on the arithmetic of picard-fuchs equationhaolee/nacho_notes.pdf · 2020. 4. 22. · families of...
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On the Arithmetic of Picard-Fuchs equation
Ignacio Darago
No theory seminar
April 23 2020
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Goal
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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!
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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.
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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.
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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).
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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
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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.
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Example: Legendre family of elliptic curves
Solving the differential equation.Frobenius method: power series expansion
σ(z) =∞
∑n=0
anzn, σ(0) = a0 = 1.
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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.
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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!).
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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).
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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.
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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.
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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 .
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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.
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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.
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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.
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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 .
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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!
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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.
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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.
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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”).