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.