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Elliptic Curves Elliptic curves Bjorn Poonen MIT Arnold Ross Lecture May 31, 2019 Plane curves Degree 1 (lines) 3x + 7y + 6 = 0 Degree 2 (conics) 2x2 + 9xy + 3y 2 3x + 7y + 6 = 0 Degree 3 (cubic curves) 4x3 + 5x2y + xy 2 + 8y 3 2x2 + 9xy + 3y 2 3x + 7y + 6 = 0 Elliptic curves are special cubic curves. 3 4 ; 5 5 12 5 − ; 13 13 A 4 3 ; − 5 5 (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) 12 5 − ; 13 13 A 4 3 ; − 5 5 (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) 3 4 ; 5 5 3 4 ; 5 5 A 4 3 ; − 5 5 (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) 12 5 − ; 13 13 3 4 ; 5 5 12 5 − ; 13 13 A 4 3 ; − 5 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) (0; −1) 3 4 ; 5 5 12 5 − ; 13 13 A (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 3 ; − 5 5 3 4 ; 5 5 12 5 − ; 13 13 A 4 3 ; − 5 5 (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) 3 4 ; 5 5 12 5 − ; 13 13 A 4 3 ; − 5 5 (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) 3 4 ; 5 5 12 5 − ; 13 13 A 4 3 ; − 5 5 (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) 3 4 ; 5 5 12 5 − ; 13 13 A 4 3 ; − 5 5 (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) 3 4 ; 5 5 12 5 − ; 13 13 A 4 3 ; − 5 5 (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) 3 4 ; 5 5 12 5 − ; 13 13 A 4 3 ; − 5 5 (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) 3 4 ; 5 5 12 5 − ; 13 13 A 4 3 ; − 5 5 (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) 3 4 ; 5 5 12 5 − ; 13 13 A 4 3 ; − 5 5 (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) 3 4 ; 5 5 12 5 − ; 13 13 A 4 3 ; − 5 5 (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) 3 4 ; 5 5 12 5 − ; 13 13 A 4 3 ; − 5 5 (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) 3 4 ; 5 5 12 5 − ; 13 13 A 4 3 ; − 5 5 (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) 3 4 ; 5 5 12 5 − ; 13 13 A 4 3 ; − 5 5 (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) 3 4 ; 5 5 12 5 − ; 13 13 A 4 3 ; − 5 5 (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) 3 4 ; 5 5 12 5 − ; 13 13 A 4 3 ; − 5 5 (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) 3 4 ; 5 5 12 5 − ; 13 13 A 4 3 ; − 5 5 (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) 3 4 ; 5 5 12 5 − ; 13 13 A 4 3 ; − 5 5 (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) 3 4 ; 5 5 12 5 − ; 13 13 A 4 3 ; − 5 5 (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) 3 4 ; 5 5 12 5 − ; 13 13 A 4 3 ; − 5 5 (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) 3 4 ; 5 5 12 5 − ; 13 13 A 4 3 ; − 5 5 (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) 3 4 ; 5 5 12 5 − ; 13 13 A 4 3 ; − 5 5 (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) 3 4 ; 5 5 12 5 − ; 13 13 A 4 3 ; − 5 5 (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) 3 4 ; 5 5 12 5 − ; 13 13 A 4 3 ; − 5 5 (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) 3 4 ; 5 5 12 5 − ; 13 13 A 4 3 ; − 5 5 (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) 3 4 ; 5 5 12 5 − ; 13 13 A 4 3 ; − 5 5 (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) 3 4 ; 5 5 12 5 − ; 13 13 A 4 3 ; − 5 5 (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) 3 4 ; 5 5 12 5 − ; 13 13 A 4 3 ; − 5 5 (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) 3 4 ; 5 5 12 5 − ; 13 13 A 4 3 ; − 5 5 (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) 3 4 ; 5 5 12 5 − ; 13 13 A 4 3 ; − 5 5 (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).
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