In the paper "Points of Order 13 on Elliptic Curves" by Mazur-Tate, they say in the introduction:

It seemed ... that if such an abelian variety $J$, which has bad reduction at only one prime and has a sizeable number of endomorphisms, has a point of order $19$, it is not entitled to have any other points.

where $J$ is the Jacobian of $X_1(13)$. Can this heuristic be made more precise? They go on to prove in the paper that $J$ has exactly $19$ rational points but I am more interested in the intuition behind the statement.

  • 1
    $\begingroup$ I don't know about the cardinality of torsion, but the rank of the group of $K$-rational points of an abelian variety is bounded above in terms of the primes of bad reduction, and other quantities depending on $\mathrm{dim}(A)$, $\Delta_K$ and $[K \colon \mathbb{Q}]$. This is proved by Ooe and Top here: math.rug.nl/~top/Ooe.pdf $\endgroup$ Nov 7 '20 at 17:09

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