Introduction to Arithmetic Algebraic Geometry · Algebraic Geometry The set X(C) or X(K) is called...
Transcript of Introduction to Arithmetic Algebraic Geometry · Algebraic Geometry The set X(C) or X(K) is called...
Introduction toArithmetic Algebraic Geometry
Sungkon Chang
The Anne and Sigmund HudsonMathematics and ComputingLuncheon Colloquium Series
Diophantine Equations
Let Z denote the set of integers.
Diophantine Equations:
2x2 +3y2 = 4z3 +5w3
where x,y,z,w ∈ Z.
Diophantine Equations
Let Z denote the set of integers.
Diophantine Equations:
x21 + y2
1 = x32 + y3
2 = x43 + y4
3
where xk’s and yk’s ∈ Z.
Rational Solutions
Let Q denote the set of rational numbers.
2x2 +3y2 = 4z3 +5w3
where x,y,z,w ∈Q.
Q is a field.
a±b, a ·b, a/b ∈Q where a,b ∈Q.
K-Rational Solutions
Let K be a field.
X : 2x2 +3y2 = 4z3 +5w3
X(K) = (x,y,z,w) ∈ K4 : 2x2 +3y2 = 4z3 +5w3.
K-Rational Solutions
Let K be a field.
X : 2x2 +3y2 = 4z3 +5w3
X(K) = (x,y,z,w) ∈ K4 : 2x2 +3y2 = 4z3 +5w3.
Question: Determine whether X(K) is empty.
K-Rational Solutions
Let K be a field.
X : 2x2 +3y2 = 4z3 +5w3
X(K) = (x,y,z,w) ∈ K4 : 2x2 +3y2 = 4z3 +5w3.
Question: Determine whether X(K) is empty.
Is X(Q) empty?
K-Rational Solutions
X : f (x,y) = 0where f (x,y) is a polynomial
with Q-coeff.
X(Q) = (x,y) ∈Q2 : f (x,y) = 0.
Question: Determine whether X(Q) is empty.
K-Rational Solutions
X : f (x,y) = 0where f (x,y) is a polynomial
with K-coeff.
X(K) = (x,y) ∈ K2 : f (x,y) = 0.
Question: Determine whether X(K) is empty.
Geometry
X : x2 + y2 = 1
t 7→ x =1− t2
1+ t2 , y =2t
1+ t2 .
Q∼= X(Q)\(−1,0)
X is called a rational curve.
Geometry
X : 64yx4 +56x3y+16x4 +16x3−4y2 +72y2x3 +16y2x2
+8y2x+40y3x3 +12y3x+32y3x2 +64y3x4
+96y2x4 +4y4x+16y4x4 +8y4x3 +16y4x2−4y3 +3y4 = 0.
Geometry
X : 64yx4 +56x3y+16x4 +16x3−4y2 +72y2x3 +16y2x2
+8y2x+40y3x3 +12y3x+32y3x2 +64y3x4
+96y2x4 +4y4x+16y4x4 +8y4x3 +16y4x2−4y3 +3y4 = 0.
x =t2−1
t4 + t +1, y =
2tt4− t +1
.
Geometry
X : x3 + y3 = 1
Is this a rational curve?x = f (t) and y = g(t)
for rational functions f (t) and g(t)?
Geometry
X : x3 + y3 = 1
Is this a rational curve?x = f (t) and y = g(t)
for rational functions f (t) and g(t)?
(a/c)3 +(b/c)3 = 1⇒ a3 +b3 = c3.
By Fermat’s Last Theorem,there are no parametric solutions.
Geometry
X : x3 + y3 = 1
Is this a rational curve?x = f (t) and y = g(t)
for rational functions f (t) and g(t)?
Geometry
X : x3 + y3 = 1
Is this a rational curve?x = f (t) and y = g(t)
for rational functions f (t) and g(t)?
Geometry of X(C)
Geometry
X : x3 + y3 = 1
Is this a rational curve?x = f (t) and y = g(t)
for rational functions f (t) and g(t)?
Geometry
Topology: continuous maps
Differential Geometry: differentiable maps
Algebraic Geometry: rational maps
Geometry
Topology: continuous maps
Differential Geometry: differentiable maps
Algebraic Geometry: rational maps
X(K)−→ Y (K)
Algebraic Geometry
X : x3 + y3 = 1Y : x6 +6x4y2 +12x2y4 +8y6
+27x3−27x2y+9xy2− y3 = 1.
Y : (x2 +2y2)3 +(3x− y)3 = 1
Y (R)−→ X(R) given by(x,y) 7→ (x2 +2y2,3x− y).
Algebraic Geometry
X : x3 + y3 = 1Y : x6 +6x4y2 +12x2y4 +8y6
+27x3−27x2y+9xy2− y3 = 1.
Y : (x2 +2y2)3 +(3x− y)3 = 1
Y (R)−→ X(R) given by(x,y) 7→ (x2 +2y2,3x− y).
Y (Q)−→ X(Q).
Algebraic Geometry
X : x3 + y3 = 1Y : x6 +6x4y2 +12x2y4 +8y6
+27x3−27x2y+9xy2− y3 = 1.
Y : (x2 +2y2)3 +(3x− y)3 = 1
Y (R)−→ X(R) given by(x,y) 7→ (x2 +2y2,3x− y).
Y (Q)−→ X(Q).
Y (Q) is empty.
Algebraic Geometry
X : x3 + y3 = 1Y : 2x3−3x2y+15xy2−7y3 +3x4y+3x5y2
+x6y3−18x3y2−18x4y3−6x5y4 +36x2y3 +36x3y4 +12x4y5
−24y4x−24y5x2−8y6x3−3xy−3x2y2− x3y3 = 1.
(x+y1+xy
)3+(x−2y)3 = 1
Y (C)\(±1,∓1) −→ X(C) given by
(x,y) 7→(
x+y1+xy, x−2y
).
Algebraic Geometry
X : x3 + y3 = 1Y : 2x3−3x2y+15xy2−7y3 +3x4y+3x5y2
+x6y3−18x3y2−18x4y3−6x5y4 +36x2y3 +36x3y4 +12x4y5
−24y4x−24y5x2−8y6x3−3xy−3x2y2− x3y3 = 1.
(x+y1+xy
)3+(x−2y)3 = 1
Y (C)\(±1,∓1) −→ X(C) given by
(x,y) 7→(
x+y1+xy, x−2y
).
Y (Q) = (±1,∓1)
Algebraic Geometry
The set X(C) or X(K) is called an algebraic setwhere K = K.
Fundamental Theorem of Algebra:All polynomials with C-coeff
have all their zeros in C, i.e.,C is algebraically closed.
Algebraic Geometry
The set X(C) or X(K) is called an algebraic setwhere K = K.
For each field K, there isa (large) smallest field extension K
which is algebraically closed.
Algebraic Geometry
The set X(C) or X(K) is called an algebraic setwhere K = K.
For each field K, there isa (large) smallest field extension K
which is algebraically closed.
Examples: finite fields, fields of functions,number fields, p-adic fields, ...
Algebraic Geometry treats all fields of all characteristics.
Algebraic Geometry
The set X(C) or X(K) is called an algebraic setwhere K = K.
Example: The set of equivalent classes ofcurves of genus 1 with C-coeff.
Algebraic Geometry
The set X(C) or X(K) is called an algebraic setwhere K = K.
Example: The set of equivalent classes ofcurves of genus 1 with C-coeff.
Algebraic Geometry
The set X(C) or X(K) is called an algebraic setwhere K = K.
Moduli Problem (Representability):
The category of equivalent classes ofcertain algebraic sets
↔ an algebraic set X (C).
X (C) is called a moduli space.
Algebraic Geometry
The set X(C) or X(K) is called an algebraic setwhere K = K.
Arithmetic Geometry Problem:
• The moduli space for the category ofequivalent classes of curves of genus 1 with Q-coeff.
• The moduli space for the category ofequivalent classes of curves of genus 1 with Q-coeff.,
which contain a rational point.
X (Q) ∼= [C] : C(Q) 6= /0, genus 1
Algebraic Geometry
The set X(C) or X(K) is called an algebraic setwhere K = K.
Arithmetic Geometry Problem:
Given a curve of genus 1 with Q-coeff.,determine whether it has a rational point.
Algebraic Geometry
The ring of functions on an algebraic set
Examples:The set of polynomial functions K[x] forms a ring R:
f ,g ∈ R⇒ f ±g, f ·g ∈ R.
f (x) = x3 + x2 + x+1 ∈Q[x].f : Q−→Qf : R−→ Rf : C−→ Cf : K −→ K
Algebraic Geometry
The ring of functions on an algebraic set
Example: f (x) = x3 + x2 + x+1.
D = 0,1. Then, f |D ≡−(x−1)+4x |D.
Algebraic Geometry
The ring of functions on an algebraic set
Example: f (x) = x3 + x2 + x+1.
D = 0,1. Then, f |D ≡−(x−1)+4x |D.
For any polynomial f (x) ∈ C[x],
f |D ≡ ax+b |D.
Let R be the ring of functions on D.
C[x]−→ R, but C[x] 6∼= R.
Algebraic Geometry
The ring of functions on an algebraic set
Example: D = 0,1, C[x], R = f |D : f ∈ C[x]
Note: f (x) = x(x−1) ·Q(x)+ r(x). So,f (x)≡ r(x) as functions on D.
We use the division algorithm.
Algebraic Geometry
The ring of functions on an algebraic set
Example: D = 0,1, C[x], R = f |D : f ∈ C[x]
Note: f (x) = x(x−1) ·Q(x)+ r(x). So,f (x)≡ r(x) as functions on D.
We use the division algorithm.
R∼= C×C
Ring of Functions
Example: D = (x,y) ∈ R2 : x2 + y2 = 1 and R[x,y].
R = f |D : f ∈ R[x,y].
f (x,y) = 3x3 +3x2y+ x2 +2y2x+4y3
+y2−2x−3y−1 ∈ R[x,y]f (x,y)≡ x3 + y3 on D
Ring of Functions
Example: D = (x,y) ∈ R2 : x2 + y2 = 1 and R[x,y].
R = f |D : f ∈ R[x,y].
f (x,y) = 3x3 +3x2y+ x2 +2y2x+4y3
+y2−2x−3y−1 ∈ R[x,y]f (x,y)≡ x3 + y3 on D
f (x,y) = (x2 + y2−1)(2x+3y+1)+ x3 + y3.f (x,y)≡ (x+ y)(1− xy) on D
Ring of Functions
Example: D = (x,y) ∈ R2 : x2 + y2 = 1 and R[x,y].
R = f |D : f ∈ R[x,y].
The ring R has no zero divisors:f ·g≡ 0 on D ⇒ f or g≡ 0 on D.
Ring of Functions
Example: D = (x,y) ∈ R2 : x2 + y2 = 1 and R[x,y].
R = f |D : f ∈ R[x,y].
The ring R has no zero divisors:f ·g≡ 0 on D ⇒ f or g≡ 0 on D.
R → R(t)where R(t) = f /g : f ,g ∈ R[t],
x 7→ 2t1+t2 , y 7→ 1−t2
1+t2 .
R∼= R[
2t1+t2 ,
1−t2
1+t2
].
Ring of Functions
Example: D = (x,y) ∈ R2 : x3 + y3 = 1 and R[x,y].
R = f |D : f ∈ R[x,y].
The ring R has no zero divisors:f ·g≡ 0 on D ⇒ f or g≡ 0 on D.
There are no injective ring homomorphisms: R→Q(t).
Ring of Functions and Topology
Example: D = (x,y) ∈Q2 : x2 + y2 = 1.
R = f |D : f ∈Q[x,y].
Topology: local geometry,defined by declaring open neighberhoods.
Ring of Functions and Topology
Example: D = (x,y) ∈Q2 : x2 + y2 = 1.
R = f |D : f ∈Q[x,y].
Topology: ∀ P = (a,b) ∈ D,
RP := f ∈ R : f (P) = 0.
Ring of Functions and Topology
Example: D = (x,y) ∈Q2 : x2 + y2 = 1.
R = f |D : f ∈Q[x,y].
Topology: ∀ P = (a,b) ∈ D,
RP := f ∈ R : f (P) = 0.
RP = (x−a) · f +(y−b) ·g : f ,g ∈ R.• This set of linear combinations is called an ideal,
denoted by 〈x−a,y−b〉.• The ideals RP’s are all maximal, and
D↔Max(R)?
Ring of Functions and Topology
Example: D = (x,y) ∈Q2 : x2 + y2 = 1.
R = f |D : f ∈Q[x,y].
Topology: ∀ P = (a,b) ∈ D,
RP := f ∈ R : f (P) = 0.
RP = (x−a) · f +(y−b) ·g : f ,g ∈ R.• This set of linear combinations are called an ideal,
denoted by 〈x−a,y−b〉.• The ideals RP’s are all maximal, and
D→Max(R).〈x2−1/2,y2−1/2〉 & 〈x2 +1,y2−2〉 are maximal.
Ring of Functions and Topology
Example: D = (x,y) ∈Q2 : x2 + y2 = 1.
R = f |D : f ∈Q[x,y].
Topology: ∀ V ⊂ D,
RV := f ∈ R : f (P) = 0, ∀P ∈V.
Ring of Functions and Topology
Example: D = (x,y) ∈Q2 : x2 + y2 = 1.
R = f |D : f ∈Q[x,y].
Topology: ∀ V ⊂ D,
RV := f ∈ R : f (P) = 0, ∀P ∈V.
RV = 〈 f1, . . . , fn〉.
For two subsets V1 and V2 of D with V1 ⊂V2,RV2 ⊂ RV1.
Ring of Functions and Topology
Example: D = (x,y) ∈Q2 : x2 + y2 = 1.
R = f |D : f ∈Q[x,y].
Topology: ∀ V ⊂ D,
RV := f ∈ R : f (P) = 0, ∀P ∈V.
RV = 〈 f1, . . . , fn〉.
For two subsets V1 and V2 of D with V1 ⊂V2,RV2 ⊂ RV1.
The chains of ideals ⇒ topology.
Ring of Functions and Topology
Example: D = (x,y) ∈Q2 : x2 + y2 = 1.
R = f |D : f ∈Q[x,y].
Zariski Topology:• Each ideal in Max(R) forms a (singleton) closed subset.• Let I be an ideal.
J ∈Max(R) : I ⊂ J forms a closed subset.Note that VJ ⊂VI.
Example: I = 〈(x−1)(x−4/5)〉 is contained in〈x−1,y〉, 〈x−4/5,y−3/5〉, and 〈x−4/5,y+3/5〉.
Ring of Functions and Topology
Example: D = (x,y) ∈Q2 : x2 + y2 = 1.
R = f |D : f ∈Q[x,y].
Zariski Topology:• Each ideal in Max(R) forms a (singleton) closed subset.• Let I be an ideal.
J ∈Max(R) : I ⊂ J forms a closed subset.Note that VJ ⊂VI.
Example: I = 〈(x2−3)(x2−5)〉 is contained in〈x2−3,y2 +3〉 and 〈x2−5,y2 +4〉.
Ring of Functions and Topology
Example: D = (x,y) ∈Q2 : x2 + y2 = 1.
R = f |D : f ∈Q[x,y].
Zariski Topology:• Each ideal in Max(R) forms a (singleton) closed subset.• Let I be an ideal.
J ∈Max(R) : I ⊂ J forms a closed subset.Note that VJ ⊂VI.
The Zariski Topology is a topology on Max(R).
Ring of Functions and Geometry
DimensionLet R be the ring of functions on an algebraic set.
• Topology on Max(R).
• The chain of ideals → the underlying geometry.
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J1
J2
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J3 J4 J5 J6
Ring of Functions and Geometry
DimensionLet R be the ring of functions on an algebraic set.
• Topology on Max(R).
• The chain of ideals → the underlying geometry.
0→ p1 → p2 → ··· →Q
where Q ∈Max(R).A chain of prime ideals
Ring of Functions and Geometry
DimensionLet R be the ring of functions on an algebraic set.
• Topology on Max(R).
• The chain of ideals → the underlying geometry.
Example:D = (x,y,z) ∈ R3 : x2 + y2 + z2 = 1, R = f |D : f ∈ R[x,y,z].
0→ p = 〈z〉 →Q = 〈x−1,y,z〉0→ p = 〈z〉 →Q′ = 〈x− 3
5,y−45,z〉
Ring of Functions and Geometry
DimensionLet R be the ring of functions on an algebraic set.
• Topology on Max(R).
• The chain of ideals → the underlying geometry.
Example:D = (x,y,z) ∈ R3 : x2 + y2 + z2 = 1, R = f |D : f ∈ R[x,y,z].
0→ p = 〈z〉 →Q = 〈x−1,y,z〉0→ p = 〈z〉 →Q′ = 〈x− 3
5,y−45,z〉
〈z〉 corresponds to 〈x−a,y−b,z〉 : a2 +b2 = 1.
Ring of Functions and Geometry
DimensionLet R be the ring of functions on an algebraic set.
• Topology on Max(R).
• The chain of ideals → the underlying geometry.
Example:D = (x,y,z) ∈ R3 : x2 + y2 + z2 = 1, R = f |D : f ∈ R[x,y,z].
0→ 〈z〉 → 〈x−1,y,z〉0→ 〈z〉 → 〈x−3/5,y−4/5,z〉
〈z〉 corresponds to a curve (of dimension 1).
Ring of Functions and Geometry
DimensionLet R be the ring of functions on an algebraic set.
• Topology on Max(R).
• The chain of ideals → the underlying geometry.
Example:
Ring of Functions and Geometry
DimensionLet R be the ring of functions on an algebraic set.
• Topology on Max(R).
• The chain of ideals → the underlying geometry.
0→ p1 → p2 → ··· →Q
where Q ∈Max(R).
The length of this chain suggeststhe dimension of the algebraic set,
called the Krull dimension of R.
Ring of Functions and Geometry
DimensionLet R be the ring of functions on an algebraic set.
• Topology on Max(R).
• The chain of ideals → the underlying geometry.
Spec(R) := p⊂ Spec(R) : p is prime idealwith Zariski topology.
Ring of Functions and Geometry
Singularity/Regularity:
dimK Q/Q2 6= dimR ⇒ singular at Q.
Genus: dimK ΩX
where ΩX is the canonical differentials on X .
MATH5900U: Special Topics Course
Integrality of a Ring of Functions
Factorization of Ideals/Galois Covers
Integrality of a ring of functions
Ideal Class Group/Picard Group
Arithmetic Geometry
Global fields K:number fields and function fields of a curve over a finite field.
• K is the field of quotient of a Dedekind domain S.(Notherian, dimension 1, integrally closed).
• S/pS forms a finite field where p is prime.
Example: Let k be the field F3 with 3 elements.• T = (x,y) ∈ F3 : x3 + y3 = 1 and S = f |D : f ∈ k[x,y].
Let K be the field of quotients of S.
Arithmetic Geometry
Global fields K:number fields and function fields of a curve over a finite field.
• K is the field of quotient of a Dedekind domain S.(Notherian, dimension 1, integrally closed).
• S/pS forms a finite field where p is prime.
Example: Let k be the field F3 with 3 elements.• T = (x,y) ∈ F3 : x3 + y3 = 1 and S = f |D : f ∈ k[x,y].
Let K be the field of quotients of S.S ∼= k[t] and K ∼= k(t).
• S/(t2 +1)S ∼= F9.
Arithmetic Geometry
Curves X :
• g(X) = 0: X(K) = /0 or parametrized.
• g(X) = 1: X(K) = /0 orX(K) forms a finitely generated abelian group,
i.e., an elliptic curve.
• g(X) > 2:Mordell-Conjecture: X(Q) is finite.
proved by Gerd Faltings, 1986 Fields Medal
Arithmetic Geometry
Curves X :
• g(X) = 0: X(K) = /0 or parametrized.
• g(X) = 1: X(K) = /0 orX(K) forms a finitely generated abelian group,
i.e., an elliptic curve.
• g(X) > 2:Mordell-Conjecture: X(Q) is finite.
proved by Gerd Faltings, 1986 Fields Medal
Conjecture: For g > 1, there is a constant cg such that#X(Q) < cg for all curves X with genus g.
Arithmetic Geometry
Let k be a finite field w/ #k = q, and kn be a finite ext of k.Let X be a curve with k-coeff, and Nn := #X(kn).
Arithmetic Geometry
Let k be a finite field w/ #k = q, and kn be a finite ext of k.Let X be a curve with k-coeff, and Nn := #X(kn).
• Hasse: X an elliptic curve ⇒|N1− (q+1)| ≤ 2
√q.
• Weil: X a curve with genus g ⇒|Nn− (qn +1)| ≤ 2g
√q.
and he proved a lot more...
Arithmetic Geometry
Let k be a finite field w/ #k = q, and kn be a finite ext of k.Let X be a curve with k-coeff, and Nn := #X(kn).
Weil: X a curve with genus g ⇒ |Nn− (qn +1)| ≤ 2g√
q.
Z(T ) := exp(∑∞n=1 Nn T n/n), a generating function.
Arithmetic Geometry
Let k be a finite field w/ #k = q, and kn be a finite ext of k.Let X be a curve with k-coeff, and Nn := #X(kn).
Weil: X a curve with genus g ⇒ |Nn− (qn +1)| ≤ 2g√
q.
Z(T ) := exp(∑∞n=1 Nn T n/n), a generating function.
Then,
Z(T ) =f (T )
(1−qT )(1−T ).
Arithmetic Geometry
Let k be a finite field w/ #k = q, and kn be a finite ext of k.Let X be a curve with k-coeff, and Nn := #X(kn).
Weil: X a curve with genus g ⇒ |Nn− (qn +1)| ≤ 2g√
q.
Z(T ) := exp(∑∞n=1 Nn T n/n) =
f (T )(1−qT )(1−T )
.
• f (T ) has coefficients in Z with degree 2g.
• f (T ) = ∏2gi=1(1−ωiT ) where ωi ∈ C.
• Riemann Hypothesis for curves over a finite field:
Arithmetic Geometry
Let k be a finite field w/ #k = q, and kn be a finite ext of k.Let X be a curve with k-coeff, and Nn := #X(kn).
Weil: X a curve with genus g ⇒ |Nn− (qn +1)| ≤ 2g√
q.
Z(T ) := exp(∑∞n=1 Nn T n/n) =
f (T )(1−qT )(1−T )
.
• f (T ) has coefficients in Z with degree 2g.
• f (T ) = ∏2gi=1(1−ωiT ) where ωi ∈ C.
• Riemann Hypothesis for curves over a finite field:|ωi|= q1/2.
Arithmetic Geometry
Let k be a finite field w/ #k = q, and kn be a finite ext of k.Let X be a curve with k-coeff, and Nn := #X(kn).
Weil: X a curve with genus g ⇒ |Nn− (qn +1)| ≤ 2g√
q.
Z(T ) := exp(∑∞n=1 Nn T n/n) =
f (T )(1−qT )(1−T )
.
• Riemann Hypothesis for curves over a finite field:|ωi|= q1/2.
log Z(T ) = ∑∞n=1
(qn +1−∑
2gi=1 ωi
)T n
n ⇒ Nn = qn +1−∑2gi=1 ωi.
Therefore, |Nn− (qn +1)| ≤ 2g√
q.
Arithmetic Geometry
Let k be a finite field w/ #k = q, and kn be a finite ext of k.Let X be a curve with k-coeff, and Nn := #X(kn).
Weil: X a curve with genus g ⇒ |Nn− (qn +1)| ≤ 2g√
q.
Z(T ) := exp(∑∞n=1 Nn T n/n) =
f (T )(1−qT )(1−T )
.
• f (T ) has coefficients in Z with degree 2g.
• f (T ) = ∏2gi=1(1−ωiT ) where ωi ∈ C.
• RH for curves over a finite field: |ωi|= q1/2.
• Functional Equations: Z(1/qT ) = (qT )1−gZ(T ).
Arithmetic Geometry
Let k be a finite field w/ #k = q, and kn be a finite ext of k.Let X be a curve with k-coeff, and Nn := #X(kn).
Weil: X a curve with genus g ⇒ |Nn− (qn +1)| ≤ 2g√
q.
Z(T ) := exp(∑∞n=1 Nn T n/n) =
f (T )(1−qT )(1−T )
.
Example: Let X be a compact smooth cubic curve.Then, X(k) must be nonempty.
f (T ) = 1+(N1−q−1)T +qT 2, so f (1) = N1,while f (T ) = (1−ω1T )(1−ω2T )⇒
f (1) = (1−ω1)(1−ω2) 6= 0 since |ωi|=√
q.
Arithmetic Geometry
Let k be a finite field w/ #k = q, and kn be a finite ext of k.Let X be a curve with k-coeff, and Nn := #X(kn).
Weil: X a curve with genus g ⇒ |Nn− (qn +1)| ≤ 2g√
q.
Z(T ) := exp(∑∞n=1 Nn T n/n) =
f (T )(1−qT )(1−T )
.
Example: Let X be a compact smooth cubic curve.Then, X(k) must be nonempty.
f (T ) = 1+(N1− p−1)T + pT 2 = 1+apT + pT 2.
F ap : p gave birth to Taniyama-Shimura-Weil Conj.
Arithmetic Geometry
Let k be a finite field w/ #k = q, and kn be a finite ext of k.Let X be a curve with k-coeff, and Nn := #X(kn).
Weil Conjecture: X a smooth projective variety of dimension n.
Z(T ) := exp(∑∞n=1 Nn T n/n).
(1) Rationality: Z(T ) ∈Q(T ).
(2) Functional Equation: Z(1/qn T ) =±qnε/2 T ε Z(T ).
(3) Rieman Hypothesis:
Z(T ) =P1(T ) · · ·P2n−1(T )P0(T ) · · ·P2n(T )
where Pi(T ) ∈ Z[T ], Pi(T ) = ∏ j(1−αi jT ) with∣∣αi j
∣∣ = qi/2
for 1≤ i≤ 2n−1,
and P0(T ) = 1−T and P2n(T ) = 1−qnT .
Arithmetic Geometry
Let k be a finite field w/ #k = q, and kn be a finite ext of k.Let X be a curve with k-coeff, and Nn := #X(kn).
Weil Conjecture: X a smooth projective variety of dimension n.
Z(T ) := exp(∑∞n=1 Nn T n/n).
Rationality, Functional Equation, Rieman Hypothesis
• Weil, 1949: proved for abelian varieties.
• Dwork, 1960: the rationality, using p-adic functional analysis.
• Artin, Grothendieck, et. al.: `-adic cohomology theory;
proved the functional equation.
• Deligne, 1973: proved the Riemann Hypothesis, Fields Medal
Arithmetic Geometry
Higher Dimensional X :
• An abelian variety X(K) is a compact algebraic group.Mordell-Weil Theorem:
X(K) is a finitely generated abelian group.
• Let C be a curve.Abel-Jacobi: C(C) → X(C) where X is an abelian variety.Weil, et. al.: C(K) → X(K) s.t. C(K) → X(K).
Arithmetic Geometry
Higher Dimensional X :
• An abelian variety X(K) is a compact algebraic group.Mordell-Weil Theorem:
X(K) is a finitely generated abelian group.
• Let C be a curve.Abel-Jacobi: C(C) → X(C) where X is an abelian variety.Weil, et. al.: C(K) → X(K) s.t. C(K) → X(K).
X is called a Jacobian variety of C.