For zoom details contact Pieter Moree (moree@mpim-bonn.mpg.de)

I will present a rank 0 and 1 p-converse theorem for CM elliptic curves defined over the rationals in the case where p is ramified in the CM field. This theorem has applications to two classical problems of arithmetic: it verifies Sylvester's conjecture on primes expressible as a sum of two rational cubes, and, combined with known Selmer corank distribution results, establishes Goldfeld's conjecture for the congruent number family. The proof of the p-converse theorem relies on formulating and proving a new Iwasawa main conjecture. The proof of the Iwasawa main conjecture involves new methods and constructions arising from interplays between Iwasawa-theoretic objects and relative p-adic Hodge theory on the infinite-level Shimura curve.

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