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Given an extension of number fields $E \subset F$ and a projective variety$X$ over $F$, we compare the problem of counting the number of rational pointsof bounded height on $X$ with that of its Weil restriction over $E$. Inparticular, we consider the compatibility with respect to the Weil restrictionof conjectural asymptotic formulae due to Manin and others. Using our methodswe prove several new cases of these conjectures. We also construct newcounterexamples over every number field.

Rational points of bounded height and the Weil restriction