Download Waldspurger formula over function fields

Waldspurger formula over function fields
Fu-Tsun Wei
(Joint work with Chih-Yun Chuang)
Pan Asia Number Theory Conference 2016
2016 July 14
Outline
1. Statement of the main theorem.
2. Central critical values of Rankin-Selberg L-functions.
3. Gross points on definite Shimura curves and the theta
element.
4. Proof of the main theorem (sketch).
Outline
1. Statement of the main theorem.
2. Central critical values of Rankin-Selberg L-functions.
3. Gross points on definite Shimura curves and the theta
element.
4. Proof of the main theorem (sketch).
Outline
1. Statement of the main theorem.
2. Central critical values of Rankin-Selberg L-functions.
3. Gross points on definite Shimura curves and the theta
element.
4. Proof of the main theorem (sketch).
Outline
1. Statement of the main theorem.
2. Central critical values of Rankin-Selberg L-functions.
3. Gross points on definite Shimura curves and the theta
element.
4. Proof of the main theorem (sketch).
Basic settings
k: a global function field with odd characteristic.
A: the adele ring of k.
D: a quaternion algebra over k.
DA := D ⊗k A.
K : a quadratic field extension over k, together with an
embedding ι : K ֒→ D.
KA := K ⊗k A.
Basic settings
k: a global function field with odd characteristic.
A: the adele ring of k.
D: a quaternion algebra over k.
DA := D ⊗k A.
K : a quadratic field extension over k, together with an
embedding ι : K ֒→ D.
KA := K ⊗k A.
Basic settings
k: a global function field with odd characteristic.
A: the adele ring of k.
D: a quaternion algebra over k.
DA := D ⊗k A.
K : a quadratic field extension over k, together with an
embedding ι : K ֒→ D.
KA := K ⊗k A.
Basic settings
k: a global function field with odd characteristic.
A: the adele ring of k.
D: a quaternion algebra over k.
DA := D ⊗k A.
K : a quadratic field extension over k, together with an
embedding ι : K ֒→ D.
KA := K ⊗k A.
Basic settings
k: a global function field with odd characteristic.
A: the adele ring of k.
D: a quaternion algebra over k.
DA := D ⊗k A.
K : a quadratic field extension over k, together with an
embedding ι : K ֒→ D.
KA := K ⊗k A.
Basic settings
k: a global function field with odd characteristic.
A: the adele ring of k.
D: a quaternion algebra over k.
DA := D ⊗k A.
K : a quadratic field extension over k, together with an
embedding ι : K ֒→ D.
KA := K ⊗k A.
Toric period integrals
Let Π be an automorphic cuspidal representation of GL2 (A),
with a unitary central character ωΠ . Assume that Π corresponds
to an automorphic representation ΠD of DA× via the
Jacquet-Langlands correspondence. Given a unitary Hecke
character χ : K × \KA× → C× , suppose ωΠ · χA× ≡ 1.
For each f ∈ ΠD , define
Z
Pχ (f ) :=
K × A× \KA×
f ι(a) · χ(a) d × a.
Toric period integrals
Let Π be an automorphic cuspidal representation of GL2 (A),
with a unitary central character ωΠ . Assume that Π corresponds
to an automorphic representation ΠD of DA× via the
Jacquet-Langlands correspondence. Given a unitary Hecke
character χ : K × \KA× → C× , suppose ωΠ · χA× ≡ 1.
For each f ∈ ΠD , define
Z
Pχ (f ) :=
K × A× \KA×
f ι(a) · χ(a) d × a.
Toric period integrals
Here the chosen measure d × a satisfies
vol(K × A× \KA× , d × a) = 2L(1, ξK /k ),
where ξK /k is the quadratic character associated to K /k.
e D be the contragredient representation of ΠD . Define
Let Π
e D → C by
Pχ : ΠD ⊗ Π
Pχ (f ⊗ f̃ ) := Pχ (f ) · Pχ−1 (f̃ ).
Toric period integrals
e D ) as ⊗v ΠD (resp. ⊗v Π
e D ), we choose a
Writing ΠD (resp. Π
v
v
D
D
e
local pairing h·, ·iv : Πv × Πv → C for each place v of k so that
h·, ·iPet =
2L(1, Π, Ad) Y
· h·, ·iv .
ζk (2)
v
e D → C is the pairing induced from the
Here h·, ·iPet : ΠD × Π
Petersson inner product (with respect to the Tamagawa
measure on DA× /A× ).
Toric period integrals
eD
For each place v of k, define Pχ,v : ΠD
v ⊗ Πv → C by
Pχ,v (fv ⊗ f̃v ) :=
Lv (1, ξK /k ) · Lv (1, Π, Ad)
Lv ( 1 , Π × χ) · ζk ,v (2)
Z 2
hΠD (ι(av ))fv , f̃v iv · χv (av ) d × av .
·
Kv× /kv×
It is observed that when v is “good,” one has Pv (fv ⊗ f̃v ) = 1.
e D → C by
We may define Pχ : ΠD ⊗ Π
Pχ := ⊗v Pχ,v .
Waldspurger formula over function fields
Theorem 1 (Chuang-W.)
Let Π be an automorphic cuspidal representation of GL2 (A),
with a unitary central character ωΠ . Given a unitary Hecke
character χ : K × \KA× → C× , suppose ωΠ · χA× ≡ 1. Then
1
Pχ = L( , Π × χ) · Pχ .
2
It is known ([Tunnell] and [Waldspurger], also [Gross-Prasad])
that for each place v of k, Pχ,v 6≡ 0 if and only if
ǫv (Πv × χv ) = χv (−1)ξK /k ,v (−1)ǫv (D).
(∗)
Here ǫv (Πv × χv ) is the local root number of Lv (s, Πv × χv ), and
ǫv (D) is the Hasse invariant of D.
Waldspurger formula over function fields
Theorem 1 (Chuang-W.)
Let Π be an automorphic cuspidal representation of GL2 (A),
with a unitary central character ωΠ . Given a unitary Hecke
character χ : K × \KA× → C× , suppose ωΠ · χA× ≡ 1. Then
1
Pχ = L( , Π × χ) · Pχ .
2
It is known ([Tunnell] and [Waldspurger], also [Gross-Prasad])
that for each place v of k, Pχ,v 6≡ 0 if and only if
ǫv (Πv × χv ) = χv (−1)ξK /k ,v (−1)ǫv (D).
(∗)
Here ǫv (Πv × χv ) is the local root number of Lv (s, Πv × χv ), and
ǫv (D) is the Hasse invariant of D.
Waldspurger formula over function fields
Theorem 1 (Chuang-W.)
Let Π be an automorphic cuspidal representation of GL2 (A),
with a unitary central character ωΠ . Given a unitary Hecke
character χ : K × \KA× → C× , suppose ωΠ · χA× ≡ 1. Then
1
Pχ = L( , Π × χ) · Pχ .
2
It is known ([Tunnell] and [Waldspurger], also [Gross-Prasad])
that for each place v of k, Pχ,v 6≡ 0 if and only if
ǫv (Πv × χv ) = χv (−1)ξK /k ,v (−1)ǫv (D).
(∗)
Here ǫv (Πv × χv ) is the local root number of Lv (s, Πv × χv ), and
ǫv (D) is the Hasse invariant of D.
Non-vanishing criterion of L(1/2, Π × χ)
Corollary
Q
Suppose v ǫv (Πv × χv ) = 1. Let D be the quaternion algebra
over k satisfying (∗) for every place v of k. Take an embedding
ι : K ֒→ D. Then L(1/2, Π × χ) is non-vanishing if and only if
there exists f ∈ ΠD such that
Z
f ι(a) χ(a) d × a 6= 0.
Pχ (f ) =
K × A× \KA×
Gross-type formula of L(1/2, Π × χ)
From now on, for simplicity we assume that k = Fq (T ) with q
odd, the central character of Π is trivial, and the conductor of Π
is n∞, where n is a square-free ideal of A = Fq [T ].
√
Let K = k( D), where D ∈ A is square-free with non-zero even
degree and the leading coefficients√
of D is not a square in F√
q
(then ∞ is inert in K ). Let OK := A[ D] and Oc := A + c · A[ D]
for each ideal c of A. Every character χ of Pic(Oc ) can be
viewed as a Hecke character on K × \KA× via the isomorphism
b × · K ×.
Pic(Oc ) ∼
= K × \KA× /O
c
∞
Gross-type formula of L(1/2, Π × χ)
From now on, for simplicity we assume that k = Fq (T ) with q
odd, the central character of Π is trivial, and the conductor of Π
is n∞, where n is a square-free ideal of A = Fq [T ].
√
Let K = k( D), where D ∈ A is square-free with non-zero even
degree and the leading coefficients√
of D is not a square in F√
q
(then ∞ is inert in K ). Let OK := A[ D] and Oc := A + c · A[ D]
for each ideal c of A. Every character χ of Pic(Oc ) can be
viewed as a Hecke character on K × \KA× via the isomorphism
b × · K ×.
Pic(Oc ) ∼
= K × \KA× /O
c
∞
Gross-type formula of L(1/2, Π × χ)
For simplicity, assume (n, cD) = 1. Write n = n+ · n− , where
n− =
Y
p|n,
p inert in K
p
and
n+ :=
n
.
n−
Assume that the number of prime factors of n− is odd. Let D be
the quaternion algebra over k which is ramified precisely at ∞
and primes p dividing n− . Choose an Eichler A-order
Rn+ ,n− ⊂ D of type (n+ , n− ), together with an optimal
embedding ι : Oc ֒→ Rn+ ,n− .
Gross-type formula of L(1/2, Π × χ)
Let
φΠ : GL2 (k)\ GL2 (A)/K0 (n∞) → C
and
b ×+ − O × ) → C
φΠD : D × \DA× /(R
D∞
n ,n
be newforms associated to Π and ΠD , respectively. Assume
that φΠ is normalized. Then the central critical value
L(1/2, Π × χ) can be expressed as follows:
Gross-type formula of L(1/2, Π × χ)
Theorem 2 (Chuang-W.)
Suppose χ is primitive of conductor c. Then
1
L( , Π × χ) =
2
kφΠ kPet
kck · |D|1/2
P
2
[A]∈Pic(Oc ) φΠD ι(A) χ([A])
·
.
kφΠD kPet
Here kck := #(A/c) and |D| := q deg D .
Gross-type formula of L(1/2, Π × χ)
Remark:
1. In fact, we are able to derive such a formula for L( 12 , Π × χ)
only under the assumptions that:
(i) the central character of Π is unramified everywhere;
(ii) n is square-free.
2. This formula was known in the following special cases:
(i) k is rational, n is prime, D is irreducible, and c = 1 (Papikian
2005);
(ii) k is rational, n is square-free, D is irreducible, and c = 1
(W.-Yu 2011);
(iii) n is square-free and c = 1 (Chuang-W.-Yu).
Gross-type formula of L(1/2, Π × χ)
Remark:
1. In fact, we are able to derive such a formula for L( 12 , Π × χ)
only under the assumptions that:
(i) the central character of Π is unramified everywhere;
(ii) n is square-free.
2. This formula was known in the following special cases:
(i) k is rational, n is prime, D is irreducible, and c = 1 (Papikian
2005);
(ii) k is rational, n is square-free, D is irreducible, and c = 1
(W.-Yu 2011);
(iii) n is square-free and c = 1 (Chuang-W.-Yu).
Theta element
Let Gc := Pic(Oc ). There exists a unique element
X
L = LcΠ,K =
cσ · σ ∈ C[Gc ]
σ∈Gc
so that for each character χ : Gc → C× , we have
χ(LcΠ,K ) =
Lc (1/2, Π × χ)
.
kφΠ kPet
From the above Gross-type formula, we may describe LcΠ,K
explicitly by using the “Gross points” on definite Shimura
curves.
Definite Shimura curve
Let Y be the genus 0 curve over k so that the points of Y over
any k-algebra M are
Y (M) = {x ∈ D ⊗k M : Tr(x) = Nr(x) = 0}/M × ,
where Tr and Nr are respectively the reduce trace and the
reduced norm on D. The group D × acts on Y (from the left) by
conjugation.
Definition
The definite Shimura curve X = Xn+ ,n− of type (n+ , n− ) is
b n+ ,n− .
X := D × \ Y × DA∞,× /R
Definite Shimura curve
Let Y be the genus 0 curve over k so that the points of Y over
any k-algebra M are
Y (M) = {x ∈ D ⊗k M : Tr(x) = Nr(x) = 0}/M × ,
where Tr and Nr are respectively the reduce trace and the
reduced norm on D. The group D × acts on Y (from the left) by
conjugation.
Definition
The definite Shimura curve X = Xn+ ,n− of type (n+ , n− ) is
b n+ ,n− .
X := D × \ Y × DA∞,× /R
Let I1 , ..., In be representatives of right ideal classes of
R = Rn+ ,n− . Then
n
a
Ri× \Y
X=
i=1
where for each i, Ri is the left order of Ii . Hence X is a finite
disjoint union of genus 0 curves, and the components
correspond canonically to left ideal classes of R. Therefore we
may identify Pic(X ) with the free abelian group generated by
the double cosets in
b n+ ,n− .
D × \DA∞,× /R
Moreover, φΠD can be viewed as an element in Hom(Pic(X ), C)
by extending additively.
Gross points
There is a canonical identification of Y (K ) with Hom(K , D). We
call a point x = [y, g] ∈ X a Gross point of conductor c over K if
"
#
∞,× b ×
x ∈ Image Y (K ) × D
/R → X (K )
A
satisfying that
b = ιy (Oc ).
ιy (K ) ∩ g −1 Rg
Here ιy is the embedding of K into D corresponding to y. We
have a natural free action of Gc = Pic(Oc ) on the set of Gross
points of conductor c over K .
Gross points
We assumed that (n, cD) = 1. Fix a Gross point x = x(c) ∈ X of
conductor c over K . For each divisor c′ of c, we take the unique
Gross point x(c′ ) ∈ X of conductor c′ over K which occurs in
Tc/c′ x(c) . It is observed that for p | c′ ,
Nc′ /(c′ /p) (x(c′ ) )


Tp x(c′ /p) − x(c′ /p) ,



T x ′ − (x σq1 + x σq2 ),
p (c /p)
(c′ /p)
(c′ /p)
=
σq

Tp x(c′ /p) − x(c′ /p) ,



Tp x(c′ /p) ,
if p | (c′ /p) in K ,
if p ∤ (c′ /p) and split in K ,
if p ∤ (c′ /p) and ramified in K ,
if p ∤ (c′ /p) and inert in K .
Gross points
We also assumed that Π has trivial central character. For each
prime p | c, take αp to be a root of X 2 − ap (Π)X + kpk, where ap
is the “Hecke eigenvalue of φΠ at p.” For each divisor c′ of c, put
Y ord (c′ )
αc′ :=
αp p .
p|c′
Let
z(c) := α−1
c ·
X
c′ |c
µ(c′ )α−1
c′ · [x(c/c′ ) ] ∈ Pic(X ) ⊗Z C.
Gross points
We also assumed that Π has trivial central character. For each
prime p | c, take αp to be a root of X 2 − ap (Π)X + kpk, where ap
is the “Hecke eigenvalue of φΠ at p.” For each divisor c′ of c, put
Y ord (c′ )
αc′ :=
αp p .
p|c′
Let
z(c) := α−1
c ·
X
c′ |c
µ(c′ )α−1
c′ · [x(c/c′ ) ] ∈ Pic(X ) ⊗Z C.
Gross points
Then:
Proposition
For c′ | c, we have
φΠD


Y


Nc/c′ (z(c) ) = φΠD ( ep ) · z(c′ )  ,
p|c,
p∤c′
where ep ∈ C[Gc′ ] is defined by

−1
−1

(1 − αp · σq1 )(1 − αp · σq2 ),
ep := (1 − α−1
p · σq ),


−1
(1 − αp ) · (1 + α−1
p ),
if p splits in K ,
if p is ramified in K ,
if p is inert in K .
Gross points
For each character χ on Gc′ and p ∤ c′ , it is observed that
1
χ(ep ) · χ(ep ) = Lp ( , Π × χ)−1 .
2
Therefore by Theorem 2, we obtain:
Proposition
Suppose Π has trivial central character and square-free
conductor n∞, with (n, D) = 1. For each ideal c of A coprime to
n, we have that for each character χ : Gc → C× ,
P
σ∈Gc
σ )χ(σ)2
φΠD (z(c)
kφΠD kPet · |D|1/2
=
Lc (1/2, Π × χ)
.
kφΠ kPet
Gross points
For each character χ on Gc′ and p ∤ c′ , it is observed that
1
χ(ep ) · χ(ep ) = Lp ( , Π × χ)−1 .
2
Therefore by Theorem 2, we obtain:
Proposition
Suppose Π has trivial central character and square-free
conductor n∞, with (n, D) = 1. For each ideal c of A coprime to
n, we have that for each character χ : Gc → C× ,
P
σ∈Gc
σ )χ(σ)2
φΠD (z(c)
kφΠD kPet · |D|1/2
=
Lc (1/2, Π × χ)
.
kφΠ kPet
Theta element
Recall that we let LcΠ,K ∈ C[Gc ] be the unique element so that
for each character χ : Gc → C× , we have
χ(LcΠ,K ) =
Lc (1/2, Π × χ)
.
kφΠ kPet
From the above Proposition, we may express LcΠ,K as follows:
Theorem 3 (Chuang-W.)
Suppose Π has trivial central character and square-free
conductor n∞, with (n, D) = 1. For each ideal c of A coprime to
n, we have
P
P
−1
σ
σ
σ∈Gc φΠD (z(c) ) · σ
σ∈Gc φΠD (z(c) ) · σ
c
.
LΠ,K =
kφΠD kPet · |D|1/2
Waldspurger formula over function fields
Recall the main theorem:
Theorem 1 (Chuang-W.)
Let Π be an automorphic cuspidal representation of GL2 (A),
with a unitary central character ωΠ . Given a unitary Hecke
character χ : K × \KA× → C× , suppose ωΠ · χA× ≡ 1. Then
1
Pχ = L( , Π × χ) · Pχ .
2
Proof of Theorem 1: Rankin-Selberg method
Write D = K + Kj, where j 2 = γ ∈ k × , we may decompose the
quadratic space (D, NrD/k ) into:
(D, NrD/k ) = (V1 , Q1 ) ⊕ (V2 , Q2 ),
where (V1 , Q1 ) = (K , NK /k ) and (V2 , P
Q2 ) = (K , −γ · NK /k ).
Given ϕ ∈ S(DA ), we may write ϕ = j ϕ1,j ⊕ ϕ2,j where
ϕi,j ∈ S(Vi (A)). For f ∈ Π, put
=
Z(f , ϕ, s)
XZ
j
+
GL2 K
+
(k )A× \ GL2 K
(A)
f (g)θχV1 (g, ϕ1,j )E (g, s, ϕ2,j )dg
Rankin-Selberg method
Applying Rankin-Selberg method, we have:
Proposition
Suppose ϕ and f are pure tensors, then
Y
Z(f , ϕ, s) =
Zv (fv , ϕv , s),
v
where Zv (fv , ϕv , s) is equal to
Z Z
NK /k (a) 0 1 D 1
Wfv
κv
ωv (κv )ϕv (ā) d κ1v
0
1
Kv× SL2 (Ov )
s− 12
·χv (a)| NK /k (a)|v
d ×a
Rankin-Selberg method
Remark: when v is “good,” one has
Zv (fv , ϕv , s) =
Lv (s, Π × χ)
.
Lv (2s, ξK /k )
Thus
L(2s, ξK /k ) · Z(f , ϕ, s) = L(s, Π × χ) ·
Y
v
Zvo (fv , ϕv , s),
where
Zvo (fv , ϕv , s) :=
Lv (2s, ξK /k )
· Zv (fv , ϕv , s).
Lv (s, Π × χ)
L(1, ξK /k ) · Z(f , ϕ, 1/2) ↔ Pχ
Siegel-Weil formula:
1
E (g, , ϕ2 ) = L(1, ξK /k )−1 · θ1VK2 (g, ϕ2 ),
2
∀ϕ2 ∈ S(V2 (A)).
Seesaw identity:
+K
[GO(V1 ) × GO(V2 )]
GL
 2
_
diagonal
+K
+K
❯❯❯❯
✐✐✐✐
❯❯❯❯
❯✐❯✐❯✐❯✐✐✐
✐
❯❯❯❯
✐✐
❯
✐✐✐✐
[GL2 × GL2 ]
_
+
GO(D) K .
L(1, ξK /k ) · Z(f , ϕ, 1/2) ↔ Pχ
Siegel-Weil formula:
1
E (g, , ϕ2 ) = L(1, ξK /k )−1 · θ1VK2 (g, ϕ2 ),
2
∀ϕ2 ∈ S(V2 (A)).
Seesaw identity:
+K
[GO(V1 ) × GO(V2 )]
GL
 2
_
diagonal
+K
+K
❯❯❯❯
✐✐✐✐
❯❯❯❯
❯✐❯✐❯✐❯✐✐✐
✐
❯❯❯❯
✐✐
❯
✐✐✐✐
[GL2 × GL2 ]
_
+
GO(D) K .
L(1, ξK /k ) · Z(f , ϕ, 1/2) ↔ Pχ
Therefore we get
L(1, ξK /k ) · Z(f , ϕ, 1/2)
Z
Z
=
θ D (h1 , h2 ; f , ϕ) · χ(h1 h2−1 )dh1 dh2 ,
K × A× \A×
K
K × A× \A×
K
where for b1 , b2 ∈ DA× ,
θ D (b1 , b2 ; f , ϕ)
Z
f g1 α(b1 b2−1 ) · θ VD g1 α(b1 b2−1 ), [b1 , b2 ]; ϕ dg1 .
:=
SL2 (k )\ SL2 (Ak )
Here α(b) :=
1
0
for every b ∈ A×
D.
0 NrD/k (b)
Shimizu correspondence
Put
Then
ΘD (Π) := θ D (·, ·; f , ϕ) | f ∈ Π, ϕ ∈ S(DA )) .
Shimizu correspondence
In particular,
eD.
ΘD (Π) = ΠD ⊗ Π
L(1, ξK /k ) · Z(f , ϕ, 1/2) = Pχ θ D (·, ·; f , ϕ) .
Local Shimizu correspondence
Given fv ∈ Πv and ϕv ∈ S(Dv ), put
Z
′
θv (bv , bv ; fv , ϕv ) =
·
for bv , bv′ ∈ Dv× , and
θvo (bv , bv′ ; fv , ϕv ) :=
Wfv gv1 α(bv bv′−1 )
U(kv )\ SL2 (kv )
ωvD (gv1 α(bv bv′−1 ), [bv , bv′ ])ϕv (1)dgv1
ζv (2)
· θv (bv , bv′ ; fv , ϕv ).
Lv (1, Πv , Ad)
Local Shimizu correspondence
Proposition
1. θvo (bv , bv′ ; fv , ϕv ) = 1 when v is “good.”
2. For b1 , b2 ∈ A×
D,
Z
θfD (bb1 , bb2 ; ϕ)db
×
D × A× \DA
=
2L(1, Π, Ad) Y o
·
θv (b1,v , b2,v ; fv , ϕv ).
ζk (2)
v
o
3. Let ΘD
v (Πv ) be the space consisting of θv (·, ·; fv , ϕv ) for
fv ∈ Πv and ϕv ∈ S(Dv ). Then
∼ D eD
ΘD
v (Πv ) = Πv ⊗ Πv .
Zvo (fv , ϕv , 1/2) ↔ Pχ,v
It is observed that
1
Zv ( ; fv , ϕv ) =
2
Z
Kv× /kv×
θv (hv , 1; fv , ϕv )χv (hv )d × hv .
Therefore
Zvo (fv , ϕv , 1/2) =
Lv (1, ξK /k ) · Lv (1, Π, Ad)
Lv ( 1 , Π × χ) · ζk ,v (2)
Z 2
θvo (hv , 1; fv , ϕv )χv (hv )d × hv
·
×
×
Kv /kv
= Pχ,v θvo (·, ·; fv , ϕv ) .
Q.E.D.
The end. Thank you for your attention!