Introduction to Arithmetic Algebraic Geometry

93 Pages·2008·1.33 MB·English

Introduction to
Arithmetic Algebraic Geometry
Sungkon Chang
The Anne and Sigmund Hudson
Mathematics and Computing
Luncheon Colloquium Series
Diophantine Equations
Z
Let denote the set of integers.
Diophantine Equations:
2x2+3y2 = 4z3+5w3
Z
where x,y,z,w ∈ .
Diophantine Equations
Z
Let denote the set of integers.
Diophantine Equations:
x2+y2 = x3+y3 = x4+y4
1 1 2 2 3 3
Z
where x ’s and y ’s ∈ .
k k
Rational Solutions
Q
Let denote the set of rational numbers.
2x2+3y2 = 4z3+5w3
Q
where x,y,z,w ∈ .
Rational Solutions
Q
Let denote the set of rational numbers.
2x2+3y2 = 4z3+5w3
Q
where x,y,z,w ∈ .
Q
is a field.
Q Q
a±b, a·b, a/b ∈ where a,b ∈ .
K-Rational Solutions
Let K be a field.
2x2+3y2 = 4z3+5w3
where x,y,z,w ∈ K.
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.
Q
Is X( ) empty?
K-Rational Solutions
X : f(x,y) = 0
where f(x,y) is a polynomial
Q
with -coeff.
X(Q) = {(x,y) ∈ Q2 : f(x,y) = 0}.
Q
Question: Determine whether X( ) is empty.