:
Proposition 4.1. Let the notations be as in Sections 1 and 2 and n > 1. The
generalized Duke-Imamoglu-Ikeda lift (2n,  ) can not be constructed by a series of
theta liftings as described above.
Proof. If one starts out with the metaplectic group eSL2 all subsequent groups G i
will be orthogonal groups of quadratic spaces in odd dimension or metaplectic groups
eSpm (with a genuine representation of eSpm on it) so that we will never arrive at Sp2n ,
with the exception that the initial step may lead to SO(3, 2) identified with PGSp2
or to SO(2, 1) identified with PGL2 (in which case we obtain a Saito-Kurokawa lift
respectively a Shimura lift). After this initial step there have to appear correspon-
dences which raise the rank of the group, and again by [14, 8] all local representa-
tions occurring will (at each finite place v of E) have at most two terms 
v
j j
j with
j 2 (1=2)Z n Z among their Bernstein-Zelevinsky data, so 
v
(2n,  ) can never occur
for n > 1.
If one starts with SL2 the representation v(2n,  ) can not occur if n > 1 for the
same reason.
References
[1] A.N. Andrianov: Quadratic Forms and Hecke Operators, Grundlehren der Mathematischen
Wissenschaften 286, Springer, Berlin, 1987.
[2] M. Asgari and R. Schmidt: Siegel modular forms and representations, Manuscripta Math. 104
(2001), 173–200.
[3] E. Freitag: Siegelsche Modulfunktionen, Grundlehren der Mathematischen Wissenschaften 254,
Springer, Berlin, 1983.
[4] R. Howe: -series and invariant theory; in Automorphic Forms, Representations and L-Functions
(Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 1, Proc. Sympos.
Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, 275–285.
LOCAL THETA CORRESPONDENCE AND DII LIFTING 971
[5] T. Ikeda: On the lifting of elliptic cusp forms to Siegel cusp forms of degree 2n, Ann. of Math.
(2) 154 (2001), 641–681.
[6] H.H. Kim and F. Shahidi: Functorial products for GL2  GL3 and functorial symmetric cube
for GL2, C.R. Acad. Sci. Paris Sér. I Math. 331 (2000), 599–604.
[7] W. Kohnen and R. Salvati Manni: Linear relations between theta series, Osaka J. Math. 41
(2004), 353–356.
[8] S.S. Kudla: On the local theta-correspondence, Invent. Math. 83 (1986), 229–255.
[9] H. Maaß: Modulformen zu indefiniten quadratischen Formen, Math. Scand. 17 (1965), 41–55.
[10] C. Mœglin, M.-F. Vignéras and J.-L. Waldspurger: Correspondances de Howe sur un Corps
p-Adique, Lecture Notes in Math. 1291, Springer, Berlin, 1987.
[11] C. Mœglin: Non nullité de certains relévements par séries théta, J. Lie Theory 7 (1997),
201–229.
[12] C. Mœglin: Quelques propriétés de base des séries théta, J. Lie Theory 7 (1997), 231–238.
[13] A. Paul: On the Howe correspondence for symplectic-orthogonal dual pairs, J. Funct. Anal.
228 (2005), 270–310.
[14] S. Rallis: Langlands’ functoriality and the Weil representation, Amer. J. Math. 104 (1982),
469–515.
[15] S. Rallis: On the Howe duality conjecture, Compositio Math. 51 (1984), 333–399.
[16] R. Schmidt: On the spin L-function of Ikeda’s lifts, Comment. Math. Univ. St. Pauli 52 (2003),
1–46.
[17] C.L. Siegel: Indefinite quadratische Formen und Modulfunktionen; in Studies and Essays
Presented to R. Courant on his 60th Birthday, January 8, 1948, Interscience Publisher, Inc.,
New York, 1948, 395–406.
[18] A. Weil: Sur la formule de Siegel dans la théorie des groupes classiques, Acta Math. 113
(1965), 1–87.
[19] H. Yoshida: The action of Hecke operators on theta series; in Algebraic and Topological The-
ories (Kinosaki, 1984), Kinokuniya, Tokyo, 1986, 197–238.
Fachrichtung 6.1 Mathematik
Universität des Saarlandes (Geb. E2.4)
Postfach 151150, 66041 Saarbrücken
Germany
e-mail: schulzep@math.uni-sb.de


Local theta correspondence and the lifting of Duke, Imamoglu and Ikeda

Kawaguchi, Ryo

The gonality conjecture for curves on certain toric surfaces

カツドウ　ホウコク

ミズノ, アキコ

水野, 亜紀子

ヒグチ　イチヨウ　コノコ　アリフレタコト　ヲ　ワダイ　ト　スル　ワタシ

フカザワ, アイ

深澤, 愛

ヒョウキ　ケンキュウ　ニ　ムケテ　ノ　カノウセイ

Osaka University

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