L-functions and periods of automorphic motives

Deligne conjecture for GL(n) × GL(m) Automorphic periods Applications of the IINH conjecture for endoscopic forms Factorization conjecture

L-functions and periods of automorphic motives Joint work in progress with Harald Grobner and Lin Jie

Michael Harris

Schloss Elmau, April 2018

Michael Harris

L-functions and periods of automorphic motives

Deligne conjecture for GL(n) × GL(m) Automorphic periods Applications of the IINH conjecture for endoscopic forms Factorization conjecture

Notation

We let F be a CM field, F + its maximal totally real subfield, c ∈ Gal(F/F + ). (Usually F + = Q to save notation.) Let H = Hn = GL(n)F . Π a cuspidal automorphic representation of Hn . We always assume Π cohomological (with coefficients in W = W(Π)) ∼

Π∨ −→ Πc

Michael Harris

L-functions and periods of automorphic motives

Deligne conjecture for GL(n) × GL(m) Automorphic periods Applications of the IINH conjecture for endoscopic forms Factorization conjecture

Notation

We let F be a CM field, F + its maximal totally real subfield, c ∈ Gal(F/F + ). (Usually F + = Q to save notation.) Let H = Hn = GL(n)F . Π a cuspidal automorphic representation of Hn . We always assume Π cohomological (with coefficients in W = W(Π)) ∼

Π∨ −→ Πc

Michael Harris

L-functions and periods of automorphic motives

Deligne conjecture for GL(n) × GL(m) Automorphic periods Applications of the IINH conjecture for endoscopic forms Factorization conjecture

Notation

We let F be a CM field, F + its maximal totally real subfield, c ∈ Gal(F/F + ). (Usually F + = Q to save notation.) Let H = Hn = GL(n)F . Π a cuspidal automorphic representation of Hn . We always assume Π cohomological (with coefficients in W = W(Π)) ∼

Π∨ −→ Πc

Michael Harris

L-functions and periods of automorphic motives

Deligne conjecture for GL(n) × GL(m) Automorphic periods Applications of the IINH conjecture for endoscopic forms Factorization conjecture

Notation

We let F be a CM field, F + its maximal totally real subfield, c ∈ Gal(F/F + ). (Usually F + = Q to save notation.) Let H = Hn = GL(n)F . Π a cuspidal automorphic representation of Hn . We always assume Π cohomological (with coefficients in W = W(Π)) ∼

Π∨ −→ Πc

Michael Harris

L-functions and periods of automorphic motives

Deligne conjecture for GL(n) × GL(m) Automorphic periods Applications of the IINH conjecture for endoscopic forms Factorization conjecture

Notation

Let H 0 = Hm , m < n, let Π0 , W 0 = W(Π0 ) on H 0 with same hypotheses, not necessarily cuspidal. W and W 0 : finite-dimensional irreducible representations of H, H 0 . Highest weights a1 ≥ a2 · · · ≥ an ; a01 ≥ a02 · · · ≥ a0m (for each real place of F + ).

Michael Harris

L-functions and periods of automorphic motives

Deligne conjecture for GL(n) × GL(m) Automorphic periods Applications of the IINH conjecture for endoscopic forms Factorization conjecture

Notation

Let H 0 = Hm , m < n, let Π0 , W 0 = W(Π0 ) on H 0 with same hypotheses, not necessarily cuspidal. W and W 0 : finite-dimensional irreducible representations of H, H 0 . Highest weights a1 ≥ a2 · · · ≥ an ; a01 ≥ a02 · · · ≥ a0m (for each real place of F + ).

Michael Harris

L-functions and periods of automorphic motives

Deligne conjecture for GL(n) × GL(m) Automorphic periods Applications of the IINH conjecture for endoscopic forms Factorization conjecture

Notation

Let H 0 = Hm , m < n, let Π0 , W 0 = W(Π0 ) on H 0 with same hypotheses, not necessarily cuspidal. W and W 0 : finite-dimensional irreducible representations of H, H 0 . Highest weights a1 ≥ a2 · · · ≥ an ; a01 ≥ a02 · · · ≥ a0m (for each real place of F + ).

Michael Harris

L-functions and periods of automorphic motives

Deligne conjecture for GL(n) × GL(m) Automorphic periods Applications of the IINH conjecture for endoscopic forms Factorization conjecture

Notation

Let H 0 = Hm , m < n, let Π0 , W 0 = W(Π0 ) on H 0 with same hypotheses, not necessarily cuspidal. W and W 0 : finite-dimensional irreducible representations of H, H 0 . Highest weights a1 ≥ a2 · · · ≥ an ; a01 ≥ a02 · · · ≥ a0m (for each real place of F + ).

Michael Harris

L-functions and periods of automorphic motives

Deligne conjecture for GL(n) × GL(m) Automorphic periods Applications of the IINH conjecture for endoscopic forms Factorization conjecture

Deligne conjecture for Rankin-Selberg motives Associated motives M(Π), M(Π0 ) of rank n, m with coefficients in E(Π), E(Π0 ). Set E = E(Π) ⊗ E(Π0 ) L(s, M(Π) ⊗ M(Π0 )) = L(s −

n+m−2 , Π × Π0 ) 2

(Rankin-Selberg on right). Conjecture (Deligne conjecture) For any critical value s0 of L(s, M(Π) ⊗ M(Π0 )) there is an element c(s0 , Π, Π0 ) ∈ E ⊗ C× such that L(s0 , M(Π) ⊗ M(Π0 )) ∼ c(s0 , Π, Π0 ) Michael Harris

L-functions and periods of automorphic motives

Deligne conjecture for GL(n) × GL(m) Automorphic periods Applications of the IINH conjecture for endoscopic forms Factorization conjecture

Deligne conjecture for Rankin-Selberg motives Associated motives M(Π), M(Π0 ) of rank n, m with coefficients in E(Π), E(Π0 ). Set E = E(Π) ⊗ E(Π0 ) L(s, M(Π) ⊗ M(Π0 )) = L(s −

n+m−2 , Π × Π0 ) 2

(Rankin-Selberg on right). Conjecture (Deligne conjecture) For any critical value s0 of L(s, M(Π) ⊗ M(Π0 )) there is an element c(s0 , Π, Π0 ) ∈ E ⊗ C× such that L(s0 , M(Π) ⊗ M(Π0 )) ∼ c(s0 , Π, Π0 ) Michael Harris

L-functions and periods of automorphic motives

Deligne conjecture for GL(n) × GL(m) Automorphic periods Applications of the IINH conjecture for endoscopic forms Factorization conjecture

Deligne conjecture for Rankin-Selberg motives Associated motives M(Π), M(Π0 ) of rank n, m with coefficients in E(Π), E(Π0 ). Set E = E(Π) ⊗ E(Π0 ) L(s, M(Π) ⊗ M(Π0 )) = L(s −

n+m−2 , Π × Π0 ) 2

(Rankin-Selberg on right). Conjecture (Deligne conjecture) For any critical value s0 of L(s, M(Π) ⊗ M(Π0 )) there is an element c(s0 , Π, Π0 ) ∈ E ⊗ C× such that L(s0 , M(Π) ⊗ M(Π0 )) ∼ c(s0 , Π, Π0 ) Michael Harris

L-functions and periods of automorphic motives

Deligne conjecture for GL(n) × GL(m) Automorphic periods Applications of the IINH conjecture for endoscopic forms Factorization conjecture

Deligne conjecture for Rankin-Selberg motives Associated motives M(Π), M(Π0 ) of rank n, m with coefficients in E(Π), E(Π0 ). Set E = E(Π) ⊗ E(Π0 ) L(s, M(Π) ⊗ M(Π0 )) = L(s −

n+m−2 , Π × Π0 ) 2

(Rankin-Selberg on right). Conjecture (Deligne conjecture) For any critical value s0 of L(s, M(Π) ⊗ M(Π0 )) there is an element c(s0 , Π, Π0 ) ∈ E ⊗ C× such that L(s0 , M(Π) ⊗ M(Π0 )) ∼ c(s0 , Π, Π0 ) Michael Harris

L-functions and periods of automorphic motives

Deligne conjecture for GL(n) × GL(m) Automorphic periods Applications of the IINH conjecture for endoscopic forms Factorization conjecture

Deligne period for Rankin-Selberg motives Assume F + = Q for simplicity. Form of Deligne period c(s0 , Π, Pi0 ) depends on relative position of (ai ) and (a0j ). Example (Standard position) m = n − 1, ai ≥ a0i ≥ ai+1 ∀i. Let M = M(Π). For 0 ≤ i ≤ n there are (motivic) invariants Qi (M), defined via cohomological structures MB , MdR that are free E-modules of the appropriateQrank. Define Pi (M) = ik=0 Qk (M) Also Qj (M 0 ), Pj (M 0 ), 0 ≤ j ≤ m. (For τ : F + → R, get Prτ (M), Qr,τ (M), etc.)

Michael Harris

L-functions and periods of automorphic motives

Deligne conjecture for GL(n) × GL(m) Automorphic periods Applications of the IINH conjecture for endoscopic forms Factorization conjecture

Deligne period for Rankin-Selberg motives Assume F + = Q for simplicity. Form of Deligne period c(s0 , Π, Pi0 ) depends on relative position of (ai ) and (a0j ). Example (Standard position) m = n − 1, ai ≥ a0i ≥ ai+1 ∀i. Let M = M(Π). For 0 ≤ i ≤ n there are (motivic) invariants Qi (M), defined via cohomological structures MB , MdR that are free E-modules of the appropriateQrank. Define Pi (M) = ik=0 Qk (M) Also Qj (M 0 ), Pj (M 0 ), 0 ≤ j ≤ m. (For τ : F + → R, get Prτ (M), Qr,τ (M), etc.)

Michael Harris

L-functions and periods of automorphic motives

Deligne conjecture for GL(n) × GL(m) Automorphic periods Applications of the IINH conjecture for endoscopic forms Factorization conjecture

Deligne period for Rankin-Selberg motives Assume F + = Q for simplicity. Form of Deligne period c(s0 , Π, Pi0 ) depends on relative position of (ai ) and (a0j ). Example (Standard position) m = n − 1, ai ≥ a0i ≥ ai+1 ∀i. Let M = M(Π). For 0 ≤ i ≤ n there are (motivic) invariants Qi (M), defined via cohomological structures MB , MdR that are free E-modules of the appropriateQrank. Define Pi (M) = ik=0 Qk (M) Also Qj (M 0 ), Pj (M 0 ), 0 ≤ j ≤ m. (For τ : F + → R, get Prτ (M), Qr,τ (M), etc.)

Michael Harris

L-functions and periods of automorphic motives

Deligne conjecture for GL(n) × GL(m) Automorphic periods Applications of the IINH conjecture for endoscopic forms Factorization conjecture

Deligne period for Rankin-Selberg motives Assume F + = Q for simplicity. Form of Deligne period c(s0 , Π, Pi0 ) depends on relative position of (ai ) and (a0j ). Example (Standard position) m = n − 1, ai ≥ a0i ≥ ai+1 ∀i. Let M = M(Π). For 0 ≤ i ≤ n there are (motivic) invariants Qi (M), defined via cohomological structures MB , MdR that are free E-modules of the appropriateQrank. Define Pi (M) = ik=0 Qk (M) Also Qj (M 0 ), Pj (M 0 ), 0 ≤ j ≤ m. (For τ : F + → R, get Prτ (M), Qr,τ (M), etc.)

Michael Harris

L-functions and periods of automorphic motives

Deligne conjecture for GL(n) × GL(m) Automorphic periods Applications of the IINH conjecture for endoscopic forms Factorization conjecture

Deligne period for Rankin-Selberg motives Assume F + = Q for simplicity. Form of Deligne period c(s0 , Π, Pi0 ) depends on relative position of (ai ) and (a0j ). Example (Standard position) m = n − 1, ai ≥ a0i ≥ ai+1 ∀i. Let M = M(Π). For 0 ≤ i ≤ n there are (motivic) invariants Qi (M), defined via cohomological structures MB , MdR that are free E-modules of the appropriateQrank. Define Pi (M) = ik=0 Qk (M) Also Qj (M 0 ), Pj (M 0 ), 0 ≤ j ≤ m. (For τ : F + → R, get Prτ (M), Qr,τ (M), etc.)

Michael Harris

L-functions and periods of automorphic motives

Deligne conjecture for GL(n) × GL(m) Automorphic periods Applications of the IINH conjecture for endoscopic forms Factorization conjecture

Deligne period for Rankin-Selberg motives Assume F + = Q for simplicity. Form of Deligne period c(s0 , Π, Pi0 ) depends on relative position of (ai ) and (a0j ). Example (Standard position) m = n − 1, ai ≥ a0i ≥ ai+1 ∀i. Let M = M(Π). For 0 ≤ i ≤ n there are (motivic) invariants Qi (M), defined via cohomological structures MB , MdR that are free E-modules of the appropriateQrank. Define Pi (M) = ik=0 Qk (M) Also Qj (M 0 ), Pj (M 0 ), 0 ≤ j ≤ m. (For τ : F + → R, get Prτ (M), Qr,τ (M), etc.)

Michael Harris

L-functions and periods of automorphic motives

Deligne conjecture for GL(n) × GL(m) Automorphic periods Applications of the IINH conjecture for endoscopic forms Factorization conjecture

Deligne period for Rankin-Selberg motives Assume F + = Q for simplicity. Form of Deligne period c(s0 , Π, Pi0 ) depends on relative position of (ai ) and (a0j ). Example (Standard position) m = n − 1, ai ≥ a0i ≥ ai+1 ∀i. Let M = M(Π). For 0 ≤ i ≤ n there are (motivic) invariants Qi (M), defined via cohomological structures MB , MdR that are free E-modules of the appropriateQrank. Define Pi (M) = ik=0 Qk (M) Also Qj (M 0 ), Pj (M 0 ), 0 ≤ j ≤ m. (For τ : F + → R, get Prτ (M), Qr,τ (M), etc.)

Michael Harris

L-functions and periods of automorphic motives

Deligne conjecture for GL(n) × GL(m) Automorphic periods Applications of the IINH conjecture for endoscopic forms Factorization conjecture

Deligne period for Rankin-Selberg motives Assume F + = Q for simplicity. Form of Deligne period c(s0 , Π, Pi0 ) depends on relative position of (ai ) and (a0j ). Example (Standard position) m = n − 1, ai ≥ a0i ≥ ai+1 ∀i. Let M = M(Π). For 0 ≤ i ≤ n there are (motivic) invariants Qi (M), defined via cohomological structures MB , MdR that are free E-modules of the appropriateQrank. Define Pi (M) = ik=0 Qk (M) Also Qj (M 0 ), Pj (M 0 ), 0 ≤ j ≤ m. (For τ : F + → R, get Prτ (M), Qr,τ (M), etc.)

Michael Harris

L-functions and periods of automorphic motives

Deligne conjecture for GL(n) × GL(m) Automorphic periods Applications of the IINH conjecture for endoscopic forms Factorization conjecture

Deligne period for Rankin-Selberg motives Assume F + = Q for simplicity. Form of Deligne period c(s0 , Π, Pi0 ) depends on relative position of (ai ) and (a0j ). Example (Standard position) m = n − 1, ai ≥ a0i ≥ ai+1 ∀i. Let M = M(Π). For 0 ≤ i ≤ n there are (motivic) invariants Qi (M), defined via cohomological structures MB , MdR that are free E-modules of the appropriateQrank. Define Pi (M) = ik=0 Qk (M) Also Qj (M 0 ), Pj (M 0 ), 0 ≤ j ≤ m. (For τ : F + → R, get Prτ (M), Qr,τ (M), etc.)

Michael Harris

L-functions and periods of automorphic motives

Deligne conjecture for GL(n) × GL(m) Automorphic periods Applications of the IINH conjecture for endoscopic forms Factorization conjecture

Deligne period for Rankin-Selberg motives The following is an application of linear algebra to the cohomology of M(Π) ⊗ M(Π0 ) Proposition For 0 ≤ i ≤ n, 0 ≤ j ≤ m, there are integers a(s0 ), sp(i, Π, Π0 ), sp(j, Π0 , Π) depending only on Π∞ , Π0∞ , such that Y Y 0 0 Pj (M 0 )sp(j,Π ,Π) c(s0 , Π, Π0 ) ∼ (2πi)a(s0 ) Pi (M)sp(i,Π,Π ) i

j

Example (Standard position) sp(i, Π, Π0 ) = sp(j, Π0 , Π) = 1 ∀i, j sp(0, Π, Π0 ) = sp(n, Π, Π0 ) = 0. Michael Harris

except

L-functions and periods of automorphic motives

Deligne conjecture for GL(n) × GL(m) Automorphic periods Applications of the IINH conjecture for endoscopic forms Factorization conjecture

Deligne period for Rankin-Selberg motives The following is an application of linear algebra to the cohomology of M(Π) ⊗ M(Π0 ) Proposition For 0 ≤ i ≤ n, 0 ≤ j ≤ m, there are integers a(s0 ), sp(i, Π, Π0 ), sp(j, Π0 , Π) depending only on Π∞ , Π0∞ , such that Y Y 0 0 c(s0 , Π, Π0 ) ∼ (2πi)a(s0 ) Pi (M)sp(i,Π,Π ) Pj (M 0 )sp(j,Π ,Π) i

j

Example (Standard position) sp(i, Π, Π0 ) = sp(j, Π0 , Π) = 1 ∀i, j sp(0, Π, Π0 ) = sp(n, Π, Π0 ) = 0. Michael Harris

except

L-functions and periods of automorphic motives

Deligne conjecture for GL(n) × GL(m) Automorphic periods Applications of the IINH conjecture for endoscopic forms Factorization conjecture

Deligne period for Rankin-Selberg motives The following is an application of linear algebra to the cohomology of M(Π) ⊗ M(Π0 ) Proposition For 0 ≤ i ≤ n, 0 ≤ j ≤ m, there are integers a(s0 ), sp(i, Π, Π0 ), sp(j, Π0 , Π) depending only on Π∞ , Π0∞ , such that Y Y 0 0 c(s0 , Π, Π0 ) ∼ (2πi)a(s0 ) Pi (M)sp(i,Π,Π ) Pj (M 0 )sp(j,Π ,Π) i

j

Example (Standard position) sp(i, Π, Π0 ) = sp(j, Π0 , Π) = 1 ∀i, j sp(0, Π, Π0 ) = sp(n, Π, Π0 ) = 0. Michael Harris

except

L-functions and periods of automorphic motives

Deligne conjecture for GL(n) × GL(m) Automorphic periods Applications of the IINH conjecture for endoscopic forms Factorization conjecture

Relative positions and GGP Assume m = n − 1. To (Π∞ , Π0∞ ) associate discrete series Vogan L-packet dp(W, W 0 ) for {U(V) × U(V 0 )}, dim V = dim V 0 + 1 = n V 0 ⊂ V. Fact (GGP, Beuzart-Plessis, H. He) To each relative position of W, W 0 there is a unique configuration 0 r(W, W 0 ) = (π∞ , π∞ ) ∈ dp(W, W 0 ) 0 , C) 6= 0. such that HomU(V 0 )∞ (π∞ ⊗ π∞

Thus (π, π 0 ) in the global L-packet for (Π, Π0 ) has a non-zero period 0 ⇒ (π∞ , π∞ ) = r(W, W 0 ) ∈ dp(W, W 0 ). Michael Harris

L-functions and periods of automorphic motives

Deligne conjecture for GL(n) × GL(m) Automorphic periods Applications of the IINH conjecture for endoscopic forms Factorization conjecture

Relative positions and GGP Assume m = n − 1. To (Π∞ , Π0∞ ) associate discrete series Vogan L-packet dp(W, W 0 ) for {U(V) × U(V 0 )}, dim V = dim V 0 + 1 = n V 0 ⊂ V. Fact (GGP, Beuzart-Plessis, H. He) To each relative position of W, W 0 there is a unique configuration 0 r(W, W 0 ) = (π∞ , π∞ ) ∈ dp(W, W 0 ) 0 , C) 6= 0. such that HomU(V 0 )∞ (π∞ ⊗ π∞

Thus (π, π 0 ) in the global L-packet for (Π, Π0 ) has a non-zero period 0 ⇒ (π∞ , π∞ ) = r(W, W 0 ) ∈ dp(W, W 0 ). Michael Harris

L-functions and periods of automorphic motives

Deligne conjecture for GL(n) × GL(m) Automorphic periods Applications of the IINH conjecture for endoscopic forms Factorization conjecture

Relative positions and GGP Assume m = n − 1. To (Π∞ , Π0∞ ) associate discrete series Vogan L-packet dp(W, W 0 ) for {U(V) × U(V 0 )}, dim V = dim V 0 + 1 = n V 0 ⊂ V. Fact (GGP, Beuzart-Plessis, H. He) To each relative position of W, W 0 there is a unique configuration 0 r(W, W 0 ) = (π∞ , π∞ ) ∈ dp(W, W 0 ) 0 , C) 6= 0. such that HomU(V 0 )∞ (π∞ ⊗ π∞

Thus (π, π 0 ) in the global L-packet for (Π, Π0 ) has a non-zero period 0 ⇒ (π∞ , π∞ ) = r(W, W 0 ) ∈ dp(W, W 0 ). Michael Harris

L-functions and periods of automorphic motives

Deligne conjecture for GL(n) × GL(m) Automorphic periods Applications of the IINH conjecture for endoscopic forms Factorization conjecture

Relative positions and GGP Assume m = n − 1. To (Π∞ , Π0∞ ) associate discrete series Vogan L-packet dp(W, W 0 ) for {U(V) × U(V 0 )}, dim V = dim V 0 + 1 = n V 0 ⊂ V. Fact (GGP, Beuzart-Plessis, H. He) To each relative position of W, W 0 there is a unique configuration 0 r(W, W 0 ) = (π∞ , π∞ ) ∈ dp(W, W 0 ) 0 , C) 6= 0. such that HomU(V 0 )∞ (π∞ ⊗ π∞

Thus (π, π 0 ) in the global L-packet for (Π, Π0 ) has a non-zero period 0 ⇒ (π∞ , π∞ ) = r(W, W 0 ) ∈ dp(W, W 0 ). Michael Harris

L-functions and periods of automorphic motives

Deligne conjecture for GL(n) × GL(m) Automorphic periods Applications of the IINH conjecture for endoscopic forms Factorization conjecture

Relative positions and GGP Assume m = n − 1. To (Π∞ , Π0∞ ) associate discrete series Vogan L-packet dp(W, W 0 ) for {U(V) × U(V 0 )}, dim V = dim V 0 + 1 = n V 0 ⊂ V. Fact (GGP, Beuzart-Plessis, H. He) To each relative position of W, W 0 there is a unique configuration 0 r(W, W 0 ) = (π∞ , π∞ ) ∈ dp(W, W 0 ) 0 , C) 6= 0. such that HomU(V 0 )∞ (π∞ ⊗ π∞

Thus (π, π 0 ) in the global L-packet for (Π, Π0 ) has a non-zero period 0 ⇒ (π∞ , π∞ ) = r(W, W 0 ) ∈ dp(W, W 0 ). Michael Harris

L-functions and periods of automorphic motives

Deligne conjecture for GL(n) × GL(m) Automorphic periods Applications of the IINH conjecture for endoscopic forms Factorization conjecture

Relative positions and GGP Assume m = n − 1. To (Π∞ , Π0∞ ) associate discrete series Vogan L-packet dp(W, W 0 ) for {U(V) × U(V 0 )}, dim V = dim V 0 + 1 = n V 0 ⊂ V. Fact (GGP, Beuzart-Plessis, H. He) To each relative position of W, W 0 there is a unique configuration 0 r(W, W 0 ) = (π∞ , π∞ ) ∈ dp(W, W 0 ) 0 , C) 6= 0. such that HomU(V 0 )∞ (π∞ ⊗ π∞

Thus (π, π 0 ) in the global L-packet for (Π, Π0 ) has a non-zero period 0 ⇒ (π∞ , π∞ ) = r(W, W 0 ) ∈ dp(W, W 0 ). Michael Harris

L-functions and periods of automorphic motives

Deligne conjecture for GL(n) × GL(m) Automorphic periods Applications of the IINH conjecture for endoscopic forms Factorization conjecture

Ichino-Ikeda conjecture for standard position (W, W 0 ) in standard position ⇒ V, V 0 definite. The following is due to W. Zhang, with a local refinement at ∞ due to Lin: Theorem (Ichino-Ikeda N. Harris conjecture, definite case) If (W, W 0 ) in standard position and some local conditions then L( 21 , Π × Π0 ) = d(n) × ratio of periods . L(1, Π × Π0 , As± ) (Elementary factor d(n); the ratio of periods is in E.) An algebraic version, with = replaced by up to rational factors was proved by Grobner-Lin, with no local conditions. Michael Harris

L-functions and periods of automorphic motives

Deligne conjecture for GL(n) × GL(m) Automorphic periods Applications of the IINH conjecture for endoscopic forms Factorization conjecture

Ichino-Ikeda conjecture for standard position (W, W 0 ) in standard position ⇒ V, V 0 definite. The following is due to W. Zhang, with a local refinement at ∞ due to Lin: Theorem (Ichino-Ikeda N. Harris conjecture, definite case) If (W, W 0 ) in standard position and some local conditions then L( 12 , Π × Π0 ) = d(n) × ratio of periods . L(1, Π × Π0 , As± ) (Elementary factor d(n); the ratio of periods is in E.) An algebraic version, with = replaced by up to rational factors was proved by Grobner-Lin, with no local conditions. Michael Harris

L-functions and periods of automorphic motives

Deligne conjecture for GL(n) × GL(m) Automorphic periods Applications of the IINH conjecture for endoscopic forms Factorization conjecture

Ichino-Ikeda conjecture for standard position (W, W 0 ) in standard position ⇒ V, V 0 definite. The following is due to W. Zhang, with a local refinement at ∞ due to Lin: Theorem (Ichino-Ikeda N. Harris conjecture, definite case) If (W, W 0 ) in standard position and some local conditions then L( 12 , Π × Π0 ) = d(n) × ratio of periods . L(1, Π × Π0 , As± ) (Elementary factor d(n); the ratio of periods is in E.) An algebraic version, with = replaced by up to rational factors was proved by Grobner-Lin, with no local conditions. Michael Harris

L-functions and periods of automorphic motives

Deligne conjecture for GL(n) × GL(m) Automorphic periods Applications of the IINH conjecture for endoscopic forms Factorization conjecture

Ichino-Ikeda conjecture for standard position (W, W 0 ) in standard position ⇒ V, V 0 definite. The following is due to W. Zhang, with a local refinement at ∞ due to Lin: Theorem (Ichino-Ikeda N. Harris conjecture, definite case) If (W, W 0 ) in standard position and some local conditions then L( 12 , Π × Π0 ) = d(n) × ratio of periods . L(1, Π × Π0 , As± ) (Elementary factor d(n); the ratio of periods is in E.) An algebraic version, with = replaced by up to rational factors was proved by Grobner-Lin, with no local conditions. Michael Harris

L-functions and periods of automorphic motives

Deligne conjecture for GL(n) × GL(m) Automorphic periods Applications of the IINH conjecture for endoscopic forms Factorization conjecture

Ichino-Ikeda conjecture for standard position (W, W 0 ) in standard position ⇒ V, V 0 definite. The following is due to W. Zhang, with a local refinement at ∞ due to Lin: Theorem (Ichino-Ikeda N. Harris conjecture, definite case) If (W, W 0 ) in standard position and some local conditions then L( 12 , Π × Π0 ) = d(n) × ratio of periods . L(1, Π × Π0 , As± ) (Elementary factor d(n); the ratio of periods is in E.) An algebraic version, with = replaced by up to rational factors was proved by Grobner-Lin, with no local conditions. Michael Harris

L-functions and periods of automorphic motives

Deligne conjecture for GL(n) × GL(m) Automorphic periods Applications of the IINH conjecture for endoscopic forms Factorization conjecture

Ichino-Ikeda conjecture for standard position (W, W 0 ) in standard position ⇒ V, V 0 definite. The following is due to W. Zhang, with a local refinement at ∞ due to Lin: Theorem (Ichino-Ikeda N. Harris conjecture, definite case) If (W, W 0 ) in standard position and some local conditions then L( 12 , Π × Π0 ) = d(n) × ratio of periods . L(1, Π × Π0 , As± ) (Elementary factor d(n); the ratio of periods is in E.) An algebraic version, with = replaced by up to rational factors was proved by Grobner-Lin, with no local conditions. Michael Harris

L-functions and periods of automorphic motives

Deligne conjecture for GL(n) × GL(m) Automorphic periods Applications of the IINH conjecture for endoscopic forms Factorization conjecture

Automorphic Deligne conjecture for standard position Theorem (Grobner-H.) If (W, W 0 ) in standard position, then (under a mild regularity hypothesis) Y Y 0 0 Pj (Π0 )sp(j,Π ,Π) . L(s0 , Π × Π0 ) ∼ (2πi)a(s0 ) Pi (Π)sp(i,Π,Π ) i

j

Here Pi (Π) is the Petersson norm of an arithmetic holomorphic form in dp(W, Π) on U(V) with signature (i, n − i) and likewise for Pj (Π0 ) (Grobner: for general CM field.) Conjecture (Factorization Conjecture) Pi (Π) = Pi (M(Π)). Q In other words, Pi (Π) = ik=0 Qk (M(Π)). Michael Harris

L-functions and periods of automorphic motives

Deligne conjecture for GL(n) × GL(m) Automorphic periods Applications of the IINH conjecture for endoscopic forms Factorization conjecture

Automorphic Deligne conjecture for standard position Theorem (Grobner-H.) If (W, W 0 ) in standard position, then (under a mild regularity hypothesis) Y Y 0 0 Pj (Π0 )sp(j,Π ,Π) . Pi (Π)sp(i,Π,Π ) L(s0 , Π × Π0 ) ∼ (2πi)a(s0 ) i

j

Here Pi (Π) is the Petersson norm of an arithmetic holomorphic form in dp(W, Π) on U(V) with signature (i, n − i) and likewise for Pj (Π0 ) (Grobner: for general CM field.) Conjecture (Factorization Conjecture) Pi (Π) = Pi (M(Π)). Q In other words, Pi (Π) = ik=0 Qk (M(Π)). Michael Harris

L-functions and periods of automorphic motives

Deligne conjecture for GL(n) × GL(m) Automorphic periods Applications of the IINH conjecture for endoscopic forms Factorization conjecture

Automorphic Deligne conjecture for standard position Theorem (Grobner-H.) If (W, W 0 ) in standard position, then (under a mild regularity hypothesis) Y Y 0 0 Pj (Π0 )sp(j,Π ,Π) . Pi (Π)sp(i,Π,Π ) L(s0 , Π × Π0 ) ∼ (2πi)a(s0 ) i

j

Here Pi (Π) is the Petersson norm of an arithmetic holomorphic form in dp(W, Π) on U(V) with signature (i, n − i) and likewise for Pj (Π0 ) (Grobner: for general CM field.) Conjecture (Factorization Conjecture) Pi (Π) = Pi (M(Π)). Q In other words, Pi (Π) = ik=0 Qk (M(Π)). Michael Harris

L-functions and periods of automorphic motives

Deligne conjecture for GL(n) × GL(m) Automorphic periods Applications of the IINH conjecture for endoscopic forms Factorization conjecture

Automorphic Deligne conjecture for standard position Theorem (Grobner-H.) If (W, W 0 ) in standard position, then (under a mild regularity hypothesis) Y Y 0 0 Pj (Π0 )sp(j,Π ,Π) . Pi (Π)sp(i,Π,Π ) L(s0 , Π × Π0 ) ∼ (2πi)a(s0 ) i

j

Here Pi (Π) is the Petersson norm of an arithmetic holomorphic form in dp(W, Π) on U(V) with signature (i, n − i) and likewise for Pj (Π0 ) (Grobner: for general CM field.) Conjecture (Factorization Conjecture) Pi (Π) = Pi (M(Π)). Q In other words, Pi (Π) = ik=0 Qk (M(Π)). Michael Harris

L-functions and periods of automorphic motives

Deligne conjecture for GL(n) × GL(m) Automorphic periods Applications of the IINH conjecture for endoscopic forms Factorization conjecture

Automorphic Deligne conjecture for standard position Theorem (Grobner-H.) If (W, W 0 ) in standard position, then (under a mild regularity hypothesis) Y Y 0 0 Pj (Π0 )sp(j,Π ,Π) . Pi (Π)sp(i,Π,Π ) L(s0 , Π × Π0 ) ∼ (2πi)a(s0 ) i

j

Here Pi (Π) is the Petersson norm of an arithmetic holomorphic form in dp(W, Π) on U(V) with signature (i, n − i) and likewise for Pj (Π0 ) (Grobner: for general CM field.) Conjecture (Factorization Conjecture) Pi (Π) = Pi (M(Π)). Q In other words, Pi (Π) = ik=0 Qk (M(Π)). Michael Harris

L-functions and periods of automorphic motives

Deligne conjecture for GL(n) × GL(m) Automorphic periods Applications of the IINH conjecture for endoscopic forms Factorization conjecture

Automorphic Deligne conjecture for standard position Theorem (Grobner-H.) If (W, W 0 ) in standard position, then (under a mild regularity hypothesis) Y Y 0 0 Pj (Π0 )sp(j,Π ,Π) . Pi (Π)sp(i,Π,Π ) L(s0 , Π × Π0 ) ∼ (2πi)a(s0 ) i

j

Here Pi (Π) is the Petersson norm of an arithmetic holomorphic form in dp(W, Π) on U(V) with signature (i, n − i) and likewise for Pj (Π0 ) (Grobner: for general CM field.) Conjecture (Factorization Conjecture) Pi (Π) = Pi (M(Π)). Q In other words, Pi (Π) = ik=0 Qk (M(Π)). Michael Harris

L-functions and periods of automorphic motives

Deligne conjecture for GL(n) × GL(m) Automorphic periods Applications of the IINH conjecture for endoscopic forms Factorization conjecture

Automorphic Deligne conjecture for standard position Theorem (Grobner-H.) If (W, W 0 ) in standard position, then (under a mild regularity hypothesis) Y Y 0 0 Pj (Π0 )sp(j,Π ,Π) . Pi (Π)sp(i,Π,Π ) L(s0 , Π × Π0 ) ∼ (2πi)a(s0 ) i

j

Here Pi (Π) is the Petersson norm of an arithmetic holomorphic form in dp(W, Π) on U(V) with signature (i, n − i) and likewise for Pj (Π0 ) (Grobner: for general CM field.) Conjecture (Factorization Conjecture) Pi (Π) = Pi (M(Π)). Q In other words, Pi (Π) = ik=0 Qk (M(Π)). Michael Harris

L-functions and periods of automorphic motives

Deligne conjecture for GL(n) × GL(m) Automorphic periods Applications of the IINH conjecture for endoscopic forms Factorization conjecture

The Q-periods

The motive M(Π) is realized in the cohomology of the Shimura variety Sh(U(V1 )) for signature (1, n − 1). Fix a πf in the (finite adelic) L-packet on U(V1 ) attached to Π Hodge decomposition: HdR M(Π) ⊗ C = ⊕nk=1 H!k−1 (Sh(U(V)), V k )[πf ] ⊗ C (coherent cohomology) for specific automorphic vector bundles V k in bijection with the L-packet for U(1, n − 1) attached to Π∞ . Each space on the right of rank 1 over E ⊗ C.

Michael Harris

L-functions and periods of automorphic motives

Deligne conjecture for GL(n) × GL(m) Automorphic periods Applications of the IINH conjecture for endoscopic forms Factorization conjecture

The Q-periods

The motive M(Π) is realized in the cohomology of the Shimura variety Sh(U(V1 )) for signature (1, n − 1). Fix a πf in the (finite adelic) L-packet on U(V1 ) attached to Π Hodge decomposition: HdR M(Π) ⊗ C = ⊕nk=1 H!k−1 (Sh(U(V)), V k )[πf ] ⊗ C (coherent cohomology) for specific automorphic vector bundles V k in bijection with the L-packet for U(1, n − 1) attached to Π∞ . Each space on the right of rank 1 over E ⊗ C.

Michael Harris

L-functions and periods of automorphic motives

Deligne conjecture for GL(n) × GL(m) Automorphic periods Applications of the IINH conjecture for endoscopic forms Factorization conjecture

The Q-periods

The motive M(Π) is realized in the cohomology of the Shimura variety Sh(U(V1 )) for signature (1, n − 1). Fix a πf in the (finite adelic) L-packet on U(V1 ) attached to Π Hodge decomposition: HdR M(Π) ⊗ C = ⊕nk=1 H!k−1 (Sh(U(V)), V k )[πf ] ⊗ C (coherent cohomology) for specific automorphic vector bundles V k in bijection with the L-packet for U(1, n − 1) attached to Π∞ . Each space on the right of rank 1 over E ⊗ C.

Michael Harris

L-functions and periods of automorphic motives

Deligne conjecture for GL(n) × GL(m) Automorphic periods Applications of the IINH conjecture for endoscopic forms Factorization conjecture

The Q-periods

The motive M(Π) is realized in the cohomology of the Shimura variety Sh(U(V1 )) for signature (1, n − 1). Fix a πf in the (finite adelic) L-packet on U(V1 ) attached to Π Hodge decomposition: HdR M(Π) ⊗ C = ⊕nk=1 H!k−1 (Sh(U(V)), V k )[πf ] ⊗ C (coherent cohomology) for specific automorphic vector bundles V k in bijection with the L-packet for U(1, n − 1) attached to Π∞ . Each space on the right of rank 1 over E ⊗ C.

Michael Harris

L-functions and periods of automorphic motives

Deligne conjecture for GL(n) × GL(m) Automorphic periods Applications of the IINH conjecture for endoscopic forms Factorization conjecture

The Q-periods

The motive M(Π) is realized in the cohomology of the Shimura variety Sh(U(V1 )) for signature (1, n − 1). Fix a πf in the (finite adelic) L-packet on U(V1 ) attached to Π Hodge decomposition: HdR M(Π) ⊗ C = ⊕nk=1 H!k−1 (Sh(U(V)), V k )[πf ] ⊗ C (coherent cohomology) for specific automorphic vector bundles V k in bijection with the L-packet for U(1, n − 1) attached to Π∞ . Each space on the right of rank 1 over E ⊗ C.

Michael Harris

L-functions and periods of automorphic motives

Deligne conjecture for GL(n) × GL(m) Automorphic periods Applications of the IINH conjecture for endoscopic forms Factorization conjecture

The Q-periods

HdR M(Π) ⊗ C = ⊕nk=1 H!k−1 (Sh(U(V)), V k )[πf ] ⊗ C Define Qk (M(Π)) to be the (normalized) Petersson norm of a generator ωk ∈ H!k−1 (Sh(U(V1 )), V k )[πf ]. Then the Factorization Conjecture is a concrete period relation between automorphic periods on inner forms of U(V1 ).

Michael Harris

L-functions and periods of automorphic motives

Deligne conjecture for GL(n) × GL(m) Automorphic periods Applications of the IINH conjecture for endoscopic forms Factorization conjecture

The Q-periods

HdR M(Π) ⊗ C = ⊕nk=1 H!k−1 (Sh(U(V)), V k )[πf ] ⊗ C Define Qk (M(Π)) to be the (normalized) Petersson norm of a generator ωk ∈ H!k−1 (Sh(U(V1 )), V k )[πf ]. Then the Factorization Conjecture is a concrete period relation between automorphic periods on inner forms of U(V1 ).

Michael Harris

L-functions and periods of automorphic motives

Deligne conjecture for GL(n) × GL(m) Automorphic periods Applications of the IINH conjecture for endoscopic forms Factorization conjecture

Main Theorems Theorem (Grobner-H-Lin) Assume the full IINH conjecture for unitary groups and a non-vanishing conjecture for central values. Assume W sufficiently regular. (ai − ai+1 > n suffices.). Then the Factorization Conjecture holds up to constants that depend only on Π∞ , Π0∞ . Theorem (Grobner-H-Lin) Under the assumptions of the previous theorem, the automorphic version of Deligne’s conjecture for critical values L(s, M(Π) ⊗ M(Π0 )) holds up to constants that depend only on Π∞ , Π0∞ .

Michael Harris

L-functions and periods of automorphic motives

Deligne conjecture for GL(n) × GL(m) Automorphic periods Applications of the IINH conjecture for endoscopic forms Factorization conjecture

Main Theorems Theorem (Grobner-H-Lin) Assume the full IINH conjecture for unitary groups and a non-vanishing conjecture for central values. Assume W sufficiently regular. (ai − ai+1 > n suffices.). Then the Factorization Conjecture holds up to constants that depend only on Π∞ , Π0∞ . Theorem (Grobner-H-Lin) Under the assumptions of the previous theorem, the automorphic version of Deligne’s conjecture for critical values L(s, M(Π) ⊗ M(Π0 )) holds up to constants that depend only on Π∞ , Π0∞ .

Michael Harris

L-functions and periods of automorphic motives

Deligne conjecture for GL(n) × GL(m) Automorphic periods Applications of the IINH conjecture for endoscopic forms Factorization conjecture

Main Theorems Theorem (Grobner-H-Lin) Assume the full IINH conjecture for unitary groups and a non-vanishing conjecture for central values. Assume W sufficiently regular. (ai − ai+1 > n suffices.). Then the Factorization Conjecture holds up to constants that depend only on Π∞ , Π0∞ . Theorem (Grobner-H-Lin) Under the assumptions of the previous theorem, the automorphic version of Deligne’s conjecture for critical values L(s, M(Π) ⊗ M(Π0 )) holds up to constants that depend only on Π∞ , Π0∞ .

Michael Harris

L-functions and periods of automorphic motives

Deligne conjecture for GL(n) × GL(m) Automorphic periods Applications of the IINH conjecture for endoscopic forms Factorization conjecture

Main Theorems Theorem (Grobner-H-Lin) Assume the full IINH conjecture for unitary groups and a non-vanishing conjecture for central values. Assume W sufficiently regular. (ai − ai+1 > n suffices.). Then the Factorization Conjecture holds up to constants that depend only on Π∞ , Π0∞ . Theorem (Grobner-H-Lin) Under the assumptions of the previous theorem, the automorphic version of Deligne’s conjecture for critical values L(s, M(Π) ⊗ M(Π0 )) holds up to constants that depend only on Π∞ , Π0∞ .

Michael Harris

L-functions and periods of automorphic motives

Deligne conjecture for GL(n) × GL(m) Automorphic periods Applications of the IINH conjecture for endoscopic forms Factorization conjecture

Main Theorems Theorem (Grobner-H-Lin) Assume the full IINH conjecture for unitary groups and a non-vanishing conjecture for central values. Assume W sufficiently regular. (ai − ai+1 > n suffices.). Then the Factorization Conjecture holds up to constants that depend only on Π∞ , Π0∞ . Theorem (Grobner-H-Lin) Under the assumptions of the previous theorem, the automorphic version of Deligne’s conjecture for critical values L(s, M(Π) ⊗ M(Π0 )) holds up to constants that depend only on Π∞ , Π0∞ .

Michael Harris

L-functions and periods of automorphic motives

Deligne conjecture for GL(n) × GL(m) Automorphic periods Applications of the IINH conjecture for endoscopic forms Factorization conjecture

Adding characters, opposite parity Assume n − m ≡ 1 (mod 2). Assuming IINH conjecture in general, we obtain an expression for L( 12 , Π × Π0 ) even if Π, Π0 Eisenstein. Choose N >> n and Hecke characters χi , i = 1, . . . , N − n; χ0j , j = 1, . . . , N − m so that M M ΠN = Π ⊕ χi ; Π0N−1 = Π0 ⊕ χ0j i

j

are in standard position.(This requires a regularity hypothesis.) Conjecture (Non-vanishing conjecture) The characters can be chosen so that the following L-values do not vanish: 1 1 1 L( , Π ⊗ χ0j ), L( , Π0 ⊗ χi ), L( , χi χ0j ), ∀ i, j. 2 2 2 Michael Harris

L-functions and periods of automorphic motives

Deligne conjecture for GL(n) × GL(m) Automorphic periods Applications of the IINH conjecture for endoscopic forms Factorization conjecture

Adding characters, opposite parity Assume n − m ≡ 1 (mod 2). Assuming IINH conjecture in general, we obtain an expression for L( 21 , Π × Π0 ) even if Π, Π0 Eisenstein. Choose N >> n and Hecke characters χi , i = 1, . . . , N − n; χ0j , j = 1, . . . , N − m so that M M ΠN = Π ⊕ χi ; Π0N−1 = Π0 ⊕ χ0j i

j

are in standard position.(This requires a regularity hypothesis.) Conjecture (Non-vanishing conjecture) The characters can be chosen so that the following L-values do not vanish: 1 1 1 L( , Π ⊗ χ0j ), L( , Π0 ⊗ χi ), L( , χi χ0j ), ∀ i, j. 2 2 2 Michael Harris

L-functions and periods of automorphic motives

Deligne conjecture for GL(n) × GL(m) Automorphic periods Applications of the IINH conjecture for endoscopic forms Factorization conjecture

Adding characters, opposite parity Assume n − m ≡ 1 (mod 2). Assuming IINH conjecture in general, we obtain an expression for L( 21 , Π × Π0 ) even if Π, Π0 Eisenstein. Choose N >> n and Hecke characters χi , i = 1, . . . , N − n; χ0j , j = 1, . . . , N − m so that M M ΠN = Π ⊕ χi ; Π0N−1 = Π0 ⊕ χ0j i

j

are in standard position.(This requires a regularity hypothesis.) Conjecture (Non-vanishing conjecture) The characters can be chosen so that the following L-values do not vanish: 1 1 1 L( , Π ⊗ χ0j ), L( , Π0 ⊗ χi ), L( , χi χ0j ), ∀ i, j. 2 2 2 Michael Harris

L-functions and periods of automorphic motives

Deligne conjecture for GL(n) × GL(m) Automorphic periods Applications of the IINH conjecture for endoscopic forms Factorization conjecture

Adding characters, opposite parity Assume n − m ≡ 1 (mod 2). Assuming IINH conjecture in general, we obtain an expression for L( 21 , Π × Π0 ) even if Π, Π0 Eisenstein. Choose N >> n and Hecke characters χi , i = 1, . . . , N − n; χ0j , j = 1, . . . , N − m so that M M ΠN = Π ⊕ χi ; Π0N−1 = Π0 ⊕ χ0j i

j

are in standard position.(This requires a regularity hypothesis.) Conjecture (Non-vanishing conjecture) The characters can be chosen so that the following L-values do not vanish: 1 1 1 L( , Π ⊗ χ0j ), L( , Π0 ⊗ χi ), L( , χi χ0j ), ∀ i, j. 2 2 2 Michael Harris

L-functions and periods of automorphic motives

Deligne conjecture for GL(n) × GL(m) Automorphic periods Applications of the IINH conjecture for endoscopic forms Factorization conjecture

Adding characters, opposite parity Assume n − m ≡ 1 (mod 2). Assuming IINH conjecture in general, we obtain an expression for L( 21 , Π × Π0 ) even if Π, Π0 Eisenstein. Choose N >> n and Hecke characters χi , i = 1, . . . , N − n; χ0j , j = 1, . . . , N − m so that M M ΠN = Π ⊕ χi ; Π0N−1 = Π0 ⊕ χ0j i

j

are in standard position.(This requires a regularity hypothesis.) Conjecture (Non-vanishing conjecture) The characters can be chosen so that the following L-values do not vanish: 1 1 1 L( , Π ⊗ χ0j ), L( , Π0 ⊗ χi ), L( , χi χ0j ), ∀ i, j. 2 2 2 Michael Harris

L-functions and periods of automorphic motives

Deligne conjecture for GL(n) × GL(m) Automorphic periods Applications of the IINH conjecture for endoscopic forms Factorization conjecture

Applying the IINH conjecture Let U(N) ⊃ U(N − 1) denote definite unitary groups of the indicated dimensions. Under the above conjecture, there is [KMSW] an (endoscopic) 0 L-packet πN × πN−1 on U(N) × U(N − 1) with base change 0 ΠN × ΠN−1 such that the period integral Z 0 : πN ⊗ πN−1 →C [U(N−1)]

is not identically zero and (when normalized by the Petersson norms) gives the IINH ratio of L-values.

Michael Harris

L-functions and periods of automorphic motives

Deligne conjecture for GL(n) × GL(m) Automorphic periods Applications of the IINH conjecture for endoscopic forms Factorization conjecture

Applying the IINH conjecture Let U(N) ⊃ U(N − 1) denote definite unitary groups of the indicated dimensions. Under the above conjecture, there is [KMSW] an (endoscopic) 0 L-packet πN × πN−1 on U(N) × U(N − 1) with base change 0 ΠN × ΠN−1 such that the period integral Z 0 →C : πN ⊗ πN−1 [U(N−1)]

is not identically zero and (when normalized by the Petersson norms) gives the IINH ratio of L-values.

Michael Harris

L-functions and periods of automorphic motives

Deligne conjecture for GL(n) × GL(m) Automorphic periods Applications of the IINH conjecture for endoscopic forms Factorization conjecture

Applying the IINH conjecture Let U(N) ⊃ U(N − 1) denote definite unitary groups of the indicated dimensions. Under the above conjecture, there is [KMSW] an (endoscopic) 0 L-packet πN × πN−1 on U(N) × U(N − 1) with base change 0 ΠN × ΠN−1 such that the period integral Z 0 : πN ⊗ πN−1 →C [U(N−1)]

is not identically zero and (when normalized by the Petersson norms) gives the IINH ratio of L-values.

Michael Harris

L-functions and periods of automorphic motives

Deligne conjecture for GL(n) × GL(m) Automorphic periods Applications of the IINH conjecture for endoscopic forms Factorization conjecture

Applying the IINH conjecture Let U(N) ⊃ U(N − 1) denote definite unitary groups of the indicated dimensions. Under the above conjecture, there is [KMSW] an (endoscopic) 0 L-packet πN × πN−1 on U(N) × U(N − 1) with base change 0 ΠN × ΠN−1 such that the period integral Z 0 : πN ⊗ πN−1 →C [U(N−1)]

is not identically zero and (when normalized by the Petersson norms) gives the IINH ratio of L-values.

Michael Harris

L-functions and periods of automorphic motives

Deligne conjecture for GL(n) × GL(m) Automorphic periods Applications of the IINH conjecture for endoscopic forms Factorization conjecture

Applying the IINH conjecture Let U(N) ⊃ U(N − 1) denote definite unitary groups of the indicated dimensions. Under the above conjecture, there is [KMSW] an (endoscopic) 0 L-packet πN × πN−1 on U(N) × U(N − 1) with base change 0 ΠN × ΠN−1 such that the period integral Z 0 : πN ⊗ πN−1 →C [U(N−1)]

is not identically zero and (when normalized by the Petersson norms) gives the IINH ratio of L-values.

Michael Harris

L-functions and periods of automorphic motives

Deligne conjecture for GL(n) × GL(m) Automorphic periods Applications of the IINH conjecture for endoscopic forms Factorization conjecture

Central value, opposite parity

The factorization 1 1 L( , ΠN × Π0N−1 ) = L( , Π × Π0 )· 2 2 Y 1 1 0 1 0 L( , Π ⊗ χj )L( , Π ⊗ χi ), L( , χi χ0j ) 2 2 2 i,j

gives the desired result for L( 21 , π × π 0 ) by induction.

Michael Harris

L-functions and periods of automorphic motives

(9)

Deligne conjecture for GL(n) × GL(m) Automorphic periods Applications of the IINH conjecture for endoscopic forms Factorization conjecture

Near-central value, same parity

For m ≡ n (mod 2), the automorphic Deligne conjecture was proved by Lin for L(1, Π × Π0 ) in her thesis using Eisenstein cohomology and the methods of Grobner-H. especially Shahidi’s computation of Whittaker coefficients. A regularity hypothesis is needed, but not the non-vanishing conjecture nor the IINH conjecture.

Michael Harris

L-functions and periods of automorphic motives

Deligne conjecture for GL(n) × GL(m) Automorphic periods Applications of the IINH conjecture for endoscopic forms Factorization conjecture

Near-central value, same parity

For m ≡ n (mod 2), the automorphic Deligne conjecture was proved by Lin for L(1, Π × Π0 ) in her thesis using Eisenstein cohomology and the methods of Grobner-H. especially Shahidi’s computation of Whittaker coefficients. A regularity hypothesis is needed, but not the non-vanishing conjecture nor the IINH conjecture.

Michael Harris

L-functions and periods of automorphic motives

Deligne conjecture for GL(n) × GL(m) Automorphic periods Applications of the IINH conjecture for endoscopic forms Factorization conjecture

Near-central value, same parity

For m ≡ n (mod 2), the automorphic Deligne conjecture was proved by Lin for L(1, Π × Π0 ) in her thesis using Eisenstein cohomology and the methods of Grobner-H. especially Shahidi’s computation of Whittaker coefficients. A regularity hypothesis is needed, but not the non-vanishing conjecture nor the IINH conjecture.

Michael Harris

L-functions and periods of automorphic motives

Deligne conjecture for GL(n) × GL(m) Automorphic periods Applications of the IINH conjecture for endoscopic forms Factorization conjecture

Near-central value, same parity

For m ≡ n (mod 2), the automorphic Deligne conjecture was proved by Lin for L(1, Π × Π0 ) in her thesis using Eisenstein cohomology and the methods of Grobner-H. especially Shahidi’s computation of Whittaker coefficients. A regularity hypothesis is needed, but not the non-vanishing conjecture nor the IINH conjecture.

Michael Harris

L-functions and periods of automorphic motives

Deligne conjecture for GL(n) × GL(m) Automorphic periods Applications of the IINH conjecture for endoscopic forms Factorization conjecture

Near-central value, same parity

For m ≡ n (mod 2), the automorphic Deligne conjecture was proved by Lin for L(1, Π × Π0 ) in her thesis using Eisenstein cohomology and the methods of Grobner-H. especially Shahidi’s computation of Whittaker coefficients. A regularity hypothesis is needed, but not the non-vanishing conjecture nor the IINH conjecture.

Michael Harris

L-functions and periods of automorphic motives

Deligne conjecture for GL(n) × GL(m) Automorphic periods Applications of the IINH conjecture for endoscopic forms Factorization conjecture

General critical values, either parity

To deduce the result for general critical values, one uses the method of Harder-Raghuram.

Michael Harris

L-functions and periods of automorphic motives

Deligne conjecture for GL(n) × GL(m) Automorphic periods Applications of the IINH conjecture for endoscopic forms Factorization conjecture

Rational cup products Assuming the automorphic Deligne conjecture, we obtain the Factorization conjecture by considering the pair U(V1 ) ⊃ U(V10 ) (signatures (1, n − 1) and (1, n − 2)). For certain π × π 0 in the corresponding L-packet, the period integral can be interpreted as a cup product in coherent cohomology. Z 0 0 P(π, π ) = fπ · fπ0 0 = Tr(ωk ∪ ωn−1−k )∈Q [U(V10 )]

if fπ and fπ0 0 define arithmetic coherent cohomology classes Under the regularity hypothesis, one finds enough examples of such π using work of Salamanca-Riba and the Burger-Sarnak method.

Michael Harris

L-functions and periods of automorphic motives

Deligne conjecture for GL(n) × GL(m) Automorphic periods Applications of the IINH conjecture for endoscopic forms Factorization conjecture

Rational cup products Assuming the automorphic Deligne conjecture, we obtain the Factorization conjecture by considering the pair U(V1 ) ⊃ U(V10 ) (signatures (1, n − 1) and (1, n − 2)). For certain π × π 0 in the corresponding L-packet, the period integral can be interpreted as a cup product in coherent cohomology. Z 0 0 P(π, π ) = fπ · fπ0 0 = Tr(ωk ∪ ωn−1−k )∈Q [U(V10 )]

if fπ and fπ0 0 define arithmetic coherent cohomology classes Under the regularity hypothesis, one finds enough examples of such π using work of Salamanca-Riba and the Burger-Sarnak method.

Michael Harris

L-functions and periods of automorphic motives

Deligne conjecture for GL(n) × GL(m) Automorphic periods Applications of the IINH conjecture for endoscopic forms Factorization conjecture

Rational cup products Assuming the automorphic Deligne conjecture, we obtain the Factorization conjecture by considering the pair U(V1 ) ⊃ U(V10 ) (signatures (1, n − 1) and (1, n − 2)). For certain π × π 0 in the corresponding L-packet, the period integral can be interpreted as a cup product in coherent cohomology. Z 0 0 P(π, π ) = fπ · fπ0 0 = Tr(ωk ∪ ωn−1−k )∈Q [U(V10 )]

if fπ and fπ0 0 define arithmetic coherent cohomology classes Under the regularity hypothesis, one finds enough examples of such π using work of Salamanca-Riba and the Burger-Sarnak method.

Michael Harris

L-functions and periods of automorphic motives

Deligne conjecture for GL(n) × GL(m) Automorphic periods Applications of the IINH conjecture for endoscopic forms Factorization conjecture

Rational cup products Assuming the automorphic Deligne conjecture, we obtain the Factorization conjecture by considering the pair U(V1 ) ⊃ U(V10 ) (signatures (1, n − 1) and (1, n − 2)). For certain π × π 0 in the corresponding L-packet, the period integral can be interpreted as a cup product in coherent cohomology. Z 0 0 P(π, π ) = fπ · fπ0 0 = Tr(ωk ∪ ωn−1−k )∈Q [U(V10 )]

if fπ and fπ0 0 define arithmetic coherent cohomology classes Under the regularity hypothesis, one finds enough examples of such π using work of Salamanca-Riba and the Burger-Sarnak method.

Michael Harris

L-functions and periods of automorphic motives

Deligne conjecture for GL(n) × GL(m) Automorphic periods Applications of the IINH conjecture for endoscopic forms Factorization conjecture

Rational cup products Assuming the automorphic Deligne conjecture, we obtain the Factorization conjecture by considering the pair U(V1 ) ⊃ U(V10 ) (signatures (1, n − 1) and (1, n − 2)). For certain π × π 0 in the corresponding L-packet, the period integral can be interpreted as a cup product in coherent cohomology. Z 0 0 P(π, π ) = fπ · fπ0 0 = Tr(ωk ∪ ωn−1−k )∈Q [U(V10 )]

if fπ and fπ0 0 define arithmetic coherent cohomology classes Under the regularity hypothesis, one finds enough examples of such π using work of Salamanca-Riba and the Burger-Sarnak method.

Michael Harris

L-functions and periods of automorphic motives

Deligne conjecture for GL(n) × GL(m) Automorphic periods Applications of the IINH conjecture for endoscopic forms Factorization conjecture

IINH conjecture, rank one The IINH conjecture then yields L( 12 , Π × Π0 ) |P(π, π 0 )| = (∗) Qk (M(Π))Qn−1−k (M(Π0 )) L(1, Π × Π0 , As± ) The numerator on the left is algebraic. The expression on the left can be written in terms of automorphic periods Pi (Π), Pj (Π0 ) by the Deligne conjecture except that the factor (∗) contains an unknown archimedean period integral. Using Burger-Sarnak, we can assume the left hand side doesn’t vanish. Comparing the expressions on the two sides gives the factorization by induction on n. Michael Harris

L-functions and periods of automorphic motives

Deligne conjecture for GL(n) × GL(m) Automorphic periods Applications of the IINH conjecture for endoscopic forms Factorization conjecture

IINH conjecture, rank one The IINH conjecture then yields L( 12 , Π × Π0 ) |P(π, π 0 )| = (∗) Qk (M(Π))Qn−1−k (M(Π0 )) L(1, Π × Π0 , As± ) The numerator on the left is algebraic. The expression on the left can be written in terms of automorphic periods Pi (Π), Pj (Π0 ) by the Deligne conjecture except that the factor (∗) contains an unknown archimedean period integral. Using Burger-Sarnak, we can assume the left hand side doesn’t vanish. Comparing the expressions on the two sides gives the factorization by induction on n. Michael Harris

L-functions and periods of automorphic motives

Deligne conjecture for GL(n) × GL(m) Automorphic periods Applications of the IINH conjecture for endoscopic forms Factorization conjecture

IINH conjecture, rank one The IINH conjecture then yields L( 12 , Π × Π0 ) |P(π, π 0 )| = (∗) Qk (M(Π))Qn−1−k (M(Π0 )) L(1, Π × Π0 , As± ) The numerator on the left is algebraic. The expression on the left can be written in terms of automorphic periods Pi (Π), Pj (Π0 ) by the Deligne conjecture except that the factor (∗) contains an unknown archimedean period integral. Using Burger-Sarnak, we can assume the left hand side doesn’t vanish. Comparing the expressions on the two sides gives the factorization by induction on n. Michael Harris

L-functions and periods of automorphic motives

Deligne conjecture for GL(n) × GL(m) Automorphic periods Applications of the IINH conjecture for endoscopic forms Factorization conjecture

IINH conjecture, rank one The IINH conjecture then yields L( 12 , Π × Π0 ) |P(π, π 0 )| = (∗) Qk (M(Π))Qn−1−k (M(Π0 )) L(1, Π × Π0 , As± ) The numerator on the left is algebraic. The expression on the left can be written in terms of automorphic periods Pi (Π), Pj (Π0 ) by the Deligne conjecture except that the factor (∗) contains an unknown archimedean period integral. Using Burger-Sarnak, we can assume the left hand side doesn’t vanish. Comparing the expressions on the two sides gives the factorization by induction on n. Michael Harris

L-functions and periods of automorphic motives

Deligne conjecture for GL(n) × GL(m) Automorphic periods Applications of the IINH conjecture for endoscopic forms Factorization conjecture

IINH conjecture, rank one The IINH conjecture then yields L( 12 , Π × Π0 ) |P(π, π 0 )| = (∗) Qk (M(Π))Qn−1−k (M(Π0 )) L(1, Π × Π0 , As± ) The numerator on the left is algebraic. The expression on the left can be written in terms of automorphic periods Pi (Π), Pj (Π0 ) by the Deligne conjecture except that the factor (∗) contains an unknown archimedean period integral. Using Burger-Sarnak, we can assume the left hand side doesn’t vanish. Comparing the expressions on the two sides gives the factorization by induction on n. Michael Harris

L-functions and periods of automorphic motives