On the Arithmetic of Picard-Fuchs equation

Ignacio Darago

No theory seminar

April 23 2020

Goal

Families of algebraic varieties

A family of algebraic varieties is a flat map π : X → S andXs := π−1(s) are the members of the family.If π is a smooth projective map then Xs are smooth projectivevarieties.

Ehresmann’s theorem

In the analytic topology, π is a locally trivial fibration.R iπ∗C is a local system with

(R iπ∗C)s = Hi (Xs , C).

The topological information doesn’t change in families but thecomplex structure does!

In the analytic topology we have an equivalence

{Local systems} Riemann-Hilbert←→ {Vector bundles with flat connection}

We get R iπ∗Ω•X an|San = Riπ∗(C)⊗C OSan with a flat connection

∇(α⊗ f ) = α⊗ df

called the Gauss-Manin connection.

Xs family of smooth projective varieties parametrized by a curve S .The OS -module R iπ∗(Ω•X |S ) is locally free of some finite rank r .If ω is a section,

ω,∇d/dzω, . . . , (∇d/dz )r ω

must be linearly dependent!If γs is a family of i-cycles the periods

℘(s) =∫

γsω

must then satisfy the Picard-Fuchs differential equation

d r℘

dz r(z) + Pr−1(z)

d r−1℘

dz r−1(z) + · · ·+ P0(z)℘ = 0.

Periods for elliptic curves

We can see elliptic curves

E : y2 = (x − a)(x − b)(x − c)

as a double-sheeted cover of P1.

The periods ℘1 and ℘2 exhibit E ' C/(π1Z + π2Z).

Example: Legendre family of elliptic curves

Consider the Legendre family

Eλ : y2 = x(x − 1)(x − λ), λ ∈ P1 r {0, 1, ∞}.

The Gauss-Manin connection can be identified with differentiationof 1-forms with respect to λ:

ω(λ) =dx√

x(x − 1)(x − λ),

∇d/dλω(λ) =1

2

dx√x(x − 1)(x − λ)3

,

(∇d/dλ)2 ω(λ) =3

4

dx√x(x − 1)(x − λ)5

Example: Legendre family of elliptic curves

We get the explicit relation

λ(λ− 1)∂2ω

∂λ2+(2λ− 1)∂ω

∂λ+

1

4ω = −1

2d

(√x(x − 1)(x − λ)

(x − λ)2

).

This means that the Picard-Fuchs equation is the well-knownGauss hypergeometric equation

z(z − 1)y ′′(z) + (2z − 1)y ′(z) + 14y(z) = 0.

Example: Legendre family of elliptic curves

Solving the differential equation.Frobenius method: power series expansion

σ(z) =∞

∑n=0

anzn, σ(0) = a0 = 1.

Example: Legendre family of elliptic curves

Applying the differential operator yields

∞

∑n=0

(z(z − 1)(n+ 2)(n+ 1)an+2 + (2z − 1)(n+ 1)an+1 +

1

4an

)zn.

Recurrence relation(n+

1

2

)2an − (n+ 1)2an+1 = 0

Explicit formula

σ(z) =∞

∑n=0

(− 12n

)2zn.

Point counts over finite fields

X smooth projective variety over Fp. We have Frobenius mapF : X → X and its fixed-points give us the Fp-valued pointsX (Fp).

Woods-Hole trace formula (Deligne, Fulton, Katz)

#X (Fp) ≡dim(X )

∑i=0

tr(F ∗ : H i (X ,OX )→ H i (X ,OX )

)mod p

Remark: this is an equality mod p, but the cohomology theory weare taking is usual sheaf cohomology (no need for fancy étale orcrystalline cohomology!).

Hypersurfaces in projective spaces

If f (x0, . . . , xn) ∈ Fp [x0, . . . , xn] is a homogeneous polynomial ofdegree d we can consider the hypersurface Z defined by f = 0inside of Pn.

0 −→ OPn(−d)·f−→ OPn −→ OZ −→ 0.

If d < n+ 1 then

H i (Z ,OZ ) = 0 for all i > 0

and so #Z (Fp) ≡ 1 (mod p).

Hypersurfaces in projective spaces

These ideas yield a proof of the following classical theorem

Chevalley-Warning theorem

Let f1, . . . , fr ∈ Fp [x0, . . . , xn] be homogeneous polynomials ofdeg(fi ) = di . If we have d = d1 + . . . + dr < n+ 1 then

#{(x0, . . . , xn) ∈ Fn+1p : f1(x0, . . . , xn) = · · · = fr (x0, . . . , xn) = 0}

is divisible by p. In particular, it has non-trivial solutions.

Hypersurfaces in projective spaces

What happens if we have a hypersurface of degree n+ 1 in Pn?The same short exact sequence provides us with an isomorphism

δ : Hn−1(Z ,OZ )→ Hn(Pn,OPn(−n− 1)).

We can compute these cohomology groups by the usual Cech coverof Pn.

Hypersurfaces in projective spaces

Reminder: If S = Fp [x0, . . . , xn] and we take the affine opencovering {xi 6= 0} then

Hn(Pn,OPn(−n− 1)) = (Sx0···xn)−n−1/n

∑i=0

(Sx0···x̂i ···xn)−n−1

= Fp ·1

x0 · · · xn

Similarly, u ∈ Hn−1(Z ,OZ ) is represented by a Cech cocycle

(h0, . . . , hn) ∈n⊕

i=0

((S/f )x0···x̂i ···xn)0 .

Hypersurfaces in projective spaces

Taking lifts hi ∈ (Sx0···x̂i ···xn)0 of hi , the element δ(u) isrepresented by the unique w ∈ (Sx0···xn)−n−1 such that

fw =n

∑i=0

(−1)ihi .

Since the action of Frobenius F ∗(u) is represented by the Cechcocycle (h0

p, . . . , hn

p) and we have that

f · f p−1wp =n

∑i=0

(−1)ihpi

it follows that the action of Frobenius on Hn−1(Z ,OZ ) is given bythe coefficient of (x0 · · · xn)p−1 in f p−1.

Hypersurfaces in projective spaces

In conclusion, for f (x0, . . . , xn) homogenous polynomial of degreen+ 1 defining a hypersurface Z ⊆ Pn, we get that

#Z (Fp) ≡ 1 + (−1)n−1coef of (x0 · · · xn)p−1 in f p−1 (mod p).

This is a very classical result.

Two elementary facts

ap−1 ≡{

0 if p | a1 if p - a

(mod p).

∑x∈Fp

xk ≡{−1 if p − 1 | k0 if p − 1 - k

(mod p).

So to count solutions to f (x0, . . . , xn) ≡ 0 (mod p), by the firstfact, we might as well look at the sum

∑x∈Fn+1p

f (x)p−1,

and the second fact tells us that all the relevant information is inthe (p − 1)-degree coefficient.

Point count of elliptic curves

For a fixed a ∈ Fp, the number of solutions of y2 ≡ a (mod p) is

given by

(a

p

)+ 1 (Legendre symbol).

If we want to count the number of points of y2 = x(x − 1)(x − λ)we can then look at

∑x∈Fp

(x(x − 1)(x − λ))p−12 + 1 (mod p).

We want to find the coefficient of xp−12 in the expansion of

((x − 1)(x − λ))p−12 .

Point count of elliptic curves

Applying the binomial theorem twice we can expand it as

(−1)p−12 ∑

k+`= p−12

( p−12

k

)( p−12

`

)λ` = (−1)

p−12

p−12

∑r=0

( p−12

r

)2λr .

But now simply notice that( p−12

r

)≡(− 12r

)(mod p)

from which we get that the number modulo p of points is

#Eλ(Fp) ≡ 1 + (−1)(−1)p−12

p−12

∑r=0

(− 12r

)2λr (mod p).

This is precisely the same expression we obtained for theholomorphic solution of the Picard-Fuchs equation!

This is not a coincidence! By Serre duality we can identify

H1(Eλ,OEλ) ' H0(Eλ, Ω1Eλ)

via the residue pairing. If we expand a 1-form

ω = dz +∞

∑r=1

crzr

for z a local coordinate, then the action of Frobenius is given bythe coefficient cp−1.

The family of 1-forms must also satisfy the linear relation given bythe Gauss-Manin connection. That is,(

λ(λ− 1) ∂2

∂λ2+ (2λ− 1) ∂

∂λ+

1

4

)(1 +

∞

∑r=1

cr (λ)zr

)=

ddz

(series expansion of the exact form).

Applying the differential operator to the zp−1 coefficient cp−1(λ)of the power series expansion of ω is the same as the coefficient ofthe derivative of the power series expansion of the exact form, andtherefore it must vanish because ddz z

p = pzp−1 = 0.

Hence

σ(λ) = tr(F ∗ : H1(Eλ,OEλ)→ H1(Eλ,OEλ))

is a formal solution to the Picard-Fuchs equation!This argument works for any family of degree n+ 1 hypersurfacesin Pn! For instance like the Dwork family

Xλ : x50 + x51 + x

52 + x

53 + x

54 − λx0x1x2x3x4x5 = 0.

This family is related to mirror symmetry (Candelas, de la Ossa,Rodriguez Villegas – “Calabi-Yau manifolds over finite fields”).