https://www.rzuser.uni-heidelberg.de/~hb3/publ/pcft.pdf

Class Field Towers

Non-isotrivial elliptic surfaces 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 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.

Families of elliptic curves with non-zero average root number

http://www.math.snu.ac.kr/~dhbyeon/06_QTE1.pdf

Quadratic twists of elliptic curves associated to the simplest cubic fields

https://hal.archives-ouvertes.fr/hal-01478267/document

/pdf/the-2-class-groups-of-cubic-fields-and-2-descents-on-29nrs3te79.pdf

Shin Nakano

Papers

The 2-class groups of cubic fields and 2-descents on elliptic curves