PDF Higher Rank Zeta Functions and Riemann Hypothesis for Elliptic Curves

Higher Rank Zeta Functions and Riemann Hypothesis for Elliptic Curves

Lin WENG

Faculty of Mathematics, Kyushu University

Arithmetic and Algebraic Geometry 2013, Tokyo

Lin WENG Zetas for Elliptic Curves

Outline

1 Pure Zetas of Curves

Parabolic Reduction, Stability & the Mass

Split Reductive Groups: Number Fields Example: Special Linear Groups

Pure Non-Abelian Zetas Zetas under Symmetry Special Uniformity of Zetas

2 Zeros of Zetas for Elliptic Curves The Riemann Hypothesis Our Proof

Counting Miracle Beta Invariants

Special and General Counting Miracles

Higher Sato-Tate

3 Global Zetas

Local Zetas for Nodal Curves Global Zetas

4 Weil Conjecture for Stacks: Semi-stable G-Bundles Behrend’s Lefschetz Trace Formula

Beta Invariants for G-Bundles END

Lin WENG Zetas for Elliptic Curves

Stability

Setting

X / F q : irreducible, reduced, regular proj curve of genus g F : function field, A : adelic ring

M X,n : moduli stack of rank n bdles of on X GL(n, F )\GL(n, A )/ K

M ss X,n (d): moduli stack of s.stable bdles of rank n, degree d m X,n (d) := P

V ∈ M

(d) 1

#Aut(V) , independent of d m ss X,n (d) := P

V ∈ M

(d) 1

#Aut(V) , dependent of d ζ b X (s): complete Artin zeta function of X

If n = n 1 + n 2 + · · · + n k , N i := n 1 + · · · + n i , N i 0 := n − N i

Lin WENG Zetas for Elliptic Curves

Mumford’s Intersection Stability

Stability

V /X : vector bundle V : semi-stable ⇔

deg(V 1 )

rank(V 1 ) ≤ deg(V )

rank(V ) , ∀V 1 ≤ V

Various Spaces

Moduli spaces do not work well Moduli stacks work

Best one: Adelic space

Tamagawa Number, Parabolic Reduction

Theorem (Tamagawa Number, Parabolic Reduction) (Weil) m X,n (d) = ζ b X (1) ζ b X (2) · · · ζ b X (n).

(Harder-Narasimhan, Desale-Ramanan and Zagier)

m

(0) q

=

n

X

k=1

(−1)

X

n₁+···+n_k=n n₁>0,···,n_k>0

1 Q

j=1

(q

− 1) ·

k

Y

j=1

m

(0).

(Weng-Zagier: in preparation)

n·m

(d ) =

n

X

k=1

X

n₁+···+n_k=n n₁>0,···,n_k>0

X

δ_i∈{0,1,...,n_i−1}

i=1,2,...,k k−1

Y

i=1

q

q

− 1 ·

k

Y

j=1

m

(δ

) q

(nj−1) 2 (g−1)

.

Here v

∈ [0, 1) ∩ Q satisfying v

≡

i

−

i+1

(mod 1)

Lin WENG Zetas for Elliptic Curves

Split Reductive Groups: Number Fields

Setting

F = Q : field of rational, A : ring of adeles G/F : split reductive group

B/F : Borel, P/F : standard parabolic subgroup P = M · N: Levi decomposition w/ M Levi factor M ∼ Q k

i=1 M i simple decomposition w/ M i ’s reductive M F ,G : moduli space of G-lattices

' G(F )\G( A )/ K

M ss F ,G : (compact) subspace of s.stable G-lattices ν F ,G := Vol

M F ,G

, ν F ss ,G := Vol

M ss F ,G

Parabolic Reduction, Stability & the Volumes

Theorem? (Weng)

Parabolic Reduction ∃ c P ∈ Q >o , e P ∈ Q >o

ν F ,G = X

P

e P · ν F ss ,P , ν F ss ,G = X

P

sgn(P) · c P · ν F,P . Here P = M · N, M ∼ Q

j M j and ν F ,P := Q

j ν F ,M

, ν F ss ,P := Q

j ν F ss ,M

j

• c P , but not yet e P : explicit expressions in terms of root system

• Different Systems of basis: ν F ss ,G ⇔ ν F ,G

• Non-Abelian versus Abelian: Group structures involved at c P

Lin WENG Zetas for Elliptic Curves

Volumes of Fundamental Domains

Theorem

(Langlands)

ν F ,G = c G · Y

i≥1

ζ b F (i) −n

(G) . Here c G = Vol

n P

α∈∆ a α α ∨ : a α ∈ [0, 1] o

n i (G) := #{α > 0, hρ, α ∨ i = i } − #{α > 0, hρ, α ∨ i = i − 1}

with ∆: simple roots, ρ = 1 2 P

α>0 α Done by taking residues of Eisenstein series

Non-Abelian versus Abelian Invariants:

Group Structures involved !!!

Example: SL n ( Z ), Stable Lattices

Definition

Λ ⊂ R n : rank n lattice.

Λ semi-stable if

Vol Λ 1 rank(Λ)

≥ Vol Λ

rank(Λ

)

, ∀Λ 1 ⊂ Λ.

M Q ,n [1]: moduli space of rank n lattices of vol 1 ' SL(n, Z )\SL(n, R )/SO(n)

M ss

Q ,n [1]: (compact) subspace of s.stable lattices v Q,n := Vol

M Q,n [1]

, v ss

Q ,n := Vol M ss

Q ,n [1]

ζ b Q (s): complete Riemann zeta function

Lin WENG Zetas for Elliptic Curves

Volume of Fund Domain, Parabolic Reduction

Theorem (Siegel)

1

n · v Q ,n = ζ b Q (1)b ζ Q (2) · · · ζ b Q (n).

(Weng)

v

=

n

X

k=1

(−1)

X

n₁+···+n_k=n n₁>0,···,n_k>0

1 Q

j=1

(n

+ n

) ·

k

Y

j=1

v

.

(Kontsevich-Soibelman)

1 n · v

=

n

X

k=1

X

n₁+···+n_k=n n₁>0,···,n_k>0

1

n

(n

+ n

) · · · (n

+ · · · + n

) · · · n

·

k

Y

j=1

v

.

Non-Abelian Zeta Function

Definition (Weng)

Pure Non-Abelian Zeta Function of X / F q :

ζ b X,n (s) := X

V∈ M

(n Z)

q h

(X ,V ) − 1

#Aut(V ) · q −s χ(X,V)

dµ, <(s) > 1.

α, β-invariants

β X,n (d) := X

V ∈ M

(d)

1

#Aut(V ) = m X ss ,n (d ),

α X,n (d) := X

V ∈ M

(d)

q h

(X ,V ) − 1

#Aut(V )

Lin WENG Zetas for Elliptic Curves

Zeta Facts

Let t = q −s , T = t n , Q = q n ,

ζ b X,n (s) =

(g−1)−1

X

m=0

α X ,n (mn) ·

T m−(g−1) + 1 QT

(g−1)−m

+ α X,n (n(g − 1)) + β X ,n (0) · (Q − 1)T (1 − QT )(1 − T )

(Initial State) ζ b X,1 (s) = ζ b X (s): complete Artin zeta (Rationality) ζ b X ,n (s) is rational in T

(Functional Eq) ζ b X,n (1 − s) = ζ b X,n (s)

(Residue) Res s=1 ζ b X,n (s) = β X,n (0)

Number Fields versus Function Fields

Parabolic Reduction & Periods

Parabolic Reduction: (i) Number Fields (Weng) v

=

n

X

k=1

(−1)

X

n₁+···+n_k=n n₁>0,···,n_k>0

1 Q

j=1

(n

+ n

) ·

k

Y

j=1

v

.

(ii) Function Fields/ F

(Zagier)

m

(0) =

n

X

k=1

X

n₁+···+n_k=n n₁>0,···,n_k>0

(−1)

Q

j=1

(q

− 1) ·

k

Y

j=1

m

(0).

Periods of G: Number Fields (Weng)

ω

(λ) := X

w∈W

1 Q

α∈∆

hwλ − ρ, α

i · Y

α>0,wα<0

ζ b

(hλ, α

i) ζ b

(hλ, α

i + 1)

Lin WENG Zetas for Elliptic Curves

Period of G/X

Setting

G/F : split reductive group, B/F : Borel P/F : maximal standard parabolic subgroup

∆: Simple roots, W : Weyl group , ρ: Weyl vector α P ∈ ∆: simple for P

{β 1 , . . . , β |P| } = ∆\{α P }

Definition (Weng) Period of G/F :

ω

(λ) := X

w∈W

1

Q

α∈∆

(1 − q

) · Y

α>0,wα<0

ζ b

(hλ, α

i) b ζ

(hλ, α

i + 1)

Period of (G, P )/F

Definition (Weng) Period of (G, P)/F :

ω X G/P (s) := Res hλ−ρ,β

1,P

i=0,hλ−ρ,β

i=0, ..., hλ−ρ,β

i=0

ω G X (λ)

:= Res hλ−ρ,β

i=0,hλ−ρ,β

i=0, ..., hλ−ρ,β

i=0

X

w∈W

1 Q

α∈∆ (1 − q hwλ−ρ,α

i ) · Y

α>0,wα<0

ζ b X (hλ, α ∨ i) ζ b X (hλ, α ∨ i + 1)

!

• Various symmetries play key roles here.

Lin WENG Zetas for Elliptic Curves

Special Uniformity of Zetas

Definition (Weng)

Zeta function ζ b X G/P for (G, P)/F : ζ b X G/P (s) := Norm

ω X G/P (s)

Zeta Facts (Weng) Functional Equation

ζ b X G/P (1 − s) = ζ b X G/P (s) Special Uniformity of Zetas

ζ b X,n (s) = · ζ b X SL

/P

(s)

Example of Special Uniformity

(SL 3 , P 2,1 )

ζ b X SL /

/P

F

(s) = ζ b X (2) ζ b X (3s)

1 − q 3−3s + ζ b X (2) ζ b X (3s − 2) 1 − q 3s + ζ b X (1) ζ b X (3s − 1)

(1 − q 3s )(1 − q 3−3s ) + ζ b X (1) ζ b X (3s − 2)

(1 − q 2 )(1 − q 3s−1 ) + ζ b X (1) ζ b X (3s) (1 − q 2 )(1 − q 2−3s ) . This is essentially rank 3 zeta function ζ b X ,3 (s).

Non-Abelian versus Abelian Invariants:

Group Structures Involved

Lin WENG Zetas for Elliptic Curves

Consequence of Special Uniformity

Alpha and Beta Invariants

α and β invariants: completely determined by (i) the Lie structure; and

(ii) Special values of Artin zetas

Say, for rank 2:

α X,2 (2m) α X,2 (0) =

m

X

i=0

q 2(m−i) α X ,1 (i) − 1 q g−1

m−1

X

i=0

q i α X,1 (i)

Intrinsic Relation among different Brill-Noether Loci

The Riemann Hypothesis

The Riemann Hypothesis

ζ b X,n (s) = 0 ⇒ <(s) = 1 2 Non-Abelian Zetas for elliptic curves:

E/F q : elliptic curve

ζ b E,n (s) = α E,n (0) + β E,n (0) (Q − 1)T (1 − T )(1 − QT )

Main Theorem (Hasse: n = 1, Weng-Zagier: n ≥ 2) ζ b E,n (s) = 0 ⇒ <(s) = 1

2

Lin WENG Zetas for Elliptic Curves

Counting Miracle

Theorem A (Counting Miracle: Weng-Zagier)

α E,n+1 (0) = X

V ∈ M

(0)

q h

(E,V ) − 1

#AutV = X

V ∈ M

(0)

1

#AutV = β E,n (0)

Corollary

E/ F q : elliptic curve

ζ b E,n (s) = β E,n−1 (0) + β E,n (0) (Q − 1)T

(1 − T )(1 − QT )

Beta Invariants

Parabolic Reduction: Zagier’s formula

β E,n (0) =

n

X

k=1

X

n

+···+n

=n n

>0,···,n

>0

(−1) k−1 Q k−1

j=1 (q n

+n

− 1) ·

k

Y

j=1 n

Y

i=1

ζ E (i).

Theorem B (Multiplicative Structure of Beta: Weng-Zagier) B E / F

(x) =:

∞

X

n=0

β B,n (0)x n x =q

−s

= Y

n≥1

ζ E (s + n)

It is a wonderful world!!!

Lin WENG Zetas for Elliptic Curves

Proof of the RH

Proof of the RH (i) (Recursion)

β E,n (0) = q n + q n+1 − a

q n − 1 β E,n−1 (0) − q n−1 − q

q n − 1 β E,n−2 (0).

(ii) (The Estimation)

√ q n − 1

√ q n + 1 ≤ β E,n (0) α E,n (0) ≤

√ q n + 1

√ q n − 1

(iii) (The Riemann Hypothesis)

ζ b E,n (s) = 0 ⇒ <(s) = 1

2

Special Counting Miracle

Atiyah Bundles E/F q : elliptic curve

Inductively define (indecomposible) Atiyah Bundle I n /E : I 1 = O E

I n = the only non-trivial extension of I n−1 by O E : 0 → O E → I n → I n−1 → 0 since

Ext 1 (I n , O E ) ' H 1 (E, I n ∨ ) ' H 0 (E , I n ∨ ) ∨ = H 0 (E , I n ) ' F q

Lin WENG Zetas for Elliptic Curves

Automorphisms

Automorphisms

Consider Atiyah bundles ⊕ s j=1 I r ⊕m

(0 < r 1 < r 2 < · · · < r s ) (Atiyah) h 0 (E, ⊕ s j=1 I r ⊕m

) = P s

j=1 m j (Weng-Zagier)

#Aut

⊕ s j=1 I r ⊕m

= q 2

P

1≤i<j≤s

r

m

m

×

s

Y

j=1

(q m

− 1)(q m

− q) · · · (q m

− q m

−1 )q m

(r

−1) .

Special Counting Miracle

Theorem A 0 (Counting Miracle: Weng-Zagier) (i) (Special Counting Miracle: Combinatorial Aspect)

α At E,n+1 (0) := X

V:Atiyah Bdl, rank=n+1

q h

(E,V ) − 1

#Aut(V )

= q

(q n − 1)(q n−1 − 1) · · · (q − 1)

= X

V:Atiyah Bdl, rank=n

1

#Aut(V ) = β E,n At (0) (ii) (Geometric Aspect) Special CM ⇔ General CM

Lin WENG Zetas for Elliptic Curves

General Counting Miracle

Geometric Aspect

Very difficulty to classify semi-stable vector bundles of degree 0 defined over F q :

Rationality Problem – intrinsic arithmetic information such as number of m-torsion points over F k q should be used.

But for V appeared in the α invariant, h 0 (E , V ) 6= 0 implies that there is at least one factor of Atiyah bundles;

Moreover, this factor admits no non-trivial morphisms to other factors

This gives the deduction without details calculation in general

Special Counting Miracle

Combinatorial Aspect: Generating Functions We have A At (x ) = B At (x ) = P ∞

n=0 ε(n)

x q

n

where ε(m) := q m

(q m − 1)(q m − q) · · · (q m − q m−1 ) and

A At (x ) :=

∞

X

s=0

X

0<r

<r

<···<r

m

>0,m

>0, ..., m

>0

ε(m 1 ) · · · ε(m s )

q N(r,m) · x r

m

+···+r

m

,

B At (x ) := 1 x

∞

X

s=0

X

0<r

<r

<···<r

m

>0, m

>0, ..., m

>0

(q m

+···+m

− 1) ε(m 1 ) · · · ε(m s ) q N(r,m)

× x r

m

+···+r

m

with N(r, m) :=

s

X

i=1

r i m i (m 1 + 2m i+1 + · · · + 2m s ).

Lin WENG Zetas for Elliptic Curves

Sugahara’s Result

Theorem (Sugahara)

X/F q : irreducible, reduced, regular projective curve of genus g:

α X,n+1 (0) = q n(g−1) · β X,n (0) Totally different approach:

motivated by the work of Reineke on Quives

Sato-Tate

Setting

p: prime, N = #X( F p ), a = p + 1 − N cos θ 1,p E := p+1−N 2 √ p , 0 ≤ θ E 1,p < π cos θ n,p E :=

−(p

−1)·

E,n−1(0)

+(p

+1) 2 √

p

Sato-Tate Conjecture

For non CM elliptic curves E, and 0 ≤ α < β ≤ π,

x→∞ lim

#{p : prime, p ≤ x , α ≤ θ 1,p E ≤ β}

#{p : prime, p ≤ x } = 1 π

Z β

α

sin 2 θ dθ.

Lin WENG Zetas for Elliptic Curves

Higher Sato-Tate

Higher Sato-Tate (Weng-Zagier?)

For non CM elliptic curves E, and 0 ≤ α < β ≤ π,

# n

p : prime, p ≤ x, sin

π 2

− α

≥ p

n − 1 · π

2 − θ

+ 1

2 √

p + 1

√ p

≥ sin

π 2

− β o

#{p : prime, p ≤ x}

−→ π 1 R β

α sin 2 θ d θ, as x → ∞

Sato-Tate ⇒ Higher Sato-Tate (Weng-Zagier) As q → ∞,

β

(0) β

(0) = 1+

(n − 1)(q − a + 1) − 2 + 3(a − 2)/q + · · ·

q

+O n

q

Local Zetas for nodal curves

Local zeta for nodal curve/ F q

X/F q : Singular but nodal curve

⇒ Semi-stable bundles make sense ⇒ Zetas for Nodal Curves:

ζ b X,n (s) = α X ,n (0) · P X,n (q −ns ) (1 − q −ns )(1 − q n(1−s) )

P X ,n (t n ): deg < 2g polynomial in T = t n with constant term 1

Lin WENG Zetas for Elliptic Curves

Global Zetas

Global Zeta: a definition X/ Q : regular curve X/ Z : a semi-stable model

X p : semi-stable reduction at p ⇒ New Global Zetas of X/ Z :

ζ b X,n (s) :=

Γ R (ns)Γ R (n(s − 1))

· Y

p

P X −1

p

,n (p −ns )

Analytic and Arithmetic Properties Central Questions

How can we meromorphic extend them?

What kinds of arithmetic they offer?

Trace Formula for Algebraic Stacks

Behrend’s Conjecture

X: smooth algebraic F q -stack defined over F q

Φ q : arithmetic Frobenius

H ∗ (X sm , Q l ): smooth cohomology of X

#X( F q ) := P

ξ∈[X] 1

#Aut(ξ) = # of F q -rational points of X q dim X · Tr Φ q | H ∗ (X sm , Q l ) = #X( F q )

• Analogue of Weil Conjecture ?!

• What are the reasons behind the RH for our pure non-abelian zeta functions ?!

Lin WENG Zetas for Elliptic Curves

β-Invariants for G-Bundles

Definition

(G, P 0 , M 0 ; P, M, N; a P = a G P ⊕ a G , λ = [λ] G P + [λ] G , a ∗ P ; Φ + P ⊃ ∆ P ; ρ P = 1 2 P

α∈Φ

α; α ∈ ∆ P , α ∨ coroot ass. to α;

{$ G α } dual basis to {α ∨ : α ∈ ∆ P })

P : collection of standard parabolic subgroups

Beta invariant of the moduli stack M ss X ,G (λ G ) of semi-stable G-bundles on X with slope λ G

β X,G (λ G ) := # M ss X,G (λ G )

β X,G total (λ G ): beta invariant of the moduli stack of all

G-bundles on X with slope λ G : Independent on λ G

β-Invariants for G-Bundles

Conjectural Formula for β-Invariants of G-Bundles (Weng) β X,G (λ G )

q dim N·(g−1) = X

P⊂G: standard parabolic

(−1) dim a

× X

λ∈Λ

, [λ]

=λ

Y

α∈∆

q 2·hρ

,α

i·<$

(λ)>

q 2·hρ

,α

i − 1 · Y

i

β X total ,M

(0)

where: < x >= 1 + [x ] − x, Λ G P := X ∗ (A 0 P ) . P

α∈∆

Z α ∨ and M ∼ Q

i M i : induced from the Lie level simple decomposition for the Levi factor M.

Motivated also by Atiyah-Bott and Laumon-Rapoport

Key: normalization to make it independent of the environment !!!

Lin WENG Zetas for Elliptic Curves

Thank You

Thank You

Tokyo, 30, 01, 2013