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A classical theorem of Minkowski and Hlawka states that there exists a lattice in R^n with packing density at least 2^{1-n}. Buser and Sarnak proved the analogue of this result in the context of complex abelian varieties. Here we give an improvement of this analogue; this shows a conjecture of Muetzel., in French

Let $X$ be a projective variety over a number field $K$ (resp. over $\mathbb{C}$). Let $H$ be the sum of ``sufficiently many positive divisors'' on $X$. We show that any set of quasi-integral points (resp. any integral curve) in $X-H$ is not Zariski dense.

We develop a strategy for bounding from above the height of rational points of modular curves with values in number fields, by functions which are polynomial in the curve's level. Our main technical tools come from effective Arakelov descriptions of modular curves and jacobians. We then fulfill this program in the following particular case: If $p$ is a not-too-small prime number, let $X_0 (p )$ be the classical modular curve of level $p$ over $\bf Q$. Assume Brumer's conjecture on the dimension of winding quotients of $J_0 (p)$. We prove that there is a function $b(p)=O(p^{5} \log p )$ (depending only on $p$) such that, for any quadratic number field $K$, the $j$-height of points in $X_0 (p ) (K)$ which are not lifts of elements of $X_0^+ (p) ({\bf Q} )$, is less or equal to $b(p)$., Comment: Revised version, with an Appendix by Pascal Autissier. (Due to some computer bug, yesterday's file was flawed: v2 was the concatenation of the old and new releases.) Up to minor differences of form, this is the text published in Algebra and Number Theory, Vol. 12 n. 9 (2018)