Families of elliptic curves with non-zero average root number

Abstract : We consider the problem of finding $1$-parameter families of elliptic curves whose root number does not average to zero as the parameter varies in $\mathbb{Z}$. We classify all such families when the degree of the coefficients (in the parameter $t$) is less than or equal to $2$ and we compute the rank over $\mathbb{Q}(t)$ of all these families. Also, we compute explicitly the average of the root numbers for some of these families highlighting some special cases. Finally, we prove some results on the possible values average root numbers can take, showing for example that all rational number in $[-1,1]$ are average root numbers for some $1$-parameter family.
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  • HAL Id : hal-01478267, version 1
  • ARXIV : 1612.03095

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Sandro Bettin, Chantal David, Christophe Delaunay. Families of elliptic curves with non-zero average root number. Journal of Number Theory, Elsevier, 2018. ⟨hal-01478267⟩

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