The Geometric Weil Representation and Pseudo-Random Vectors 3_GWR_PR... · 2014. 8. 5. · Shamgar...

The Geometric Weil Representation and Pseudo-RandomVectors

Shamgar Gurevich

Madison

August 5, 2014

Shamgar Gurevich (Madison) GWR and PR Vectors August 5, 2014 1 / 19

(0) MOTIVATION - GPS

CLIENT WANT: Coordinates of satellite and time delay (enables tocalculate distance to a satellite)?

Motivation - GPS

S , R ∈ H = Cp —Hilbert space of digital sequences, p 1000.

S , R : Fp = 0, ...., p − 1 → C.

ψ[n] = e2πip n .

Satellite transmits b · S , b ∈ 1,−1 coordinates.

FactClient receives

R [n] = b · α0 · ψ[ω0n] · S [n− τ0] +W [n], n ∈ Fp ,

α0 ∈ C attenuation, ω0 ∈ Fp Doppler, τ0 ∈ Fp delay, W ∈ H randomwhite noise.

Problem (The GPS Problem)

Design S ∈ H, and effective method to extract (b, τ0), using R and S .

Motivation - GPS

S , R : Fp = 0, ...., p − 1 → C.

ψ[n] = e2πip n .

FactClient receives

Motivation - GPS

S , R : Fp = 0, ...., p − 1 → C.

ψ[n] = e2πip n .

FactClient receives

Motivation - GPS

S , R : Fp = 0, ...., p − 1 → C.

ψ[n] = e2πip n .

FactClient receives

Motivation - GPS

S , R : Fp = 0, ...., p − 1 → C.

ψ[n] = e2πip n .

FactClient receives

Motivation - GPS

S , R : Fp = 0, ...., p − 1 → C.

ψ[n] = e2πip n .

FactClient receives

SOLUTION - Matched Filter

DefinitionMatched filter M(R, S) :

Time-Frequency︷︸︸︷Fp ×Fp → C,

M(R, S)[τ,ω] = 〈R [n] , ψ[ωn] · S [n− τ]〉 .

Question: What S to use for extracting (τ0,ω0) fromM(R, S) ?

SOLUTION - Matched Filter

DefinitionMatched filter M(R, S) :

Time-Frequency︷︸︸︷Fp ×Fp → C,

M(R, S)[τ,ω] = 〈R [n] , ψ[ωn] · S [n− τ]〉 .

Question: What S to use for extracting (τ0,ω0) fromM(R, S) ?

Solution - MATCHED FILTER

Typical solution: S = pseudo-random (PR)

Figure: |M(R, S)|, (τ0,ω0) = (50, 50).

Using FFT computeM(R,S) in O(p2 · log p) operations.

Task: Find method to construct explicit PR sequences.

Figure: |M(R, S)|, (τ0,ω0) = (50, 50).

WEIL REP’N SEQUENCES - Idea

Weil rep’n (Weil 64)ρ : SL2(Fp)→ GL(C(Fp)),

(0 −11 0

)= DFT .

Mechanism for sequences construction

T ⊂ SL2(Fp) torus

ρ : T y H = ⊕χ:T→C∗

dimHχ = 1, ϕχ ∈ Hχ,∥∥∥ϕχ

∥∥∥ = 1.

(0 −11 0

)= DFT .

T ⊂ SL2(Fp) torus

ρ : T y H = ⊕χ:T→C∗

∥∥∥ = 1.

(0 −11 0

)= DFT .

T ⊂ SL2(Fp) torus

ρ : T y H = ⊕χ:T→C∗

∥∥∥ = 1.

(0 −11 0

)= DFT .

T ⊂ SL2(Fp) torus

ρ : T y H = ⊕χ:T→C∗

∥∥∥ = 1.Shamgar Gurevich (Madison) GWR and PR Vectors August 5, 2014 6 / 19

Weil Rep’n Sequences - IDEA

Theorem (Pseudo Randomness)

For (τ,ω) 6= (0, 0) ∣∣∣M(ϕχ , ϕχ)[τ,ω]∣∣∣ ≤ 2√

Figure: M(ϕχ, ϕχ), q = 199.

BOUNDS VIA GEOMETRIC WEIL REP’N - Idea

`-adic sheaves (Grothendieck 60s)

Dbc (A1) C(Fq).

WantGeometric Weil Rep’n ρ.

Application: GeometrizingM(ϕχ, ϕχ) and obtain the bound.

Dbc (A1) C(Fq).

(I) WEIL REPRESENTATION

Heisenberg Representation

(V ,Ω) —2n-dimensional symplectic vector space over k = Fq ,char 6= 2.H = H(V ) —Heisenberg group

H = V × k ;(v , z) · (v ′, z ′) = (v + v ′, z + z ′ + 1

2Ω(v , v ′)).

1 6= ψ : k → C∗ —additive character.

Theorem (Stone—von Neumann)

There exists a unique (up to ') irreducible representationπ : H → GL(H)︸︷︷︸Heisenberg rep’n

s.t. π(z) = ψ(z) · IdH, z ∈ Z (H) = k.

(V ,Ω) —2n-dimensional symplectic vector space over k = Fq ,char 6= 2.

H = H(V ) —Heisenberg group

H = V × k ;(v , z) · (v ′, z ′) = (v + v ′, z + z ′ + 1

2Ω(v , v ′)).

s.t. π(z) = ψ(z) · IdH, z ∈ Z (H) = k.

H = V × k ;(v , z) · (v ′, z ′) = (v + v ′, z + z ′ + 1

2Ω(v , v ′)).

s.t. π(z) = ψ(z) · IdH, z ∈ Z (H) = k.

H = V × k ;

(v , z) · (v ′, z ′) = (v + v ′, z + z ′ + 12Ω(v , v ′)).

s.t. π(z) = ψ(z) · IdH, z ∈ Z (H) = k.

H = V × k ;(v , z) · (v ′, z ′) = (v + v ′, z + z ′ + 1

2Ω(v , v ′)).

s.t. π(z) = ψ(z) · IdH, z ∈ Z (H) = k.

H = V × k ;(v , z) · (v ′, z ′) = (v + v ′, z + z ′ + 1

2Ω(v , v ′)).

s.t. π(z) = ψ(z) · IdH, z ∈ Z (H) = k.

H = V × k ;(v , z) · (v ′, z ′) = (v + v ′, z + z ′ + 1

2Ω(v , v ′)).

s.t. π(z) = ψ(z) · IdH, z ∈ Z (H) = k.

WEIL REPRESENTATION

Example

n = 1, V = k × k, H = C(k),

[π(τ, 0, 0)f ](t) = f (t + τ);

[π(0,ω, 0)f ](t) = ψ(ωt)f (t);

[π(0, 0, z)f ](t) = ψ(z)f (t).

Weil Representation (Weil ’64): Take g ∈ Sp(V ) = Sp, then

πρ(g )→ πg (v , z) = π(gv , z),

i.e.,ρ(g)π(h)ρ(g)−1 = π(g [h]), for every h ∈ H. (1)

WEIL REPRESENTATION

Example

n = 1, V = k × k, H = C(k),

[π(τ, 0, 0)f ](t) = f (t + τ);

[π(0,ω, 0)f ](t) = ψ(ωt)f (t);

[π(0, 0, z)f ](t) = ψ(z)f (t).

πρ(g )→ πg (v , z) = π(gv , z),

WEIL REPRESENTATION

Example

n = 1, V = k × k, H = C(k),

[π(τ, 0, 0)f ](t) = f (t + τ);

[π(0,ω, 0)f ](t) = ψ(ωt)f (t);

[π(0, 0, z)f ](t) = ψ(z)f (t).

πρ(g )→ πg (v , z) = π(gv , z),

WEIL REPRESENTATION

Example

n = 1, V = k × k, H = C(k),

[π(τ, 0, 0)f ](t) = f (t + τ);

[π(0,ω, 0)f ](t) = ψ(ωt)f (t);

[π(0, 0, z)f ](t) = ψ(z)f (t).

πρ(g )→ πg (v , z) = π(gv , z),

WEIL REPRESENTATION

Example

n = 1, V = k × k, H = C(k),

[π(τ, 0, 0)f ](t) = f (t + τ);

[π(0,ω, 0)f ](t) = ψ(ωt)f (t);

[π(0, 0, z)f ](t) = ψ(z)f (t).

πρ(g )→ πg (v , z) = π(gv , z),

WEIL REPRESENTATION

Example

n = 1, V = k × k, H = C(k),

[π(τ, 0, 0)f ](t) = f (t + τ);

[π(0,ω, 0)f ](t) = ψ(ωt)f (t);

[π(0, 0, z)f ](t) = ψ(z)f (t).

πρ(g )→ πg (v , z) = π(gv , z),

WEIL REPRESENTATION

TheoremThere exists a unique representation

ρ : Sp → GL(H) – Weil representation,

that satisfies (1).

ProblemFormula for ρ ?

WEIL REPRESENTATION

TheoremThere exists a unique representation

ρ : Sp → GL(H) – Weil representation,

that satisfies (1).

ProblemFormula for ρ ?

Weil Rep’n - FORMULA

Formula for A ∈ End(H)?

Weyl Transform

∗C(H,ψ−1)

End(H);

W (A)[h] = 1dimHTr(Aπ[h−1 ]).

Kernel of Weil Rep’nK : Sp × V → C,

K (g , v) = W (ρ(g))[v ].

1 π[K (g , ·)] := ∑v∈V

K (g , v)π(v) = ρ(g).

2 K (g1, ·) ∗K (g2, ·) = K (g1g2, ·).3 Formula: On U = g ∈ Sp; det(g − I ) 6= 0

K (g , v) =1qn· σ[(−1)n det(g − I )] · ψ[Ω(g + I

g − I v , v)].

Proof. Geometric Weil Rep’n.

Formula for A ∈ End(H)?Weyl Transform

∗C(H,ψ−1)

End(H);

K (g , v) = W (ρ(g))[v ].

1 π[K (g , ·)] := ∑v∈V

K (g , v)π(v) = ρ(g).

g − I v , v)].

∗C(H,ψ−1)

End(H);

K (g , v) = W (ρ(g))[v ].

1 π[K (g , ·)] := ∑v∈V

K (g , v)π(v) = ρ(g).

g − I v , v)].

∗C(H,ψ−1)

End(H);

K (g , v) = W (ρ(g))[v ].

1 π[K (g , ·)] := ∑v∈V

K (g , v)π(v) = ρ(g).

g − I v , v)].

∗C(H,ψ−1)

End(H);

K (g , v) = W (ρ(g))[v ].

1 π[K (g , ·)] := ∑v∈V

K (g , v)π(v) = ρ(g).

g − I v , v)].

∗C(H,ψ−1)

End(H);

K (g , v) = W (ρ(g))[v ].

1 π[K (g , ·)] := ∑v∈V

K (g , v)π(v) = ρ(g).

g − I v , v)].

∗C(H,ψ−1)

End(H);

K (g , v) = W (ρ(g))[v ].

1 π[K (g , ·)] := ∑v∈V

K (g , v)π(v) = ρ(g).

2 K (g1, ·) ∗K (g2, ·) = K (g1g2, ·).

3 Formula: On U = g ∈ Sp; det(g − I ) 6= 0

g − I v , v)].

∗C(H,ψ−1)

End(H);

K (g , v) = W (ρ(g))[v ].

1 π[K (g , ·)] := ∑v∈V

K (g , v)π(v) = ρ(g).

g − I v , v)].

∗C(H,ψ−1)

End(H);

K (g , v) = W (ρ(g))[v ].

1 π[K (g , ·)] := ∑v∈V

K (g , v)π(v) = ρ(g).

g − I v , v)].

Proof. Geometric Weil Rep’n.Shamgar Gurevich (Madison) GWR and PR Vectors August 5, 2014 12 / 19

(II) Geometric Weil Rep’n - GEOMETRIZATION

Geometrization (Grothendieck)

Nice set

X —Algebraic variety over k = Fq .

Fr y X.X = XFr = X(k).

Nice function

Fr y`-adic Weil sheaf

−−−−−− >sheaf-to-function

f F : X → C.

Fr : Fx →FFr (x ).