Elliptic curves maximal over extensions of finite base fields

TL;DR

This paper investigates when elliptic curves over finite fields reach the maximum number of rational points over extensions, providing bounds on the extension degree and classifying such curves.

Contribution

It establishes new upper bounds on the extension degree for elliptic curves to attain the Hasse bound, and classifies isogeny classes for small degrees and large fields.

Findings

Upper bound on extension degree n for ordinary elliptic curves using logarithm estimates.

Improved upper bound n ≤ 11 for large q using Schmidt's Subspace Theorem.

Existence of infinitely many isogeny classes with n=3.

Abstract

Given an elliptic curve $E$ over a finite field $\mathbb{F}_q$ we study the finite extensions $\mathbb{F}_{q^n}$ of $\mathbb{F}_q$ such that the number of $\mathbb{F}_{q^n}$-rational points on $E$ attains the Hasse upper bound. We obtain an upper bound on the degree $n$ for $E$ ordinary using an estimate for linear forms in logarithms, which allows us to compute the pairs of isogeny classes of such curves and degree $n$ for small $q$. Using a consequence of Schmidt's Subspace Theorem, we improve the upper bound to $n\leq 11$ for sufficiently large $q$. We also show that there are infinitely many isogeny classes of ordinary elliptic curves with $n=3$.