Constructing families of moderate-rank elliptic curves over number fields

TL;DR

This paper extends a method for constructing moderate-rank elliptic curves from rational numbers to general number fields, enabling rank computation via Frobenius traces without explicit point determination.

Contribution

It generalizes a known construction of elliptic curve families to number fields, avoiding explicit point calculations and determinants of height matrices.

Findings

Constructed families of moderate-rank elliptic curves over number fields.

Rank can be computed by controlling Frobenius trace averages.

Method simplifies the process by not requiring explicit points.

Abstract

We generalize a construction of families of moderate rank elliptic curves over $\mathbb{Q}$ to number fields $K/\mathbb{Q}$. The construction, originally due to Steven J. Miller, \'Alvaro Lozano-Robledo and Scott Arms, invokes a theorem of Rosen and Silverman to show that computing the rank of these curves can be done by controlling the average of the traces of Frobenius, the construction for number fields proceeds in essentially the same way. One novelty of this method is that we can construct families of moderate rank without having to explicitly determine points and calculating determinants of height matrices.