exeter, january 22nd 2016 - julio c....

Algebraic geometry over finite fields via counting

Tim Browning

University of Bristol

Exeter, January 22nd 2016

What are the simplest varieties?

What are the simplest algebraic varieties X ⊂ PN defined over afield k?

In dimension 1: rational curves (curves of genus 0)

In higher dimension: lots of rational curves P1 → X .

Definition

A rational curve on X is a non-constant morphism P1 → X .

A morphism f : P1 → X of degree e is given by

f = (f0(u, v), . . . , fN(u, v)),

with f0, . . . , fN ∈ k[u, v ] forms of degree e, with no non-constantcommon factor in k[u, v ], such that

Fi (f0(u, v), . . . , fN(u, v)) ≡ 0, (1 6 i 6 R),

where F1, . . . ,FR ∈ k[x0, . . . , xN ] are the polynomials defining X .

Definition

f = (f0(u, v), . . . , fN(u, v)),

Fi (f0(u, v), . . . , fN(u, v)) ≡ 0, (1 6 i 6 R),

Definition

f = (f0(u, v), . . . , fN(u, v)),

Fi (f0(u, v), . . . , fN(u, v)) ≡ 0, (1 6 i 6 R),

Rationally Connected Unirational Rational

X is RC: ∀x1, x2 ∈ X there exists a P1 → X through themX is unirational: ∃ onto rational map Pdim X → XX is rational: if there is also a rational map X → Pdim X

Example

Take X = {x2 + y2 = z2} ⊂ P2. Here P1 → X is given by[s : t] 7→ [s2 − t2 : 2st : s2 + t2] and X → P1 is given by[x : y : z ] 7→ [x/z : y/z ]

Rationally Connected Unirational Rational

X is RC: ∀x1, x2 ∈ X there exists a P1 → X through themX is unirational: ∃ onto rational map Pdim X → XX is rational: if there is also a rational map X → Pdim X

Example

Take X = {x2 + y2 = z2} ⊂ P2. Here P1 → X is given by[s : t] 7→ [s2 − t2 : 2st : s2 + t2] and X → P1 is given by[x : y : z ] 7→ [x/z : y/z ]

{Rationally Connected}??) {Unirational}

Clemens−Griffiths) {Rational}

Suppose X ⊂ Pn is a non-singular hypersurface of degree d .

X is RC if n + 1 > d . When is it unirational?

Example (Harris–Mazur-Pandaripande 1998)

Assume k = k and char(k) = 0. X is unirational if n > d ↑↑ d

What about X in a “reasonable” number of variables?! Can weeven show X contains a rational surface?

Study rational curves on X ...

Rational curves

k is a field and X = {F = 0} ⊂ Pn, with

F ∈ k[x0, . . . , xn]

a non-singular form of degree d .

Let More(X ) be space of rational curves1 on X :

{f ∈ More(X )} ←→{

open set in P(n+1)(e+1)−1

de + 1 equations of degree d

1morphism f : P1 → X of degree e is given by f = (f0(u, v), . . . , fn(u, v)),with f0, . . . , fn ∈ k[u, v ] forms of degree e, with no non-constant commonfactor, such that F (f0(u, v), . . . , fn(u, v)) ≡ 0.

{f ∈ More(X )} ←→{

Questions

dim More(X )??= (n+1)(e+1)−1−(de+1) = e(n+1−d)+n−1.

Is More(X ) irreducible?

Is More(X )(k) non-empty?

Coskun and Starr (2009): Yes! (if d = 3, n > 5 and k = C).

Use analytic number theory to answer these questions when k = Fq

{f ∈ More(X )} ←→{

Questions

dim More(X )??= (n+1)(e+1)−1−(de+1) = e(n+1−d)+n−1.

Is More(X ) irreducible?

Is More(X )(k) non-empty?

Coskun and Starr (2009): Yes! (if d = 3, n > 5 and k = C).

Use analytic number theory to answer these questions when k = Fq

The players

We are interested in the following data:

k = Fq a finite field

F ∈ k[x0, . . . , xn] a non-singular form of2 degree 3

X ⊂ Pn the hypersurface defined by F = 0

More(X ) the space of rational curves of degree e that arecontained in X

Example

For the case n = 3 of cubic surfaces, Mor1(X ) has 27 componentswhich are permuted by the action of Gal(k/k).

Describing More(X ) for dim(X ) > 2 is difficult!

2For degree > 3 (or anything to do with x) see Alan Lee (2011), a PhDstudent of Trevor Wooley

The players

We are interested in the following data:

k = Fq a finite field

F ∈ k[x0, . . . , xn] a non-singular form of2 degree 3

X ⊂ Pn the hypersurface defined by F = 0

More(X ) the space of rational curves of degree e that arecontained in X

Example

For the case n = 3 of cubic surfaces, Mor1(X ) has 27 componentswhich are permuted by the action of Gal(k/k).

Describing More(X ) for dim(X ) > 2 is difficult!

2For degree > 3 (or anything to do with x) see Alan Lee (2011), a PhDstudent of Trevor Wooley

k-points

Question: When is X (k) 6= ∅?

Theorem (Chevalley–Warning, 1936)

f1, . . . , fr ⊂ k[x0, . . . , xn] homogenous polynomials such that

n >r∑

deg fj .

Then ∃ x ∈ kn+1 \ {0} such that fj (x) = 0 for 1 6 j 6 r .

Answer: If n > 3

k-points

Question: When is X (k) 6= ∅?

Theorem (Chevalley–Warning, 1936)

f1, . . . , fr ⊂ k[x0, . . . , xn] homogenous polynomials such that

n >r∑

deg fj .

Then ∃ x ∈ kn+1 \ {0} such that fj (x) = 0 for 1 6 j 6 r .

Answer: If n > 3

Question: When is #X (k) > 2?

Example (Swinnerton-Dyer)

The cubic surface

x30 + x3

1 + x32 + x2

0x1 + x21x2 + x2

2x0 + x0x1x2 + x2x23 + x2

2x3 = 0

has exactly one point over F2.

Answer: This is the only exception!

Question: When is #X (k) > 2?

Example (Swinnerton-Dyer)

The cubic surface

x30 + x3

1 + x32 + x2

0x1 + x21x2 + x2

2x0 + x0x1x2 + x2x23 + x2

2x3 = 0

has exactly one point over F2.

Answer: This is the only exception!

k-rational curves

Question: When is More(X )(k) 6= ∅?

We know n > 3⇒ ∃ p ∈ X (k)

∃ Fq-lines in X through p ⇐⇒ ∃ q ∈ X (k) such that

0 ≡ F (λp + µq) = λ2µ q.∇F (p)︸︷︷︸linear

+λµ2 p.∇F (q)︸︷︷︸quadratic

+µ3 F (q)︸︷︷︸cubic

Answer: Chevalley–Warning ⇒ Mor1(X )(k) 6= ∅ if

n > 1 + 2 + 3 = 6

k-rational curves

We know n > 3⇒ ∃ p ∈ X (k)

n > 1 + 2 + 3 = 6

k-rational curves

We know n > 3⇒ ∃ p ∈ X (k)

n > 1 + 2 + 3 = 6

Main result

Theorem (B. & Pankaj Vishe, 2015)

Assume char(k) > 3 and let e > 1. Let X ⊂ Pn be a non-singularcubic hypersurface over k , with n > 12. Then More(X ) isirreducible and has the expected dimension e(n − 2) + n − 1.

Remarks:

This + Lang–Weil ⇒ More(X )(k) 6= ∅ if #k large enough

Pugin (2011): Special case of diagonal cubic hypersurfacesa0x

30 + · · ·+ anx

3n = 0, for a0, . . . , an ∈ k∗.

What about space More(x ;X ) of rational curves of degree eon X passing through x ∈ X?

Smaller n? Larger degree hypersurfaces?

Main result

Assume char(k) > 3 and let e > 1. Let X ⊂ Pn be a non-singularcubic hypersurface over k , with n > 12. Then More(X ) isirreducible and has the expected dimension e(n − 2) + n − 1.

Remarks:

This + Lang–Weil ⇒ More(X )(k) 6= ∅ if #k large enough

Pugin (2011): Special case of diagonal cubic hypersurfacesa0x

30 + · · ·+ anx

3n = 0, for a0, . . . , an ∈ k∗.

What about space More(x ;X ) of rational curves of degree eon X passing through x ∈ X?

Smaller n? Larger degree hypersurfaces?

From geometry to counting

Assume that #k = q and put E = e(n − 2) + n − 1.

Theorem (Lang–Weil, 1953)

Let V ⊂ PN be a variety defined over k . Then

lim supq→∞

q− dim V #V (k) = # irred components of V

It therefore suffices to prove:

lim sup`→∞

q−`E # More(X )(Fq`) 6 1.

Key idea{Fq`-points on More(X )

}←→

{Fq`(t)-points on X of degree e

From geometry to counting

Assume that #k = q and put E = e(n − 2) + n − 1.

Theorem (Lang–Weil, 1953)

Let V ⊂ PN be a variety defined over k . Then

lim supq→∞

q− dim V #V (k) = # irred components of V

It therefore suffices to prove:

lim sup`→∞

q−`E # More(X )(Fq`) 6 1.

Key idea{Fq`-points on More(X )

}←→

{Fq`(t)-points on X of degree e

Function field dictionary

Idea: Count points in X (Fq(t)) of degree e using Fq(t)-version ofthe Hardy–Littlewood circle method

Q K = Fq(t)

Z O = Fq[t]p prime π ∈ O monic and irreducible

residue field Z/pZ = Fp Fπ = Fqdeg π

∞ t prime at infinityabs. values | · |, | · |p | · | = | · |∞, | · |πcompletions R,Qp K∞ = Fq((t−1)),KπFourier analysis [0, 1] T = {x ∈ K∞ : |x | < 1}add. character exp(2πi ·) ψ(x) = exp(

2πiTrFq/Fp a−1

p ) if

x =∑

i6N ai ti ∈ K∞

Here: |a/b| = qdeg a−deg b if a, b ∈ O and b 6= 0.

Recall: X = {F = 0} ⊂ Pn defined by a non-singular cubic formF ∈ Fq[x0, . . . , xn].

Estimate

N(q, e) = #

{x ∈ On+1 : F (x) = 0, max

06i6n|xi | 6 qe

}as q →∞

Can use this to make deductions about # More(X )(Fq)...

Enter the adelic circle method...

N(q, e) = #

{x ∈ On+1 : F (x) = 0, max

06i6n|xi | 6 qe

∫TS(α)dα, where S(α) =

∑x∈On+1: maxi |xi |6qe

ψ(αF (x))

by orthogonality of characters

=∑|r |<qQ

r monic

∑|a|<|r |

gcd(a,r)=1

∫|θ|<|r |−1q−Q

S(a/r + θ)dθ

by Dirichlet + ultrametric inequality (any Q > 1 )

= qe(n+1)∑|r |<qQ

r monic

|r |−(n+1)

∫|θ|<|r |−1q−Q

∑c∈On+1

Sr (c)Ir (c)dθ

by adelic Poisson summation

N(q, e) = qe(n+1)∑|r |<qQ

r monic

|r |−(n+1)

∫|θ|<|r |−1q−Q

∑c∈On+1

Sr (c)Ir (c)dθ

Sr (c) =∑|a|<|r |

gcd(a,r)=1

∑y∈On+1

|yi |<|r |

(aF (y)− c.y

)complete exp. sum

Ir (c) =

∫Tn+1

(θdqdeF (x) +

)dx oscill. integral

Main contribution: c = 0Error term: c 6= 0

N(q, e) = qe(n+1)∑|r |<qQ

r monic

|r |−(n+1)

∫|θ|<|r |−1q−Q

∑c∈On+1

Sr (c)Ir (c)dθ

Sr (c) =∑|a|<|r |

gcd(a,r)=1

∑y∈On+1

|yi |<|r |

(aF (y)− c.y

)complete exp. sum

Ir (c) =

∫Tn+1

(θdqdeF (x) +

)dx oscill. integral

Main contribution: c = 0Error term: c 6= 0

Back to varieties over finite fields

Need to analyse

Sr (c) =∑|a|<|r |

gcd(a,r)=1

∑y∈On+1

|yi |<|r |

(aF (y)− c.y

r = πν11 . . . πνs

s ⇒ Sr (c) = Sπν11

(c) . . . Sπνss

ν > 1: Elementary treatment...ν = 1: Then

Sπ(c) = |π| {|π|#Xc(Fπ)−#X (Fπ) + 1}

where Xc = X ∩ {∑n

i=0 cixi = 0} has dim(Xc) = n − 2

Back to varieties over finite fields

Need to analyse

Sr (c) =∑|a|<|r |

gcd(a,r)=1

∑y∈On+1

|yi |<|r |

(aF (y)− c.y

r = πν11 . . . πνs

s ⇒ Sr (c) = Sπν11

(c) . . . Sπνss

ν > 1: Elementary treatment...ν = 1: Then

Sπ(c) = |π| {|π|#Xc(Fπ)−#X (Fπ) + 1}

where Xc = X ∩ {∑n

i=0 cixi = 0} has dim(Xc) = n − 2

Sπ(c) = |π| {|π|#Xc(Fπ)−#X (Fπ) + 1}

For generic3 values of c, Deligne (Weil I) ⇒

Sπ(c) = (−1)n−2|π|2bn−2∑j=1

ωn−2,j + O(|π|n+1

where ωn−2,j are eigenvalues of Frobenius acting on Hn−2et (X c,Q`).

These satisfy |ωn−2,j | = |π|n−2

Hence r square-free ⇒ Sr (c) = O(|r |n+2

...this is square-root cancellation!

3i.e. [c] 6∈ X dual

Sπ(c) = |π| {|π|#Xc(Fπ)−#X (Fπ) + 1}

For generic3 values of c, Deligne (Weil I) ⇒

Sπ(c) = (−1)n−2|π|2bn−2∑j=1

ωn−2,j + O(|π|n+1

where ωn−2,j are eigenvalues of Frobenius acting on Hn−2et (X c,Q`).

These satisfy |ωn−2,j | = |π|n−2

Hence r square-free ⇒ Sr (c) = O(|r |n+2

...this is square-root cancellation!

3i.e. [c] 6∈ X dual

e-aspect?

Since we are interested in q →∞ limit, need to make all theimplied constants uniform in q in above strategy.

The circle method can also be applied to deal with e-aspect. Infact, for fixed q, we have

n > 7⇒ lime→∞

q−e(n−2)N(q, d) = c∞∏π

⇒ lime→∞

q−e(n−2)N(q, d) > 0 if X (Kπ) 6= ∅ ∀ primes π

Let K = Fq(t) and assume that char(Fq) > 3. Let X ⊂ Pn be anon-singular cubic hypersurface over K , with n > 7. Then Xsatisfies the Manin conjecture and the Hasse principle over K .

Newsflash! (Zhiyu Tian, 2015)

Hasse Principle for n > 5 and char(Fq) > 5.

e-aspect?

n > 7⇒ lime→∞

q−e(n−2)N(q, d) = c∞∏π

⇒ lime→∞

e-aspect?

n > 7⇒ lime→∞

q−e(n−2)N(q, d) = c∞∏π

⇒ lime→∞

e-aspect?

n > 7⇒ lime→∞

q−e(n−2)N(q, d) = c∞∏π

⇒ lime→∞

What next?

Higher degree.Extend investigation of More(X ) to hypersurfaces of degree > 3over finite fields (a la Alan Lee)

A Manin conjecture for Campana orbifolds.The Manin conjecture makes predictions about equidistribution ofK -rational points on projective varieties over global fields K .

There is a version emerging for integral points on Campanaorbifolds (P1,∆) with χ(∆) > 0.

Example

e-aspect: is it true that

{(x , y , z) ∈ Fq[t] :

x , y , z square-full & degree ex + y = z

}??= O(qe/2).

q-aspect: is this any easier?

What next?

Higher degree.Extend investigation of More(X ) to hypersurfaces of degree > 3over finite fields (a la Alan Lee)

A Manin conjecture for Campana orbifolds.The Manin conjecture makes predictions about equidistribution ofK -rational points on projective varieties over global fields K .

There is a version emerging for integral points on Campanaorbifolds (P1,∆) with χ(∆) > 0.

Example

e-aspect: is it true that

{(x , y , z) ∈ Fq[t] :

x , y , z square-full & degree ex + y = z

}??= O(qe/2).

q-aspect: is this any easier?

exeter, january 22nd 2016 - julio c....

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