Periods of automorphic forms and special values of...
Transcript of Periods of automorphic forms and special values of...
![Page 1: Periods of automorphic forms and special values of L-functionsmhsaito/agsymkobe07/proceedings/ichino.pdf · ¶Method to study aut L-functions ‡ • Integral representation (more](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d14d47e708231d4389625/html5/thumbnails/1.jpg)
Periods of automorphic forms and
special values of L-functions
Atsushi Ichino (Osaka City University)
August 7, 2007
1
![Page 2: Periods of automorphic forms and special values of L-functionsmhsaito/agsymkobe07/proceedings/ichino.pdf · ¶Method to study aut L-functions ‡ • Integral representation (more](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d14d47e708231d4389625/html5/thumbnails/2.jpg)
Plan
§1 Automorphic forms
§2 L-functions
§3 Periods
2
![Page 3: Periods of automorphic forms and special values of L-functionsmhsaito/agsymkobe07/proceedings/ichino.pdf · ¶Method to study aut L-functions ‡ • Integral representation (more](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d14d47e708231d4389625/html5/thumbnails/3.jpg)
§1 Automorphic forms
3
![Page 4: Periods of automorphic forms and special values of L-functionsmhsaito/agsymkobe07/proceedings/ichino.pdf · ¶Method to study aut L-functions ‡ • Integral representation (more](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d14d47e708231d4389625/html5/thumbnails/4.jpg)
²±
¯°automorphic rep + rep of G(A) with high symmetry
G : semisimple algebraic group over QA =
∏p≤∞
′Qp : adele ring of Q, loc cpt top ring
p : prime or ∞, Q∞ = RQ ↪→ A : diagonal embedding, discrete subring
π : irred automorphic rep of G(A)
G(A) =∏
p≤∞
′G(Qp) ⇒ π =
⊗p≤∞
′πp
πp : irred rep of G(Qp)
4
![Page 5: Periods of automorphic forms and special values of L-functionsmhsaito/agsymkobe07/proceedings/ichino.pdf · ¶Method to study aut L-functions ‡ • Integral representation (more](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d14d47e708231d4389625/html5/thumbnails/5.jpg)
²±
¯°automorphic rep + rep of G(A) with high symmetry
G : semisimple algebraic group over QA =
∏p≤∞
′Qp : adele ring of Q, loc cpt top ring
p : prime or ∞, Q∞ = RQ ↪→ A : diagonal embedding, discrete subring
π : irred automorphic rep of G(A)
G(A) =∏
p≤∞
′G(Qp) ⇒ π =
⊗p≤∞
′πp
πp : irred rep of G(Qp)
4-a
![Page 6: Periods of automorphic forms and special values of L-functionsmhsaito/agsymkobe07/proceedings/ichino.pdf · ¶Method to study aut L-functions ‡ • Integral representation (more](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d14d47e708231d4389625/html5/thumbnails/6.jpg)
²±
¯°automorphic rep + rep of G(A) with high symmetry
G : semisimple algebraic group over QA =
∏p≤∞
′Qp : adele ring of Q, loc cpt top ring
p : prime or ∞, Q∞ = RQ ↪→ A : diagonal embedding, discrete subring
π : irred automorphic rep of G(A)
G(A) =∏
p≤∞
′G(Qp) ⇒ π =
⊗p≤∞
′πp
πp : irred rep of G(Qp)
4-b
![Page 7: Periods of automorphic forms and special values of L-functionsmhsaito/agsymkobe07/proceedings/ichino.pdf · ¶Method to study aut L-functions ‡ • Integral representation (more](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d14d47e708231d4389625/html5/thumbnails/7.jpg)
G(A) is a loc cpt gp (∃ Haar measure).
G(Q) ⊂ G(A) : discrete subgroup
ρ : rep of G(A) on the space
L2(G(Q)\G(A))
where
[ρ(g)φ](x) = φ(xg)
for g, x ∈ G(A), φ ∈ L2(G(Q)\G(A))
Consider an irreducible subrepresentation
π ⊂ L2(G(Q)\G(A))
and call it an automorphic rep.
5
![Page 8: Periods of automorphic forms and special values of L-functionsmhsaito/agsymkobe07/proceedings/ichino.pdf · ¶Method to study aut L-functions ‡ • Integral representation (more](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d14d47e708231d4389625/html5/thumbnails/8.jpg)
G(A) is a loc cpt gp (∃ Haar measure).
G(Q) ⊂ G(A) : discrete subgroup
ρ : rep of G(A) on the space
L2(G(Q)\G(A))
where
[ρ(g)φ](x) = φ(xg)
for g, x ∈ G(A), φ ∈ L2(G(Q)\G(A))
Consider an irreducible subrepresentation
π ⊂ L2(G(Q)\G(A))
and call it an automorphic rep.
5-a
![Page 9: Periods of automorphic forms and special values of L-functionsmhsaito/agsymkobe07/proceedings/ichino.pdf · ¶Method to study aut L-functions ‡ • Integral representation (more](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d14d47e708231d4389625/html5/thumbnails/9.jpg)
G(A) is a loc cpt gp (∃ Haar measure).
G(Q) ⊂ G(A) : discrete subgroup
ρ : rep of G(A) on the space
L2(G(Q)\G(A))
where
[ρ(g)φ](x) = φ(xg)
for g, x ∈ G(A), φ ∈ L2(G(Q)\G(A))
Consider an irreducible subrepresentation
π ⊂ L2(G(Q)\G(A))
and call it an automorphic rep.
5-b
![Page 10: Periods of automorphic forms and special values of L-functionsmhsaito/agsymkobe07/proceedings/ichino.pdf · ¶Method to study aut L-functions ‡ • Integral representation (more](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d14d47e708231d4389625/html5/thumbnails/10.jpg)
Example¶ ³1 ∈ L2(G(Q)\G(A))
(Fact: G(Q)\G(A) is not nec cpt, but finite volume.)
⇒ the trivial rep is automorphic rep (high symmetry).µ ´
“Building blocks” are cuspidal automorphic rep.
L2(G(Q)\G(A)) = L2disc ⊕ L2
cont L2disc = L2
cusp ⊕ L2res
The trivial rep belongs to L2res and is non-cuspidal.
6
![Page 11: Periods of automorphic forms and special values of L-functionsmhsaito/agsymkobe07/proceedings/ichino.pdf · ¶Method to study aut L-functions ‡ • Integral representation (more](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d14d47e708231d4389625/html5/thumbnails/11.jpg)
Example¶ ³1 ∈ L2(G(Q)\G(A))
(Fact: G(Q)\G(A) is not nec cpt, but finite volume.)
⇒ the trivial rep is automorphic rep (high symmetry).µ ´
“Building blocks” are cuspidal automorphic rep.
L2(G(Q)\G(A)) = L2disc ⊕ L2
cont L2disc = L2
cusp ⊕ L2res
The trivial rep belongs to L2res and is non-cuspidal.
6-a
![Page 12: Periods of automorphic forms and special values of L-functionsmhsaito/agsymkobe07/proceedings/ichino.pdf · ¶Method to study aut L-functions ‡ • Integral representation (more](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d14d47e708231d4389625/html5/thumbnails/12.jpg)
Example¶ ³1 ∈ L2(G(Q)\G(A))
(Fact: G(Q)\G(A) is not nec cpt, but finite volume.)
⇒ the trivial rep is automorphic rep (high symmetry).µ ´
“Building blocks” are cuspidal automorphic rep.
L2(G(Q)\G(A)) = L2disc ⊕ L2
cont L2disc = L2
cusp ⊕ L2res
The trivial rep belongs to L2res and is non-cuspidal.
6-b
![Page 13: Periods of automorphic forms and special values of L-functionsmhsaito/agsymkobe07/proceedings/ichino.pdf · ¶Method to study aut L-functions ‡ • Integral representation (more](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d14d47e708231d4389625/html5/thumbnails/13.jpg)
§2 L-functions
7
![Page 14: Periods of automorphic forms and special values of L-functionsmhsaito/agsymkobe07/proceedings/ichino.pdf · ¶Method to study aut L-functions ‡ • Integral representation (more](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d14d47e708231d4389625/html5/thumbnails/14.jpg)
π =⊗
p≤∞
′πp : irred cusp aut rep of G(A) =
∏p≤∞
′G(Qp)
LG = G o Gal(Q/Q) : L-group of G
G : dual group of G (over C)
r : fin dim rep of LG
s ∈ C
L(s, π, r) :=∏
p:good
det[1 − p−s · r(c(πp))
]−1 ∏p:bad
· · ·
(Fact: almost all primes p are good.)
c(πp) ∈ LG : Satake parameter of πp at good p
®
©ªL-function is defined by an Euler product
8
![Page 15: Periods of automorphic forms and special values of L-functionsmhsaito/agsymkobe07/proceedings/ichino.pdf · ¶Method to study aut L-functions ‡ • Integral representation (more](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d14d47e708231d4389625/html5/thumbnails/15.jpg)
π =⊗
p≤∞
′πp : irred cusp aut rep of G(A) =
∏p≤∞
′G(Qp)
LG = G o Gal(Q/Q) : L-group of G
G : dual group of G (over C)
r : fin dim rep of LG
s ∈ C
L(s, π, r) :=∏
p:good
det[1 − p−s · r(c(πp))
]−1 ∏p:bad
· · ·
(Fact: almost all primes p are good.)
c(πp) ∈ LG : Satake parameter of πp at good p
®
©ªL-function is defined by an Euler product
8-a
![Page 16: Periods of automorphic forms and special values of L-functionsmhsaito/agsymkobe07/proceedings/ichino.pdf · ¶Method to study aut L-functions ‡ • Integral representation (more](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d14d47e708231d4389625/html5/thumbnails/16.jpg)
π =⊗
p≤∞
′πp : irred cusp aut rep of G(A) =
∏p≤∞
′G(Qp)
LG = G o Gal(Q/Q) : L-group of G
G : dual group of G (over C)
r : fin dim rep of LG
s ∈ C
L(s, π, r) :=∏
p:good
det[1 − p−s · r(c(πp))
]−1 ∏p:bad
· · ·
(Fact: almost all primes p are good.)
c(πp) ∈ LG : Satake parameter of πp at good p
®
©ªL-function is defined by an Euler product
8-b
![Page 17: Periods of automorphic forms and special values of L-functionsmhsaito/agsymkobe07/proceedings/ichino.pdf · ¶Method to study aut L-functions ‡ • Integral representation (more](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d14d47e708231d4389625/html5/thumbnails/17.jpg)
π =⊗
p≤∞
′πp : irred cusp aut rep of G(A) =
∏p≤∞
′G(Qp)
LG = G o Gal(Q/Q) : L-group of G
G : dual group of G (over C)
r : fin dim rep of LG
s ∈ C
L(s, π, r) :=∏
p:good
det[1 − p−s · r(c(πp))
]−1 ∏p:bad
· · ·
(Fact: almost all primes p are good.)
c(πp) ∈ LG : Satake parameter of πp at good p
®
©ªL-function is defined by an Euler product
8-c
![Page 18: Periods of automorphic forms and special values of L-functionsmhsaito/agsymkobe07/proceedings/ichino.pdf · ¶Method to study aut L-functions ‡ • Integral representation (more](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d14d47e708231d4389625/html5/thumbnails/18.jpg)
Theorem (Langlands).
L(s, π, r) is absolutely convergent for Re(s) ≫ 0.
Problem¶ ³Show that automorphic L-functions are “nice”.
• meromorphic continuation (MC) to C• functional equation (FE)
L(s, π, r) = ε(s, π, r) · L(1 − s, π∨, r)
(π∨ : contragredient of π)
• holomorphy, poles, non-vanishing . . .µ ´
9
![Page 19: Periods of automorphic forms and special values of L-functionsmhsaito/agsymkobe07/proceedings/ichino.pdf · ¶Method to study aut L-functions ‡ • Integral representation (more](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d14d47e708231d4389625/html5/thumbnails/19.jpg)
Theorem (Langlands).
L(s, π, r) is absolutely convergent for Re(s) ≫ 0.
Problem¶ ³Show that automorphic L-functions are “nice”.
• meromorphic continuation (MC) to C• functional equation (FE)
L(s, π, r) = ε(s, π, r) · L(1 − s, π∨, r)
(π∨ : contragredient of π)
• holomorphy, poles, non-vanishing . . .µ ´
9-a
![Page 20: Periods of automorphic forms and special values of L-functionsmhsaito/agsymkobe07/proceedings/ichino.pdf · ¶Method to study aut L-functions ‡ • Integral representation (more](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d14d47e708231d4389625/html5/thumbnails/20.jpg)
E : elliptic curve over QE(Q) is fin gen abelian group.
L(s, E) : L-function of E
(defined by an Euler product, abs conv for Re(s) > 32)
BSD conjecture (worth for $1,000,000).
ords=1
L(s, E) = rankE(Q)
s = 1 is the center of FE (s ↔ 2 − s),
which is out of the range of convergence.
10
![Page 21: Periods of automorphic forms and special values of L-functionsmhsaito/agsymkobe07/proceedings/ichino.pdf · ¶Method to study aut L-functions ‡ • Integral representation (more](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d14d47e708231d4389625/html5/thumbnails/21.jpg)
E : elliptic curve over QE(Q) is fin gen abelian group.
L(s, E) : L-function of E
(defined by an Euler product, abs conv for Re(s) > 32)
BSD conjecture (worth for $1,000,000).
ords=1
L(s, E) = rankE(Q)
s = 1 is the center of FE (s ↔ 2 − s),
which is out of the range of convergence.
10-a
![Page 22: Periods of automorphic forms and special values of L-functionsmhsaito/agsymkobe07/proceedings/ichino.pdf · ¶Method to study aut L-functions ‡ • Integral representation (more](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d14d47e708231d4389625/html5/thumbnails/22.jpg)
E : elliptic curve over QE(Q) is fin gen abelian group.
L(s, E) : L-function of E
(defined by an Euler product, abs conv for Re(s) > 32)
BSD conjecture (worth for $1,000,000).
ords=1
L(s, E) = rankE(Q)
s = 1 is the center of FE (s ↔ 2 − s),
which is out of the range of convergence.
10-b
![Page 23: Periods of automorphic forms and special values of L-functionsmhsaito/agsymkobe07/proceedings/ichino.pdf · ¶Method to study aut L-functions ‡ • Integral representation (more](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d14d47e708231d4389625/html5/thumbnails/23.jpg)
E : elliptic curve over QE(Q) is fin gen abelian group.
L(s, E) : L-function of E
(defined by an Euler product, abs conv for Re(s) > 32)
BSD conjecture (worth for $1,000,000).
ords=1
L(s, E) = rankE(Q)
s = 1 is the center of FE (s ↔ 2 − s),
which is out of the range of convergence.
10-c
![Page 24: Periods of automorphic forms and special values of L-functionsmhsaito/agsymkobe07/proceedings/ichino.pdf · ¶Method to study aut L-functions ‡ • Integral representation (more](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d14d47e708231d4389625/html5/thumbnails/24.jpg)
Taniyama-Shimura conjecture (proved after Wiles).E : elliptic curve over Q⇒ ∃ πE : irred cuspidal automorphic rep of GL2(A) s.t.
L(s, E) = L(s + 12, πE, st)
st : the standard 2-dim rep of GL2(C) + L-gp of GL2
®
©ªWe have analytic theory of automorphic L-functions
Integral representation:
L(s + 12, πE, st) =
∫Q×\A×
φ
(a 00 1
)|a|s da
φ ∈ πE : suitably normalized
RHS is abs conv for all s ∈ C, so ords=1
L(s, E) is well-def.
11
![Page 25: Periods of automorphic forms and special values of L-functionsmhsaito/agsymkobe07/proceedings/ichino.pdf · ¶Method to study aut L-functions ‡ • Integral representation (more](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d14d47e708231d4389625/html5/thumbnails/25.jpg)
Taniyama-Shimura conjecture (proved after Wiles).E : elliptic curve over Q⇒ ∃ πE : irred cuspidal automorphic rep of GL2(A) s.t.
L(s, E) = L(s + 12, πE, st)
st : the standard 2-dim rep of GL2(C) + L-gp of GL2
®
©ªWe have analytic theory of automorphic L-functions
Integral representation:
L(s + 12, πE, st) =
∫Q×\A×
φ
(a 00 1
)|a|s da
φ ∈ πE : suitably normalized
RHS is abs conv for all s ∈ C, so ords=1
L(s, E) is well-def.
11-a
![Page 26: Periods of automorphic forms and special values of L-functionsmhsaito/agsymkobe07/proceedings/ichino.pdf · ¶Method to study aut L-functions ‡ • Integral representation (more](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d14d47e708231d4389625/html5/thumbnails/26.jpg)
Taniyama-Shimura conjecture (proved after Wiles).E : elliptic curve over Q⇒ ∃ πE : irred cuspidal automorphic rep of GL2(A) s.t.
L(s, E) = L(s + 12, πE, st)
st : the standard 2-dim rep of GL2(C) + L-gp of GL2
®
©ªWe have analytic theory of automorphic L-functions
Integral representation:
L(s + 12, πE, st) =
∫Q×\A×
φ
(a 00 1
)|a|s da
φ ∈ πE : suitably normalized
RHS is abs conv for all s ∈ C, so ords=1
L(s, E) is well-def.
11-b
![Page 27: Periods of automorphic forms and special values of L-functionsmhsaito/agsymkobe07/proceedings/ichino.pdf · ¶Method to study aut L-functions ‡ • Integral representation (more](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d14d47e708231d4389625/html5/thumbnails/27.jpg)
Taniyama-Shimura conjecture (proved after Wiles).E : elliptic curve over Q⇒ ∃ πE : irred cuspidal automorphic rep of GL2(A) s.t.
L(s, E) = L(s + 12, πE, st)
st : the standard 2-dim rep of GL2(C) + L-gp of GL2
®
©ªWe have analytic theory of automorphic L-functions
Integral representation:
L(s + 12, πE, st) =
∫Q×\A×
φ
(a 00 1
)|a|s da
φ ∈ πE : suitably normalized
RHS is abs conv for all s ∈ C, so ords=1
L(s, E) is well-def.
11-c
![Page 28: Periods of automorphic forms and special values of L-functionsmhsaito/agsymkobe07/proceedings/ichino.pdf · ¶Method to study aut L-functions ‡ • Integral representation (more](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d14d47e708231d4389625/html5/thumbnails/28.jpg)
Method to study aut L-functions¶ ³• Integral representation (more difficult to find)
• Langlands-Shahidi method (∃ complete list)µ ´
Let’s consider L(s, πE,Symn).
Symn : symmetric power rep of GL2(C) of dim n + 1
Around 1999-2000, using LS-method,
Kim-Shahidi proved MC & FE for n ≤ 9.
In 2006, using R = T ,
Taylor proved MC & FE for all n,
and “nice” enough to prove Sato-Tate conj for most E!
I’m very shocked.
12
![Page 29: Periods of automorphic forms and special values of L-functionsmhsaito/agsymkobe07/proceedings/ichino.pdf · ¶Method to study aut L-functions ‡ • Integral representation (more](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d14d47e708231d4389625/html5/thumbnails/29.jpg)
Method to study aut L-functions¶ ³• Integral representation (more difficult to find)
• Langlands-Shahidi method (∃ complete list)µ ´
Let’s consider L(s, πE,Symn).
Symn : symmetric power rep of GL2(C) of dim n + 1
Around 1999-2000, using LS-method,
Kim-Shahidi proved MC & FE for n ≤ 9.
In 2006, using R = T ,
Taylor proved MC & FE for all n,
and “nice” enough to prove Sato-Tate conj for most E!
I’m very shocked.
12-a
![Page 30: Periods of automorphic forms and special values of L-functionsmhsaito/agsymkobe07/proceedings/ichino.pdf · ¶Method to study aut L-functions ‡ • Integral representation (more](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d14d47e708231d4389625/html5/thumbnails/30.jpg)
Method to study aut L-functions¶ ³• Integral representation (more difficult to find)
• Langlands-Shahidi method (∃ complete list)µ ´
Let’s consider L(s, πE,Symn).
Symn : symmetric power rep of GL2(C) of dim n + 1
Around 1999-2000, using LS-method,
Kim-Shahidi proved MC & FE for n ≤ 9.
In 2006, using R = T ,
Taylor proved MC & FE for all n,
and “nice” enough to prove Sato-Tate conj for most E!
I’m very shocked.
12-b
![Page 31: Periods of automorphic forms and special values of L-functionsmhsaito/agsymkobe07/proceedings/ichino.pdf · ¶Method to study aut L-functions ‡ • Integral representation (more](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d14d47e708231d4389625/html5/thumbnails/31.jpg)
Method to study aut L-functions¶ ³• Integral representation (more difficult to find)
• Langlands-Shahidi method (∃ complete list)µ ´
Let’s consider L(s, πE,Symn).
Symn : symmetric power rep of GL2(C) of dim n + 1
Around 1999-2000, using LS-method,
Kim-Shahidi proved MC & FE for n ≤ 9.
In 2006, using R = T ,
Taylor proved MC & FE for all n,
and “nice” enough to prove Sato-Tate conj for most E!
I’m very shocked.
12-c
![Page 32: Periods of automorphic forms and special values of L-functionsmhsaito/agsymkobe07/proceedings/ichino.pdf · ¶Method to study aut L-functions ‡ • Integral representation (more](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d14d47e708231d4389625/html5/thumbnails/32.jpg)
Method to study aut L-functions¶ ³• Integral representation (more difficult to find)
• Langlands-Shahidi method (∃ complete list)µ ´
Let’s consider L(s, πE,Symn).
Symn : symmetric power rep of GL2(C) of dim n + 1
Around 1999-2000, using LS-method,
Kim-Shahidi proved MC & FE for n ≤ 9.
In 2006, using R = T ,
Taylor proved MC & FE for all n,
and “nice” enough to prove Sato-Tate conj for most E!
I’m very shocked.
12-d
![Page 33: Periods of automorphic forms and special values of L-functionsmhsaito/agsymkobe07/proceedings/ichino.pdf · ¶Method to study aut L-functions ‡ • Integral representation (more](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d14d47e708231d4389625/html5/thumbnails/33.jpg)
§3 Periods
13
![Page 34: Periods of automorphic forms and special values of L-functionsmhsaito/agsymkobe07/proceedings/ichino.pdf · ¶Method to study aut L-functions ‡ • Integral representation (more](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d14d47e708231d4389625/html5/thumbnails/34.jpg)
G0 ⊂ G1 : both semisimple over Qπi : irred aut rep of Gi(A) (i = 0,1)
φi ∈ πi : aut form
Recall φi is a left Gi(Q)-invariant function on Gi(A).
Consider an integral
⟨φ1|G0, φ0⟩ :=
∫G0(Q)\G0(A)
φ1(g)φ0(g) dg ∈ C
(if it converges) and call it a period.
14
![Page 35: Periods of automorphic forms and special values of L-functionsmhsaito/agsymkobe07/proceedings/ichino.pdf · ¶Method to study aut L-functions ‡ • Integral representation (more](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d14d47e708231d4389625/html5/thumbnails/35.jpg)
G0 ⊂ G1 : both semisimple over Qπi : irred aut rep of Gi(A) (i = 0,1)
φi ∈ πi : aut form
Recall φi is a left Gi(Q)-invariant function on Gi(A).
Consider an integral
⟨φ1|G0, φ0⟩ :=
∫G0(Q)\G0(A)
φ1(g)φ0(g) dg ∈ C
(if it converges) and call it a period.
14-a
![Page 36: Periods of automorphic forms and special values of L-functionsmhsaito/agsymkobe07/proceedings/ichino.pdf · ¶Method to study aut L-functions ‡ • Integral representation (more](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d14d47e708231d4389625/html5/thumbnails/36.jpg)
E : elliptic curve over QÃ πE : irred cusp aut rep of PGL2(A)
à L(1, E) = L(12, πE, st) =
∫Q×\A×
φ
(a 00 1
)da = c
∫E(R)
ω
φ ∈ πE : suitably normalized
ω : non-zero diff form on E over Qc ∈ π−1 · Q
Forget geometry and consider only automorphic rep.
15
![Page 37: Periods of automorphic forms and special values of L-functionsmhsaito/agsymkobe07/proceedings/ichino.pdf · ¶Method to study aut L-functions ‡ • Integral representation (more](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d14d47e708231d4389625/html5/thumbnails/37.jpg)
E : elliptic curve over QÃ πE : irred cusp aut rep of PGL2(A)
à L(1, E) = L(12, πE, st) =
∫Q×\A×
φ
(a 00 1
)da = c
∫E(R)
ω
φ ∈ πE : suitably normalized
ω : non-zero diff form on E over Qc ∈ π−1 · Q
Forget geometry and consider only automorphic rep.
15-a
![Page 38: Periods of automorphic forms and special values of L-functionsmhsaito/agsymkobe07/proceedings/ichino.pdf · ¶Method to study aut L-functions ‡ • Integral representation (more](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d14d47e708231d4389625/html5/thumbnails/38.jpg)
E : elliptic curve over QÃ πE : irred cusp aut rep of PGL2(A)
à L(1, E) = L(12, πE, st) =
∫Q×\A×
φ
(a 00 1
)da = c
∫E(R)
ω
φ ∈ πE : suitably normalized
ω : non-zero diff form on E over Qc ∈ π−1 · Q
Forget geometry and consider only automorphic rep.
15-b
![Page 39: Periods of automorphic forms and special values of L-functionsmhsaito/agsymkobe07/proceedings/ichino.pdf · ¶Method to study aut L-functions ‡ • Integral representation (more](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d14d47e708231d4389625/html5/thumbnails/39.jpg)
E : elliptic curve over QÃ πE : irred cusp aut rep of PGL2(A)
à L(1, E) =L(12, πE, st) =
∫Q×\A×
φ
(a 00 1
)da= c
∫E(R)
ω
φ ∈ πE : suitably normalized
ω : non-zero diff form on E over Qc ∈ π−1 · Q
Forget geometry and consider only automorphic rep.
15-c
![Page 40: Periods of automorphic forms and special values of L-functionsmhsaito/agsymkobe07/proceedings/ichino.pdf · ¶Method to study aut L-functions ‡ • Integral representation (more](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d14d47e708231d4389625/html5/thumbnails/40.jpg)
Problem¶ ³Relate periods to special values of aut L-functions
• non-vanishing
• explicit formula
• . . .µ ´
There are many case-by-case examples.
• Integral representation
• Relative trace formula (Jacquet, Lapid . . . )
So far, there is no method to study problems in general.
16
![Page 41: Periods of automorphic forms and special values of L-functionsmhsaito/agsymkobe07/proceedings/ichino.pdf · ¶Method to study aut L-functions ‡ • Integral representation (more](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d14d47e708231d4389625/html5/thumbnails/41.jpg)
Problem¶ ³Relate periods to special values of aut L-functions
• non-vanishing
• explicit formula
• . . .µ ´
There are many case-by-case examples.
• Integral representation
• Relative trace formula (Jacquet, Lapid . . . )
So far, there is no method to study problems in general.
16-a
![Page 42: Periods of automorphic forms and special values of L-functionsmhsaito/agsymkobe07/proceedings/ichino.pdf · ¶Method to study aut L-functions ‡ • Integral representation (more](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d14d47e708231d4389625/html5/thumbnails/42.jpg)
Problem¶ ³Relate periods to special values of aut L-functions
• non-vanishing
• explicit formula
• . . .µ ´
There are many case-by-case examples.
• Integral representation
• Relative trace formula (Jacquet, Lapid . . . )
So far, there is no method to study problems in general.
16-b
![Page 43: Periods of automorphic forms and special values of L-functionsmhsaito/agsymkobe07/proceedings/ichino.pdf · ¶Method to study aut L-functions ‡ • Integral representation (more](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d14d47e708231d4389625/html5/thumbnails/43.jpg)
Gross-Prasad case:
G0 = SOn ⊂ G1 = SOn+1
[φ1 ⊗ φ0 7→ ⟨φ1|G0
, φ0⟩ =∫G0(Q)\G0(A)
φ1(g)φ0(g) dg
]∈ HomG0(A)(π1 ⊗ π0, C)
Multiplicity free¶ ³We expect that
dimC HomG0(Qp)(π1,p ⊗ π0,p, C) ≤ 1
for all p ≤ ∞.µ ´17
![Page 44: Periods of automorphic forms and special values of L-functionsmhsaito/agsymkobe07/proceedings/ichino.pdf · ¶Method to study aut L-functions ‡ • Integral representation (more](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d14d47e708231d4389625/html5/thumbnails/44.jpg)
Gross-Prasad case:
G0 = SOn ⊂ G1 = SOn+1[φ1 ⊗ φ0 7→ ⟨φ1|G0
, φ0⟩ =∫G0(Q)\G0(A)
φ1(g)φ0(g) dg
]∈ HomG0(A)(π1 ⊗ π0, C)
Multiplicity free¶ ³We expect that
dimC HomG0(Qp)(π1,p ⊗ π0,p, C) ≤ 1
for all p ≤ ∞.µ ´17-a
![Page 45: Periods of automorphic forms and special values of L-functionsmhsaito/agsymkobe07/proceedings/ichino.pdf · ¶Method to study aut L-functions ‡ • Integral representation (more](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d14d47e708231d4389625/html5/thumbnails/45.jpg)
Gross-Prasad case:
G0 = SOn ⊂ G1 = SOn+1[φ1 ⊗ φ0 7→ ⟨φ1|G0
, φ0⟩ =∫G0(Q)\G0(A)
φ1(g)φ0(g) dg
]∈ HomG0(A)(π1 ⊗ π0, C)
Multiplicity free¶ ³We expect that
dimC HomG0(Qp)(π1,p ⊗ π0,p, C) ≤ 1
for all p ≤ ∞.µ ´17-b
![Page 46: Periods of automorphic forms and special values of L-functionsmhsaito/agsymkobe07/proceedings/ichino.pdf · ¶Method to study aut L-functions ‡ • Integral representation (more](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d14d47e708231d4389625/html5/thumbnails/46.jpg)
π0 : irred cusp aut rep of G0(A) = SOn(A)π1 : irred cusp aut rep of G1(A) = SOn+1(A)
Assumption¶ ³• πi is tempered (i = 0,1).
• No local obstruction:
HomG0(Qp)(π1,p ⊗ π0,p, C) = 0 ∀p ≤ ∞µ ´Gross-Prasad conjecture (’92).
⟨φ1|G0, φ0⟩ = 0 for some φi ∈ πi ⇔ L(1
2, π1 £ π0) = 0
L(s, π1 £ π0) : associated to the tensor productof the standard rep of LG1 and LG0
s = 12 is the center of FE (s ↔ 1 − s).
18
![Page 47: Periods of automorphic forms and special values of L-functionsmhsaito/agsymkobe07/proceedings/ichino.pdf · ¶Method to study aut L-functions ‡ • Integral representation (more](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d14d47e708231d4389625/html5/thumbnails/47.jpg)
π0 : irred cusp aut rep of G0(A) = SOn(A)π1 : irred cusp aut rep of G1(A) = SOn+1(A)
Assumption¶ ³• πi is tempered (i = 0,1).
• No local obstruction:
HomG0(Qp)(π1,p ⊗ π0,p, C) = 0 ∀p ≤ ∞µ ´Gross-Prasad conjecture (’92).
⟨φ1|G0, φ0⟩ = 0 for some φi ∈ πi ⇔ L(1
2, π1 £ π0) = 0
L(s, π1 £ π0) : associated to the tensor productof the standard rep of LG1 and LG0
s = 12 is the center of FE (s ↔ 1 − s).
18-a
![Page 48: Periods of automorphic forms and special values of L-functionsmhsaito/agsymkobe07/proceedings/ichino.pdf · ¶Method to study aut L-functions ‡ • Integral representation (more](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d14d47e708231d4389625/html5/thumbnails/48.jpg)
π0 : irred cusp aut rep of G0(A) = SOn(A)π1 : irred cusp aut rep of G1(A) = SOn+1(A)
Assumption¶ ³• πi is tempered (i = 0,1).
• No local obstruction:
HomG0(Qp)(π1,p ⊗ π0,p, C) = 0 ∀p ≤ ∞µ ´Gross-Prasad conjecture (’92).
⟨φ1|G0, φ0⟩ = 0 for some φi ∈ πi ⇔ L(1
2, π1 £ π0) = 0
L(s, π1 £ π0) : associated to the tensor productof the standard rep of LG1 and LG0
s = 12 is the center of FE (s ↔ 1 − s).
18-b
![Page 49: Periods of automorphic forms and special values of L-functionsmhsaito/agsymkobe07/proceedings/ichino.pdf · ¶Method to study aut L-functions ‡ • Integral representation (more](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d14d47e708231d4389625/html5/thumbnails/49.jpg)
Difficulty:
L(12, π1 £ π0) has not been defined in general.
Best result so far¶ ³If Gi is split, πi is generic and stable, then ⇒ holds.
(Ginzburg-Jiang-Rallis ’05)µ ´
Not only the non-vanishing criterion,
we want a formula for ⟨φ1|G0, φ0⟩, at least conjecturally.
19
![Page 50: Periods of automorphic forms and special values of L-functionsmhsaito/agsymkobe07/proceedings/ichino.pdf · ¶Method to study aut L-functions ‡ • Integral representation (more](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d14d47e708231d4389625/html5/thumbnails/50.jpg)
Difficulty:
L(12, π1 £ π0) has not been defined in general.
Best result so far¶ ³If Gi is split, πi is generic and stable, then ⇒ holds.
(Ginzburg-Jiang-Rallis ’05)µ ´
Not only the non-vanishing criterion,
we want a formula for ⟨φ1|G0, φ0⟩, at least conjecturally.
19-a
![Page 51: Periods of automorphic forms and special values of L-functionsmhsaito/agsymkobe07/proceedings/ichino.pdf · ¶Method to study aut L-functions ‡ • Integral representation (more](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d14d47e708231d4389625/html5/thumbnails/51.jpg)
Difficulty:
L(12, π1 £ π0) has not been defined in general.
Best result so far¶ ³If Gi is split, πi is generic and stable, then ⇒ holds.
(Ginzburg-Jiang-Rallis ’05)µ ´
Not only the non-vanishing criterion,
we want a formula for ⟨φ1|G0, φ0⟩, at least conjecturally.
19-b
![Page 52: Periods of automorphic forms and special values of L-functionsmhsaito/agsymkobe07/proceedings/ichino.pdf · ¶Method to study aut L-functions ‡ • Integral representation (more](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d14d47e708231d4389625/html5/thumbnails/52.jpg)
Conjecture (with Tamotsu Ikeda).
|⟨φ1|G0, φ0⟩|2
∥φ1∥2 · ∥φ0∥2= 2β · C0 · LS(M∨
1 (1))
×LS(1
2, π1 £ π0)
LS(1, π1,Ad)LS(1, π0,Ad)
×∏p∈S
Ip(φ1,p, φ0,p)
∥φ1,p∥2p · ∥φ0,p∥2p
20
![Page 53: Periods of automorphic forms and special values of L-functionsmhsaito/agsymkobe07/proceedings/ichino.pdf · ¶Method to study aut L-functions ‡ • Integral representation (more](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d14d47e708231d4389625/html5/thumbnails/53.jpg)
Conjecture (with Tamotsu Ikeda).
|⟨φ1|G0, φ0⟩|2
∥φ1∥2 · ∥φ0∥2= 2β · C0 · LS(M∨
1 (1))
×LS(1
2, π1 £ π0)
LS(1, π1,Ad)LS(1, π0,Ad)
×∏p∈S
Ip(φ1,p, φ0,p)
∥φ1,p∥2p · ∥φ0,p∥2p
Our period and L-value are ⟨φ1|G0, φ0⟩ and LS(1
2, π1£π0).
20-a
![Page 54: Periods of automorphic forms and special values of L-functionsmhsaito/agsymkobe07/proceedings/ichino.pdf · ¶Method to study aut L-functions ‡ • Integral representation (more](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d14d47e708231d4389625/html5/thumbnails/54.jpg)
Conjecture (with Tamotsu Ikeda).
|⟨φ1|G0, φ0⟩|2
∥φ1∥2 · ∥φ0∥2= 2β · C0 · LS(M∨
1 (1))
×LS(1
2, π1 £ π0)
LS(1, π1,Ad)LS(1, π0,Ad)
×∏p∈S
Ip(φ1,p, φ0,p)
∥φ1,p∥2p · ∥φ0,p∥2p
S : fin set of bad primes
LS : Euler product without local factor at p ∈ S
20-b
![Page 55: Periods of automorphic forms and special values of L-functionsmhsaito/agsymkobe07/proceedings/ichino.pdf · ¶Method to study aut L-functions ‡ • Integral representation (more](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d14d47e708231d4389625/html5/thumbnails/55.jpg)
Conjecture (with Tamotsu Ikeda).
|⟨φ1|G0, φ0⟩|2
∥φ1∥2 · ∥φ0∥2= 2β · C0 · LS(M∨
1 (1))
×LS(1
2, π1 £ π0)
LS(1, π1,Ad)LS(1, π0,Ad)
×∏p∈S
Ip(φ1,p, φ0,p)
∥φ1,p∥2p · ∥φ0,p∥2p
Ad : adjoint rep of LGi on its Lie algebra
20-c
![Page 56: Periods of automorphic forms and special values of L-functionsmhsaito/agsymkobe07/proceedings/ichino.pdf · ¶Method to study aut L-functions ‡ • Integral representation (more](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d14d47e708231d4389625/html5/thumbnails/56.jpg)
Conjecture (with Tamotsu Ikeda).
|⟨φ1|G0, φ0⟩|2
∥φ1∥2 · ∥φ0∥2= 2β · C0 · LS(M∨
1 (1))
×LS(1
2, π1 £ π0)
LS(1, π1,Ad)LS(1, π0,Ad)
×∏p∈S
Ip(φ1,p, φ0,p)
∥φ1,p∥2p · ∥φ0,p∥2p
C0 : constant dep on normalization of Haar measures
M1 : Gross’ Artin motive attached to G1
20-d
![Page 57: Periods of automorphic forms and special values of L-functionsmhsaito/agsymkobe07/proceedings/ichino.pdf · ¶Method to study aut L-functions ‡ • Integral representation (more](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d14d47e708231d4389625/html5/thumbnails/57.jpg)
Conjecture (with Tamotsu Ikeda).
|⟨φ1|G0, φ0⟩|2
∥φ1∥2 · ∥φ0∥2= 2β · C0 · LS(M∨
1 (1))
×LS(1
2, π1 £ π0)
LS(1, π1,Ad)LS(1, π0,Ad)
×∏p∈S
Ip(φ1,p, φ0,p)
∥φ1,p∥2p · ∥φ0,p∥2p
Ip(φ1,p, φ0,p)
∥φ1,p∥2p · ∥φ0,p∥2p≥ 0 : local object dep only on φi,p ∈ πi,p
20-e
![Page 58: Periods of automorphic forms and special values of L-functionsmhsaito/agsymkobe07/proceedings/ichino.pdf · ¶Method to study aut L-functions ‡ • Integral representation (more](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d14d47e708231d4389625/html5/thumbnails/58.jpg)
Conjecture (with Tamotsu Ikeda).
|⟨φ1|G0, φ0⟩|2
∥φ1∥2 · ∥φ0∥2= 2β · C0 · LS(M∨
1 (1))
×LS(1
2, π1 £ π0)
LS(1, π1,Ad)LS(1, π0,Ad)
×∏p∈S
Ip(φ1,p, φ0,p)
∥φ1,p∥2p · ∥φ0,p∥2p
β ∈ Z : global object
We believe that β is related to Arthur’s conjecture
(multiplicity of rep in the space of automorphic forms).
20-f
![Page 59: Periods of automorphic forms and special values of L-functionsmhsaito/agsymkobe07/proceedings/ichino.pdf · ¶Method to study aut L-functions ‡ • Integral representation (more](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d14d47e708231d4389625/html5/thumbnails/59.jpg)
Conjecture (with Tamotsu Ikeda).
|⟨φ1|G0, φ0⟩|2
∥φ1∥2 · ∥φ0∥2= 2β · C0 · LS(M∨
1 (1))
×LS(1
2, π1 £ π0)
LS(1, π1,Ad)LS(1, π0,Ad)
×∏p∈S
Ip(φ1,p, φ0,p)
∥φ1,p∥2p · ∥φ0,p∥2p
β ∈ Z : global object
We believe that β is related to Arthur’s conjecture
(multiplicity of rep in the space of automorphic forms).
20-g
![Page 60: Periods of automorphic forms and special values of L-functionsmhsaito/agsymkobe07/proceedings/ichino.pdf · ¶Method to study aut L-functions ‡ • Integral representation (more](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d14d47e708231d4389625/html5/thumbnails/60.jpg)
Known results¶ ³• SO2 ⊂ SO3 : Waldspurger ’85
• SO3 ⊂ SO4 : Garrett ’87, Harris-Kudla ’91 . . . I.µ ´When n ≥ 4, our L-values are not well-def in general.But, for SO4 ⊂ SO5,∃ example of Bocherer-Furusawa-Schulze-Pillot ’04.
∃ non-tempered cusp aut rep of SOn(A)
Some non-tempered examples¶ ³• SO4 ⊂ SO5 : I. ’05
• SO5 ⊂ SO6 : I.-Ikedaµ ´But ∃ more difficulty to formulate a conjecture.
21
![Page 61: Periods of automorphic forms and special values of L-functionsmhsaito/agsymkobe07/proceedings/ichino.pdf · ¶Method to study aut L-functions ‡ • Integral representation (more](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d14d47e708231d4389625/html5/thumbnails/61.jpg)
Known results¶ ³• SO2 ⊂ SO3 : Waldspurger ’85
• SO3 ⊂ SO4 : Garrett ’87, Harris-Kudla ’91 . . . I.µ ´When n ≥ 4, our L-values are not well-def in general.But, for SO4 ⊂ SO5,∃ example of Bocherer-Furusawa-Schulze-Pillot ’04.
∃ non-tempered cusp aut rep of SOn(A)
Some non-tempered examples¶ ³• SO4 ⊂ SO5 : I. ’05
• SO5 ⊂ SO6 : I.-Ikedaµ ´But ∃ more difficulty to formulate a conjecture.
21-a
![Page 62: Periods of automorphic forms and special values of L-functionsmhsaito/agsymkobe07/proceedings/ichino.pdf · ¶Method to study aut L-functions ‡ • Integral representation (more](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d14d47e708231d4389625/html5/thumbnails/62.jpg)
Known results¶ ³• SO2 ⊂ SO3 : Waldspurger ’85
• SO3 ⊂ SO4 : Garrett ’87, Harris-Kudla ’91 . . . I.µ ´When n ≥ 4, our L-values are not well-def in general.But, for SO4 ⊂ SO5,∃ example of Bocherer-Furusawa-Schulze-Pillot ’04.
∃ non-tempered cusp aut rep of SOn(A)
Some non-tempered examples¶ ³• SO4 ⊂ SO5 : I. ’05
• SO5 ⊂ SO6 : I.-Ikedaµ ´But ∃ more difficulty to formulate a conjecture.
21-b
![Page 63: Periods of automorphic forms and special values of L-functionsmhsaito/agsymkobe07/proceedings/ichino.pdf · ¶Method to study aut L-functions ‡ • Integral representation (more](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d14d47e708231d4389625/html5/thumbnails/63.jpg)
Known results¶ ³• SO2 ⊂ SO3 : Waldspurger ’85
• SO3 ⊂ SO4 : Garrett ’87, Harris-Kudla ’91 . . . I.µ ´When n ≥ 4, our L-values are not well-def in general.But, for SO4 ⊂ SO5,∃ example of Bocherer-Furusawa-Schulze-Pillot ’04.
∃ non-tempered cusp aut rep of SOn(A)
Some non-tempered examples¶ ³• SO4 ⊂ SO5 : I. ’05
• SO5 ⊂ SO6 : I.-Ikedaµ ´But ∃ more difficulty to formulate a conjecture.
21-c
![Page 64: Periods of automorphic forms and special values of L-functionsmhsaito/agsymkobe07/proceedings/ichino.pdf · ¶Method to study aut L-functions ‡ • Integral representation (more](https://reader034.fdocuments.us/reader034/viewer/2022042323/5f0d14d47e708231d4389625/html5/thumbnails/64.jpg)
Thank you!
22