Elliptic curves Bjorn Poonen MIT Arnold Ross Lecture May 31, 2019 - PowerPoint PPT Presentation

Elliptic curves Bjorn Poonen MIT Arnold Ross Lecture May 31, 2019

Plane curves Degree 1 (lines) 3 x + 7 y + 6 = 0 Degree 2 (conics) 2 x 2 + 9 xy + 3 y 2 3 x + 7 y + 6 = 0 Degree 3 (cubic curves) 4 x 3 + 5 x 2 y + xy 2 + 8 y 3 2 x 2 + 9 xy + 3 y 2 3 x + 7 y + 6 = 0 Elliptic curves are special cubic curves.

Rational points on the unit circle Definition A rational point on a curve is a point whose coordinates are rational numbers (elements of Q ). Example (The unit circle x 2 + y 2 = 1)

Rational points on the unit circle Definition A rational point on a curve is a point whose coordinates are rational numbers (elements of Q ). Example (The unit circle x 2 + y 2 = 1) � 3 5 , 4 � 5

Rational points on the unit circle Definition A rational point on a curve is a point whose coordinates are rational numbers (elements of Q ). Example (The unit circle x 2 + y 2 = 1) � � − 12 13 , 5 13

Rational points on the unit circle Definition A rational point on a curve is a point whose coordinates are rational numbers (elements of Q ). Example (The unit circle x 2 + y 2 = 1) (0 , − 1)

Rational points on the unit circle Definition A rational point on a curve is a point whose coordinates are rational numbers (elements of Q ). Example (The unit circle x 2 + y 2 = 1) � 4 5 , − 3 � 5

Rational points on the unit circle Definition A rational point on a curve is a point whose coordinates are rational numbers (elements of Q ). Example (The unit circle x 2 + y 2 = 1)

Rational points on the unit circle Definition A rational point on a curve is a point whose coordinates are rational numbers (elements of Q ). Example (The unit circle x 2 + y 2 = 1)

Rational points on the unit circle Definition A rational point on a curve is a point whose coordinates are rational numbers (elements of Q ). Example (The unit circle x 2 + y 2 = 1)

Rational points on the unit circle Definition A rational point on a curve is a point whose coordinates are rational numbers (elements of Q ). Example (The unit circle x 2 + y 2 = 1)

Rational points on the unit circle Definition A rational point on a curve is a point whose coordinates are rational numbers (elements of Q ). Example (The unit circle x 2 + y 2 = 1)

Rational points on the unit circle Definition A rational point on a curve is a point whose coordinates are rational numbers (elements of Q ). Example (The unit circle x 2 + y 2 = 1)

Rational points on the unit circle Definition A rational point on a curve is a point whose coordinates are rational numbers (elements of Q ). Example (The unit circle x 2 + y 2 = 1)

Rational points on the unit circle Definition A rational point on a curve is a point whose coordinates are rational numbers (elements of Q ). Example (The unit circle x 2 + y 2 = 1)

Rational points on the unit circle Definition A rational point on a curve is a point whose coordinates are rational numbers (elements of Q ). Example (The unit circle x 2 + y 2 = 1)

Rational points on the unit circle Definition A rational point on a curve is a point whose coordinates are rational numbers (elements of Q ). Example (The unit circle x 2 + y 2 = 1)

Rational points on the unit circle Definition A rational point on a curve is a point whose coordinates are rational numbers (elements of Q ). Example (The unit circle x 2 + y 2 = 1)

Rational points on the unit circle Definition A rational point on a curve is a point whose coordinates are rational numbers (elements of Q ). Example (The unit circle x 2 + y 2 = 1)

Rational points on the unit circle Definition A rational point on a curve is a point whose coordinates are rational numbers (elements of Q ). Example (The unit circle x 2 + y 2 = 1)

Rational points on the unit circle Definition A rational point on a curve is a point whose coordinates are rational numbers (elements of Q ). Example (The unit circle x 2 + y 2 = 1)

Rational points on the unit circle Definition A rational point on a curve is a point whose coordinates are rational numbers (elements of Q ). Example (The unit circle x 2 + y 2 = 1)

Rational points on the unit circle Definition A rational point on a curve is a point whose coordinates are rational numbers (elements of Q ). Example (The unit circle x 2 + y 2 = 1)

Rational points on the unit circle Definition A rational point on a curve is a point whose coordinates are rational numbers (elements of Q ). Example (The unit circle x 2 + y 2 = 1)

Rational points on the unit circle Definition A rational point on a curve is a point whose coordinates are rational numbers (elements of Q ). Example (The unit circle x 2 + y 2 = 1)

Rational points on the unit circle Definition A rational point on a curve is a point whose coordinates are rational numbers (elements of Q ). Example (The unit circle x 2 + y 2 = 1)

Rational points on the unit circle Definition A rational point on a curve is a point whose coordinates are rational numbers (elements of Q ). Example (The unit circle x 2 + y 2 = 1)

Rational points on the unit circle Definition A rational point on a curve is a point whose coordinates are rational numbers (elements of Q ). Example (The unit circle x 2 + y 2 = 1)

Rational points on the unit circle Definition A rational point on a curve is a point whose coordinates are rational numbers (elements of Q ). Example (The unit circle x 2 + y 2 = 1)

Rational points on the unit circle Definition A rational point on a curve is a point whose coordinates are rational numbers (elements of Q ). Example (The unit circle x 2 + y 2 = 1)

Rational points on the unit circle Definition A rational point on a curve is a point whose coordinates are rational numbers (elements of Q ). Example (The unit circle x 2 + y 2 = 1)

Rational points on the unit circle Definition A rational point on a curve is a point whose coordinates are rational numbers (elements of Q ). Example (The unit circle x 2 + y 2 = 1)

Rational points on the unit circle Definition A rational point on a curve is a point whose coordinates are rational numbers (elements of Q ). Example (The unit circle x 2 + y 2 = 1)

Rational points on the unit circle Definition A rational point on a curve is a point whose coordinates are rational numbers (elements of Q ). Example (The unit circle x 2 + y 2 = 1)

Rational points on the unit circle Definition A rational point on a curve is a point whose coordinates are rational numbers (elements of Q ). Example (The unit circle x 2 + y 2 = 1)

Rational points on the unit circle Definition A rational point on a curve is a point whose coordinates are rational numbers (elements of Q ). Example (The unit circle x 2 + y 2 = 1)

Rational points on the unit circle Definition A rational point on a curve is a point whose coordinates are rational numbers (elements of Q ). Example (The unit circle x 2 + y 2 = 1)

Rational points on the unit circle Definition A rational point on a curve is a point whose coordinates are rational numbers (elements of Q ). Example (The unit circle x 2 + y 2 = 1)

Rational points on the unit circle Definition A rational point on a curve is a point whose coordinates are rational numbers (elements of Q ). Example (The unit circle x 2 + y 2 = 1)

Rational points on the unit circle Definition A rational point on a curve is a point whose coordinates are rational numbers (elements of Q ). Example (The unit circle x 2 + y 2 = 1)

Rational points on the unit circle Definition A rational point on a curve is a point whose coordinates are rational numbers (elements of Q ). Example (The unit circle x 2 + y 2 = 1)

Rational points on the unit circle Definition A rational point on a curve is a point whose coordinates are rational numbers (elements of Q ). Example (The unit circle x 2 + y 2 = 1)

Rational points on the unit circle Definition A rational point on a curve is a point whose coordinates are rational numbers (elements of Q ). Example (The unit circle x 2 + y 2 = 1)

Rational points on the unit circle Definition A rational point on a curve is a point whose coordinates are rational numbers (elements of Q ). Example (The unit circle x 2 + y 2 = 1)

Rational points on the unit circle Definition A rational point on a curve is a point whose coordinates are rational numbers (elements of Q ). Example (The unit circle x 2 + y 2 = 1)

Rational points on the unit circle Definition A rational point on a curve is a point whose coordinates are rational numbers (elements of Q ). Example (The unit circle x 2 + y 2 = 1)

Rational points on the unit circle Definition A rational point on a curve is a point whose coordinates are rational numbers (elements of Q ). Example (The unit circle x 2 + y 2 = 1)

Rational points on the unit circle Definition A rational point on a curve is a point whose coordinates are rational numbers (elements of Q ). Example (The unit circle x 2 + y 2 = 1)

Rational points on the unit circle Definition A rational point on a curve is a point whose coordinates are rational numbers (elements of Q ). Example (The unit circle x 2 + y 2 = 1)

Rational points on the unit circle Definition A rational point on a curve is a point whose coordinates are rational numbers (elements of Q ). Example (The unit circle x 2 + y 2 = 1)