Elliptic curves on some homogeneous spaces

TL;DR

This paper proves that for large enough degree, the space of morphisms from an elliptic curve to certain homogeneous spaces is irreducible, providing explicit and often optimal bounds.

Contribution

It establishes irreducibility of morphism schemes for elliptic curves into specific homogeneous spaces with explicit degree bounds, extending known results.

Findings

Irreducibility holds for sufficiently large degrees

Explicit lower bounds for degree are provided

Bounds are optimal in many cases

Abstract

Let $X$ be a minuscule homogeneous space, an odd quadric, or an adjoint homogenous space of type different from $A$ and $G_2$. Le $C$ be an elliptic curve. In this paper, we prove that for $d$ large enough, the scheme of degree $d$ morphisms from $C$ to $X$ is irreducible, giving an explicit lower bound for $d$ which is optimal in many cases.