# Upper bound for the torsion subgroup of an elliptic curve over arbitrary number fields

Let $$K$$ be a finite extension of $$\mathbb{Q}$$ of degree $$d$$ and let $$E(K)$$ be an elliptic curve over the field $$K$$ with coefficients in $$K$$. Let us fix $$d$$ and vary over all the possible $$K$$, in turn varying over all possible $$E$$. My question is the following: does there exist an integer $$N=N_d$$ such that $$N \geq \#|E(K_{\mathrm{tors}})|$$ for any $$K, E$$ and fixed $$d$$? If this is not known to be the case, then what is the general consensus regarding the number of $$K$$-rational torsion points of $$E$$? If none of this is known, is there a record for the largest known torsion order? As of right now I'm curious for $$d=2$$ and non-CM elliptic curves $$E(K)$$, but any information regarding a specific $$d > 1$$ is helpful.

• Yes, such N exists for each d; Loïc Merel proved this (with an explicit albeit large N) in a paper published by Inventiones mathematicæ in 1996. For $d=2$ The problem is completely solved. en.wikipedia.org/wiki/Torsion_conjecture Nov 10, 2022 at 4:47
• @NoamD.Elkies Your comment should be an answer (so that this question can be closed). Nov 10, 2022 at 7:21
• What does "over the field $K$ with coefficients in $K$" mean? I know what "over the field $K$" means (i.e. the curve is defined over $K$ and the basepoint is a $K$-point) — I would rather have written "an elliptic curve $E$ over the field $K$", with $E(K)$ rather denoting the group of $K$-points.
– YCor
Nov 10, 2022 at 7:37
• @NoamD.Elkies Thanks, this is a great answer. This question should be closed now. Nov 11, 2022 at 1:12
• @GHfromMO I didn't yet formulate my comment as an answer because I wanted to first look up the list for $d=2$ and to credit it properly. Nov 11, 2022 at 1:27