On the typical rank of elliptic curves over ${\mathbb Q}(T)$

preprint2022arXivOpen access

We give upper bounds for the number of rational elliptic surfaces in some families having positive rank, obtaining in particular that these form a subset of density zero. This confirms Cowan's conjecture (arXiv:2009.08622v2) in the case $m,n\leq2$ .

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On the typical rank of elliptic curves over ${\mathbb Q}(T)$

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On the typical rank of elliptic curves over Q(T){\mathbb Q}(T)Q(T)

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On the typical rank of elliptic curves over ${\mathbb Q}(T)$