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Close mobile search navigation Article Navigation. Volume Periods of Hilbert modular forms and rational points on elliptic curves Henri Darmon. Oxford Academic.

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## Elliptic Curves in Nature

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## Rational Points on Elliptic Curves | Mathematical Association of America

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Close mobile search navigation Article Navigation. Volume Periods of Hilbert modular forms and rational points on elliptic curves Henri Darmon. Oxford Academic.

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### Elliptic curves

It is also required that the curve not be singular, which in form 1 says that 4A 3 B 2 is not equal to 0. Details in any book on elliptic curves. Depending on A and B , the equation might have no rational solutions, or just a few, or infinitely many. A particular case concerns congruent numbers. Furthermore this is a simple but important point , if a, b, and c are rational numbers, then so are x and y , and y is not 0. It follows that such a point must have infinite order, which is why the congruent number problem has a connection to the Birch and Swinnerton-Dyer conjecture.