Let $f:\mathbb{N}\to\mathbb{N}$ be the map sending $n$ to the number of isomorphism classes of elliptic curves over $\mathbb{Q}$ with conductor $n$.
Is there a polynomial $P$ such that $P(f(n))>n$ for infinitely many $n$?
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$\begingroup$ Probably no such $P$ exists, but it might already be nontrivial to prove there does not exist a linear such $P$. $\endgroup$– Will SawinCommented Aug 25, 2021 at 16:44
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$\begingroup$ If you believe ranks are uniformly bounded (or even just really small in terms of the minimal discriminant) then yeah, check out Corollary 3.11 of Helfgott-Venkatesh arxiv.org/pdf/math/0405180.pdf and apply it to the handful of Mordell curves you get in the usual argument that there are only finitely many elliptic curves over a number field of given conductor. (Lemme know if I’ve confused myself though!) $\endgroup$– alpogeCommented Aug 25, 2021 at 23:01
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