Linear forms, algebraic cycles, and
derivatives of L-series
Shou-Wu Zhang
September 13, 2010
Contents
1 Modular L-functions as linear forms
2
2 Gross–Prasad conjectures
2.1 Linear forms and pairing . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Global conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Arithmetic Gross–Prasad conjecture . . . . . . . . . . . . . . . . . . . . . . .
3
3
4
5
3 Rallis Inner product formula
3.1 Local theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Inner product formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Arithmetical inner product formula . . . . . . . . . . . . . . . . . . . . . . .
6
6
8
8
Introduction
This is note for a talk given at the conference “Number Theory and Representation Theory”
in honor of the 60th birthday of Benedict Gross, to be held on June 2–June 5, 2010. In
this note, I will state some refinements of conjectures of Gross–Prasad [7, 8] and Kudla [20]
concerning the central derivatives of L-series and special cycles on Shimura varieties. The
analogues of our formulation for special values of L-series are written in terms of invariant
linear forms on autormorphic representations with inner products defined by integrations
of matrix coefficients. These formulae look simpler than but are actually equivalent to the
formulae in Ichino–Ikeda [16], Neel Harris [11], and Rallis’ inner product formula [28, 13,
14]. In the derivative situation, our formula will use another space of invariant forms with
Beilinson–Bloch height pairings. Again our formulae look simpler than but equivalent to
those given in Wei Zhang [35], and Yifeng Liu [26]. We will also state some recent work
about this formulae by Yuan–Zhang–Zhang [32, 33] and Yifeng Liu [27].
1
Acknowledgment
We would like thank W. T. Gan, B. Gross, J.-S. Li, Y. Liu, A. Venkatesh, and W. Zhang
for their help in preparation of this note.
1
Modular L-functions as linear forms
Let k be a positive integer. There is an action of GL2 (R)+ , the group of real matrix with
positive determinants, on the space of holomorphic functions on H by the following formula:
az + b det γ k/2
a b
f 7→ f |k γ,
f |k γ(z) = f
, γ=
.
c d
cz + d (cz + d)k
A function f is called a modular form, if it is invariant under some congruence subgroup Γ
and whose fourier expansion vanishes at every cusp. Let Sk denote the space of cusp forms of
weight k. Then Sk admits an action by GL2 (Q)+ , the group of two by two rational matrices
b + , the
with positive determinants. Such an action can be extended into an action by GL2 (Q)
Q
×
b =(
group of matrices over Q
p Zp ) ⊗ Q with determinants in Q>0 .
For each f ∈ Sk , define its L-series by Mellin transform
Z ∞
dy
f (iy)y s .
L(f, s) =
y
0
Then it is easy to see that this function is entire for s ∈ C. For each diagonal element
a 0
γ=
, we have
0 d
Z ∞
dy
f (iay/d)(a/d)k/2 y s
L(f |k γ, s) =
= (a/d)k/2−s L(f, s).
y
0
Let A denote the subgroup of GL2 (Q)+ of diagonal elements. For each t ∈ C, define a
character χt by
a 0
χt
= (a/d)t .
0 d
We embed A into A × GL2 (Q)+ diagonally, then f 7→ L(f, s) define an element in
`s ∈ HomA (Sk ⊗ χs−k/2 , C).
To understand this last space, we decompose Sk into a direct sum of irreducible representations π which is generated by a unique new form fπ . Write L(s, π) = L(fπ , s). Let `s,π
denote the restriction of `s on π ⊗ χs−k/2 :
`s,π ∈ HomA (π ⊗ χs−k/2 , C).
On other hand, it can be shown that the right hand side is one dimensional and is generated
one element es such that
`s = L(π, s)es .
Thus we have just explain that the L-series for newforms are just factors of the Mellin
transform over a base.
2
2
Gross–Prasad conjectures
2.1
Linear forms and pairing
Let F be a local field and (E, σ) a semisimple algebra over F of degree ≤ 2 with an involution
σ so that the fixed part of E is F , i.e., (E, σ) is either F with σ = 1, a quadratic field extension
of F with σ the non-trivial element of Gal(E/F ), or the direct sum F ⊕ F with σ switches
two factors. Let V be a hermitian space over E of dimension n with respect to σ. Let e ∈ V
with nonzero norm and let W be the complement of e in V . Define algebraic groups over G
and H as follows:
G = U (V ) × U (W ),
H = U (W )
and embed H into G diagonally. Fix a Haar measure on H. Notice that according the three
cases of (E, σ), U (V ) is either an orthogonal group, unitary group, or the general linear
group. We consider the general linear case as the split unitary case.
Let π be an admissible representation of G. The H-invariant form on π is define as
P(π) = HomH (π, C).
Then it is known that P(π) is at most one dimensional by recent work of Aizenbud, Gourevitch, Rallis, Schiffman, Sun, and Zhu [1, 30]. When π is tempered, then one defines an
element
I ∈ P(π) ⊗ P(e
π)
by matrix coefficient integrals:
Z
I(v ⊗ ve) =
H
hπ(h)v, veidh.
Sakellaridis and Venkatesh [29] have recently shown that the nonvanishing of P(π) is equivalent to the non-vanishing of I. According to the computations of Ichino–Ikeda [16] in
orthogonal case and Neel Harris [11] in unitary case, we may normalize the I in the following
way
L(1, π, ad)
I.
m :=
L(M ∨ (1), 0)L(1/2, π)
In this way,
m(v ⊗ ve) = 1
in the spherical case:
• π is unramified,
• G and H are hyperspecial with standard measure,
• v and ve are spherical elements such that hv, vei = 1.
3
Here M is the Gross motive attached to U (V ) which has the L-series
(
ζ(2) · ζ(4) · · · ζ(n − 1)
if n is odd
L(M, 0) =
ζ(2)ζ(4) · · · ζ(n − 2)L(n/2, χdisc(V ) ) if n is even
if F = E and otherwise,
(
L(1, η)ζ(2)L(3, η) · · · L(n − 2, η)ζ(n − 1) if n is odd
L(M, 0) =
L(1, η)ζ(2)L(3, η) · · · L(n − 1, η)
if n is even
We assume that π is tempered and define a bilinear form
P(π) ⊗ P(e
π ) −→ C
h·, ·i :
by
` ⊗ `e
,
` ⊗ `e ∈ P(π) ⊗ P(e
π ).
m
By work of H. He [12], when π is unitary, this pairing is positive definite.
e =
h`, `i
2.2
Global conjecture
Let F be a number field with adeles A and (E, σ) a semisimple algebra over F of degree ≤ 2
such that E σ = F . Let V be a hermitian space over (E, σ) and let W be the orthogonal
complement for some element e. Define analogously the group G and H as in the local case.
Let π = ⊗πv be an irreducible cuspidal representation of G(AF ). We define
P(π) = ⊗v P(πv ).
When it is not zero, then we define a bilinear form
h·, ·i :
P(π) ⊗ P(e
π ) −→ C
by product of local ones.
Define an element Pπ ∈ P(π) by
Z
φ(h)dh,
Pπ (φ) =
H(F )\H(A)
where the integration is taken using Tamagawa measure. Then we have the following conjecture of Gross–Prasad [7, 8] and Gan–Gross–Prasad ([5]), and the refinements by Ichino–Ikeda
[16] and Neel Harris [11]:
Conjecture 2.2.1. Assume that P(π) 6= 0. There is an integer β such that
hPπ , Pπe i =
L(M ∨ (1), 0) 1
L( , π).
2β L(1, π, ad) 2
b of the image of Arthur parameter of π,
Moreover, if S is the centralizer in the dual group G
then
2β = |S|.
4
Q
Remark 2.2.1. Decompose L(s, π) = L(s, Mi ) into a product of self-dual L-series according
to a decomposition of the Arthur parameter. Then P(π) = 0 if one of the these L-series
has odd sign of functional equation. If all of these L-series have even sign of functional
equation, then Gan, Gross, and Prasad have conjectured a unique inner form (H 0 , G0 ) of
(H, G) such that P(π 0 ) 6= 0 for the Jacquet–Langlands correspondence π 0 of π. In particular
if L(1/2, π) 6= 0 or if S is a singlet, such an inner form always exists.
Remark 2.2.2. The conjecture holds in the following cases:
1. V is hermitian of dimension ≤ 2 or orthogonal of dimension ≤ 3, by Waldspurger [31];
2. V is orthogonal of dimension 4 by Ichino [15].
Remark 2.2.3. For unitary group, H. Jacquet and S. Rallis [23] have formulated a relative
trace formula approach to attack Gross–Prasasde conjecture. The corresponding fundamental lemma has just been proved by Z. Yun [34].
2.3
Arithmetic Gross–Prasad conjecture
Let F be a totally real number field and E be either F or a quadratic CM-extension. Let
V be an incoherent hermitian space over AE and let W be the orthogonal complement of an
element e with norm in F × . We define the group G and H for U (V) × U (W) and U (W) as
algebraic groups over A and embed H into G diagonally.
We assume that V has signature (n, 0) at all archimedean place of F . Then G define a
projective system XK of Shimura varieties XK over E parameterized by compact subgroups
K of G(Af ). Let X denote the projective limit of XK which is a scheme over E with an
action by G(Af ). We define Ch∗ (X) the direct limit of Ch∗ (XK ), the Chow group of cycles
on XK . For an irreducible representation π of G with trivial archimedean components, define
Ch∗ (π) := HomG (π, Ch∗ (X)C ).
Then the so called modularity conjectures says that this group is non-vanishing only if π is
automorphic in the sense that it has automorphic lifting to the quasi-split coherent group of
G and that we have decomposition
M
Ch∗ (X)C =
π ⊗ Ch∗ (π)
π∈A (G)
Here A (G) denote the set of automorphic represenations of G. If π is cuspidal, we even
conjecture that Ch∗ (π) takes values in Ch∗ (X)00
C , the group of homologically trivial cycles.
∗
By work of Beilinson and Bloch [2, 3, 4] Ch (π) is finite dimensional with dimension equal
to ords=1/2 L(s, π) and there is a positive definite height pairing
h·, ·i :
Ch∗ (π) ⊗ Ch∗ (e
π ) −→ C.
5
In duality, we may define Ch∗ (X)K = Chdim X−∗ (XK ), and Ch∗ (X) to be the projective
limit of Ch∗ (XK ) under the projection maps as transaction. Then we have a decomposition
dual to the above:
Y
Ch∗ (X) =
π ∨ ⊗ Ch∗ (π ∨ )
π∈A (G)
where π ∨ = Hom(π, C) as G-modulo, and
Ch∗ (π ∨ ) = HomG (π ∨ , Ch∗ (X)C ) = Chdim X−∗ (e
π ).
The group H defines a projective system YK of subvarieties of XK with limit Y . The cycle Y
has a class in Ch∗ (X) and has component Yπ in π ∨ ⊗ Ch∗ (π ∨ ). It is clear that Y is invariant
in H thus we may replace π ∨ by P(π) = (π ∨ )H . Thus we have define element
Yπ ∈ P(π) ⊗ Ch∗ (e
π ).
The arithmetical Gross–Prasad conjecture formulated in Gan–Gross–Prasad [5] and refined by W. Zhang [35] is as follows:
Conjecture 2.3.1. Assume that P(π) 6= 0. Then
hYπ , Yπe i =
L(M ∨ (1), 0) 0
L (1/2, π).
2β L(1, π, ad)
Remark 2.3.1. Assume that the root number of π is -1 and S is a singlet. Then Gan, Gross,
and Prasad have conjectured a unique incoherent inner form (G0 , H0 ) of (G, H) such that
P(π 0 ) 6= 0 for Jacquet–Langlands correspondence π 0 of π.
Remark 2.3.2. The conjecture holds in the following situations:
1. V is unitary of dimension 2 by Yuan–Zhang–Zhang [32] as a generalization of Gross–
Zagier formula [9].
2. V is orthogonal of dimension 4 by Yuan–Zhang–Zhang [33].
Remark 2.3.3. Inspired by work of Jacquet–Rallis [23], W. Zhang [35] has formulated a
relative trace formula approach to attack the arithmetic Gross–Prasad conjecture for unitary
group. W. Zhang [36] also proved the corresponding fundamental lemma when n = 3.
3
Rallis Inner product formula
3.1
Local theory
Let F be a local field. We consider a reductive pairing (G, H) of following two types:
1. G = O(V ) with for V a symmetric space over F of odd dimension 2n + 1, H = Sp(F 2n )
with standard symplectic form on F 2n .
6
2. G = U (V ) for V a hermitian space over quadratic field of even dimension 2n, and
H = U (E 2n ) with a standard skew hermitian form on E 2n .
We embed G × H into Mp(V 2n ) and let R be the Weil representation of Mp(V 2n ) realized
on the Schwartz of functions on the polarization V n with inner product defined over L with
respect to Haar measure. For any irreducible representations π we consider the space of
linear forms
P(π) := ⊕σ HomH×G (π ⊗ R, σ)
where the sum is over a complete set of inequivalent irreducible representations σ of G.
Then the Howe duality says that this space is at most one dimensional. In particular there
is at most one irreducible representation σ (called theta lifting of π) contributes to P(π).
Moreover, if π is tempered, by recent work of J.-S. Li, M. Harris, and B.-Y. Sun [25], this
space is non-vanishing if and only if the matrix coefficients integrations is non-vanishing: for
e
some f ∈ π ⊗ R, fe ∈ π
e ⊗ R,
Z
hhf, feidh 6= 0.
H
Following work of J. Li [24], we may normalize the matrix integral so that it takes value one
in spherical case by
Z
L(1,
M
)
hhf, feidh
m(f, fe) =
L( 12 , π) H
where
L(1, M ) =
n
Y
i
L(i, ηE/F
)
i=1
in unitary case, and
L(1, M ) = ζ(1/2)
n
Y
ζ(2i)
i=1
in orthogonal case. It is easy to see
e C).
m ∈ HomH 2 ×∆(G) (π ⊗ π
e ⊗ R ⊗ R,
Using m, we can define a bilinear form
h·, ·i, P(π) ⊗ P(e
π ) −→ C
by
e =
h`, `i
e fe)i
h`(f ), `(
m(f, fe)
for any choice of f, fe with nonzero m-value. When π is unitary and tempered, this pairing
is positive definite.
7
3.2
Inner product formula
For cuspidal representations π and σ of H and G respectively, we define analogously the
space P(π) and linear forms as product of local datum.
One can define one element Pπ ∈ P(π) as follows:
Z
Pπ (f ⊗ φ)(g) =
θφ (g, h)f (h)dh.
[H]
Here θφ is the theta series:
θφ (g) =
X
g ∈ Mp((V ⊗ W )A ).
R(g)φ(x),
x∈V m
The Rallis Inner Product Formula is given as
Theorem 3.2.1.
hPπ , Pπe i =
1
· L(1/2, π).
2L(1, M )
For reference see Rallis [28], Kudla–Rallis [21], Li [24], and Ichino [13, 14].
3.3
Arithmetical inner product formula
Assume that F is totally real, and E/F is CM . Let V be an incoherent orthogonal over
AF of rank 2n + 1 (resp. hermitian space over AE of rank 2n) which is positive definite at
all archimedean places. Let G denote U (V) as an algebraic group over A. Then we have
a projective system of Shimura varieties XK of dimension 2n − 1 parameterized by open
compact group K of Gf as above. For each irreducible representation σ of G with trivial
archimedean components, define
Ch∗ (σ) = HomG (σ, Ch∗ (X)).
Moreover if σ has cuspidal lifting to H(A), then Ch∗ (σ) has values in homologically trivial
cycles. The Beilinson–Bloch conjecture says that this is a finite dimensional space with
dimension equal to ords=1/2 L(s, π) where π is the theta lifting of σ on H(A), and that there
is a positive definite pairing on Ch∗ (σ).
The arithmetical Rallis inner product formula of Kudla [19] refined by Y. Liu [26] is of
form
Conjecture 3.3.1. Assume that P(π) 6= 0, then the sign of functional equation of L(s, π)
is −1 and there is an explicit
Yπ ∈ P(π) ⊗ Ch(σ)
such that
hYπ , Yπe i =
1
· L0 (1/2, π)
2L(1, M )
8
Here is a sketch of construction of Yπ . For any φ ∈ S (V n ), one can construct a Kudla
generating series Zφ ∈ Chn (X)C . For g ∈ H(A) define
Zφ (g) = ZR(g)φ ∈ Chn (X).
With respect to the Weil action by G × H(A), Z defines an element
Z ∈ HomG×H(A) (R, A (H) ⊗ Chn (X)C ).
Let π be an irreducible cuspidal representation of H(A), the inner product with A (H)
defines
Yπ ∈HomG×H(A) (R ⊗ π, Chn (X)C ))
= HomG×H(A) (R ⊗ π, σ) ⊗ HomG (σ, Chn (X)C ))
= P(π) ⊗ Ch∗ (σ).
Remark 3.3.1. The conjecture hold in the case m = 1 by Kudla, Rapoport, and Yang [22]
and Yifeng Liu [27].
References
[1] A. Aizenbud, D. Gourvitch, S. Rallis, and G. Schiffmann, Multiplicity one theorems,
Arxiv: 0709.4215; to appear in Annals of Mathematics.
[2] A. Beilinson, Higher regulators and values of L-functions. J. Soviet Math., 30 (1985),
2036-2070.
[3] A. Beilinson, Height pairing between algebraic cycles. Current trends in arithmetical
algebraic geometry (Arcata, Calif., 1985), 1–24, Contemp. Math., 67, Amer. Math.
Soc., Providence, RI, 1987.
[4] S. Bloch, Height pairings for algebraic cycles. Proceedings of the Luminy conference on
algebraic K-theory (Luminy, 1983). J. Pure Appl. Algebra 34 (1984), no. 2-3, 119–145.
[5] W. T. Gan, B. Gross, and D. Prasad, Symplectic local root numbers, central critical Lvalues, and restriction problems in the represenation theory of classical groups, Preprint,
available at http://www.math.harvard.edu/ gross
[6] S. Gelbart, I. Piatetski-Shapiro and S. Rallis, Explicit Constructions of Automorphic
L-Functions, Lecture Notes in Mathematics 1254, Springer-Verlag Berlin Heidelberg
1987
[7] B. Gross and D. Prasad, On the decomposition of a representation of SOn when restricted
to SOn−1 . Canad. J. Math. 44 (1992), no. 5. 974-1002.
9
[8] B. Gross and D. Prasad, On irreducible representations of SO2n+1 × SO2m . Canad. J.
Math. 46 (1994), no. 5, 974-1002.
[9] B. Gross and D. Zagier, Heegner points and derivatives of L-series, Invent. Math. 84
(1986), no 2, 225-320.
[10] M. Harris, S. Kudla and W. Sweet, Jr., Theta dichotomy for unitary group, J. Amer.
Math. Soc. 9 (1996), 941-1004
[11] N. Harris, Thesis at UCSD, in preparation.
[12] H. He, Unitary representations and the theta correspondence for type I classical groups.
Journal of Functional Analysis, 199, 92-121 (2003)
[13] A. Ichino, A regularized Siegel-Weil formula for unitary groups, Math. Z. 247 (2004),
241-277
[14] A. Ichino, On the Siegel-Weil formula for unitary groups, Math. Z. 255 (2007), 721-729
[15] A. Ichino, Trilinear forms and the central values of triple product L-functions, Duke
Math. Journal. 145 (2), 281-307 (2008)
[16] A. Ichino and T. Ikeda, On the periods of automorphic forms on special orthogonal
groups and the Gross–Prasad conjecture, to appear in Geometric and Functional Analysis.
[17] S. Kudla, Central derivatives of Eisenstein series and height pairings, Ann. of Math.
146 (1997), 545-646
[18] S. Kudla, Derivatives of Eisenstein series and generating functions for arithmetic cycles,
Séminaire Boubaki 876, Astérisque 276, Société Mathématique de France, Paris 2002,
341-368
[19] S. Kudla, Modular forms and arithmetic geometry, in Current Developments in Mathematics 2002, International Press, Somerville 2003, 135-179
[20] S. Kudla, Special cycles and derivatives of Eisenstein series, in Heegner points and
Rankin L-series, Math. Sci. Res. Inst. Publ., 49, 243–270, Cambridge Univ. Press, Cambridge, 2004
[21] S. Kudla and S. Rallis, A regularized Siegel-Weil formula: The first term identity, Ann.
of Math. 140 (1994), 1-80
[22] S. Kudla, M. Rapoport and T. Yang, Modular Forms and Special Cycles on Shimura
Curves, Annals of Mathematics Studies 161, Princeton University Press, Princeton and
Oxford 2006
10
[23] H. Jacquet and S. Rallis, On the Gross–Prasad conjecture for unitary groups, Clay
Mathematics Proceedings.
[24] J.-S. Li, Non-vanishing theorems for the cohomology of certain arithemetic quotients, J.
Reine Angew. Math 428 (1992), 177-217
[25] J. -S. Li, M. Harris, and B. -Y. Sun, Theta correspondences for close unitary groups,
Preprint.
[26] Y. Liu, Arithmetic theta lifting and L-derivatives for unitary groups I. preprint.
[27] Y. Liu, Arithmetic theta lifting and L-derivatives for unitary groups II. preprint.
[28] S. Rallis, Injectivity properties of liftings associated to Weil representations, Comp.
Math. 52 (1984), 136-169
[29] Y. Sakelaridis and A. Venkatesh, Periods and harmonic analysis on spherical varieties,
in preparation.
[30] B. Sun and C. Zhu, Multiplicity one theorems: the Archimedean case, arXiv 0903.1413.
[31] J. - L. Waldspurger, Sur les valeurs de certaines fonctions L automorphes en leaur centre
de symmétre. Compositio Mathematica, 54 (2), 193-242 (1985)
[32] X. Yuan, S. Zhang, and W. Zhang, Heights of CM points on Shimura curves and Gross–
Zagier formula, preprint.
[33] X. Yuan, S. Zhang, and W. Zhang, Gross–Schoen cycles and triple product L-series, in
preparation.
[34] Z. Yun, The fundamental lemma of Jacquet-Rallis, to appear in Duke Math. J.
[35] W. Zhang, Relatice trace formula and arithmetical Gross–Prasad conjecture, preprint
(2009)
[36] W. Zhang, On arithmetic fundamental lemmas, submitted
11