Families of elliptic curves with non-zero average root number

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 ra...

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Bibliographic Details
Published in:Journal of number theory 2018
Main Authors: Bettin, Sandro, David, Chantal, Delaunay, Christophe
Format: Article
Language:English
Subjects:
Mathematics
Number Theory
Online Access:Get full text
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Summary: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.
ISSN:0022-314X
1096-1658