# 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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Journal articles

Cited literature [22 references]

https://hal.archives-ouvertes.fr/hal-01478267
Contributor : Christophe Delaunay <>
Submitted on : Tuesday, February 28, 2017 - 8:50:14 AM
Last modification on : Thursday, October 11, 2018 - 9:18:42 AM
Long-term archiving on : Monday, May 29, 2017 - 12:52:37 PM

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biased_families-_C2.pdf
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### Identifiers

• HAL Id : hal-01478267, version 1
• ARXIV : 1612.03095

### Citation

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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