5
Dujella's webpage contains relevant information:
Infinite families of elliptic curves with high rank and prescribed torsion
He also has pages describing rank records for individual curves and for curves defined over quadratic fields.
4
For elliptic curves, the answer is no, as Chris Wuthrich and Aron Fehm point out in the comments. In fact I think every extension with Galois group a finite simple group not of the form $PSL_2(\mathbb F_p)$ for any $p$ cannot arise this way.
For abelian varieties, the answer is positive. In fact it's sufficient to take just the $2$-torsion, and even just the ...
4
Let $E/\mathbb{Q}$ be an elliptic curve. There exist positive integers $d_p$ and $e_p$, with $d_p|e_p$, such that group $E(\mathbb{F}_p)$ is isomorphic to $\mathbb{Z}/d_p\mathbb{Z} \times \mathbb{Z}/e_p\mathbb{Z}$. Kowalski conjectured that there exists a constant $c_E>0$ such that $\sum_{p\leq x}d_p\sim c_E f_E(x)$, where $f_E(x)=x$ if $E$ has CM and $...
1
To define this correspondence you don't want to write down either side explicitly, but rather do it abstractly. Then you can calculate what it is in explicit terms if desired. The first steps are
(1) Send each cusp in $\overline{M}_n$ to its formal neighborhood in $\overline{M}_n$, and then to its punctured formal neighborhood in $M_n$.
(2) Observe that ...
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