Timeline for Rank growth of elliptic curves after cubic extensions

4 events

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Jan 10, 2013 at 14:15	comment	added	Dave M da C		Following the approach that Felipe mentions one can show that there are at least $c_E X^{1/2}$ cubic extensions which show an increase in rank (at least when $j(E) \neq 0$). It would be interesting to know how many more extensions there should be where rank grows which are not of the form $f(x) - c^2$. Perhaps, as Noam suggests, the true exponent of $X$ depends on the reduction types the curve exhibits and is not uniform in $E$ unlike the conjecture above and others like Goldfeld's.
Jan 10, 2013 at 10:43	comment	added	Chris Wuthrich		Yes, Noam, root number do not change in odd-degree Galois extensions.
Jan 10, 2013 at 2:42	comment	added	Noam D. Elkies		Well it's easy to get a lower bound $cX^\theta$ for some $c,\theta>0$, but the question is how big $\theta$ can get. For cyclic cubics we're told it might be $1/2 - \epsilon$. For unrestricted cubics, possibly even a positive proportion (i.e. $\theta = 1$), at least if $E$ has a place of bad reduction whose contribution to the root number can change from ${\bf Q}$ to $K$ $-$ or is there a reason that this can't happen (as it presumably doesn't for cyclic cubics if we're to believe the heuristic of David, Fearnley, and Kisilevsky)?
Jan 10, 2013 at 0:36	history	answered	Felipe Voloch	CC BY-SA 3.0