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|What I did in my holidays|
|Abstract: I will discuss progress I made on the subconvexity problem for automorphic L-functions while at the IAS this past year.|
|How does the Mordell-Weil rank of an elliptic curve grow in towers of number fields, if you start with a quadratic imaginary field?|
|Abstract: The Mordell-Weil theorem says that for a number field $K$, the $K$-rational solutions of an elliptic curve $E: y^2=x^3+ax+b$ form a finitely generated abelian group. One natural question is how the rank of these groups behave as $K$ varies. In infinite towers of number fields, one may guess that the rank should keep growing ('more numbers should mean new solutions'). However, this guess is not correct: When starting with $\Q$ and climbing along the tower of number fields that make up the $Z_p$-extension of $Q$, the rank stops growing, as envisioned by Mazur in the 1970's.
What happens if we start with an imaginary quadratic field instead of $\Q$, and look at the growth of the rank in a $Z_p$-extension? Our new found 'intuition' now tells us we should expect the rank to stay bounded, but this is not always the case, as shown in Bertolini's thesis. So how badly does the rank grow in general? We initiate an answer to this question in this talk. This is joint work with Antonio Lei.