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Vesselin Dimitrov
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Results and conjectures on bounds on degrees of isogenies
By the way, Hindry and Silverman have shown that for elliptic curves with integral modulus (everywhere good reduction), the number of torsion points rational over a given number field of degree $d$ is at most $\mathrm{const} \cdot d\log{d}$. This lends, perhaps, additional support to the expectation that the bound in Merel's theorem should be polynomial. (Note that the trivial bound provided by good reduction at $2$ is exponential in $d$.)
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Results and conjectures on bounds on degrees of isogenies
@Michael: This is stated, for example, in Remark 2 in Marusia Rebolledo's expository article ("Merel's theorem on the boundedness of the torsion of elliptic curves.") I am sure I have seen it in at least one other paper, but right now I can't remember the reference. But the idea is that since for individual elliptic curves the count is proportional to $[K:\mathbb{Q}]^{3/2}$ and $[K:\mathbb{Q}]^2$ in the non-CM and CM cases, respectively, it is not too wild to ask whether this count is uniform. Thus, one can ask whether the count is in fact uniformly bounded by $C(r) [K:\mathbb{Q}]^{2+r}$.
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Results and conjectures on bounds on degrees of isogenies
As far as I know, the conclusion of Merel's theorem is believed to hold for abelian varieties up to bounded dimension. In fact, the bound on the $K$-rational torsion is supposed to be polynomial in $[K:\mathbb{Q}]$. (This is wide open even for elliptic curves; Merel's theorem gives an exponential bound.)
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The height of an orbit under rational self-maps
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The height of an orbit under rational self-maps
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The height of an orbit under rational self-maps
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Are the Chern numbers of a hyperbolic-type compact complex manifold bounded in terms of the Euler number?
I ask about bounding the degree-$n$ combinations of the $c_i$ in terms of $c_n$. Now, $c_i$ has degree $i$. Since the only degree-$n$ combination involving $c_n$ is $c_n$ itself, I removed $i=n$ from the product.
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Points of minimum Arakelov height and harmonic arithmetical varieties
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Points of minimum Arakelov height and harmonic arithmetical varieties
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Points of minimum Arakelov height and harmonic arithmetical varieties
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