Published on *Max Planck Institute for Mathematics* (http://www.mpim-bonn.mpg.de)

Posted in

- Talk [1]

Speaker:

Daniel Kriz
Affiliation:

MIT
Date:

Wed, 05/01/2022 - 14:30 - 15:30 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.

**Links:**

[1] http://www.mpim-bonn.mpg.de/taxonomy/term/39

[2] http://www.mpim-bonn.mpg.de/node/246