The last few years have witnessed a number of developments in the arithmetic of elliptic curves, notably the proof that there are positive proportions of elliptic curves of rank zero and rank one for which the Bircha€”Swinnerton-Dyer conjecture is true. The proof of this landmark result relies on an appealing mix of diverse techniques arising from the newly resurgent field of arithmetic invariant theory, Iwasara theory, congruences between modular forms, and the theory of Heegner points and related Euler systems. The purpose of this workshop is to survey the proof of this theorem and to describe the new perspectives on the Bircha€”Swinnterton-Dyer conjecture which it opens up.
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