# Why should an abelian variety with few places of bad reduction and a lot of endomorphisms not have many points?

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.

• 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 Nov 7, 2020 at 17:09