Minerva Lecture III: Logic, Elliptic curves, and Diophantine stability
Minerva Lecture III: Logic, Elliptic curves, and Diophantine stability
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Barry Charles Mazur, Gerhard Gade University Professor at Harvard University
Minerva Lecture III:ÌýÌýAn introduction to aspects ofÌýmathematical logicÌýandÌýthe arithmetic of elliptic curvesÌýthat make these branches of mathematics inspiring to each other. ÌýSpecifically: algebraic curves - other than the projective line - over number fields tendÌýto acquire no new rational points over many extension fields. This feature (which I call 'diophantine stability') makes elliptic curves, in particular, useful as vehicles to establishÌýdiophantine unsolvability for many large rings. To repay the debt, mathematical logicÌýoffers consequences to the arithmetic of elliptic curves over decidable rings. I will alsoÌýdiscuss new results about diophantine stability.