A note on Galois embeddings of abelian varieties

Abstract

In this note we show that if an abelian variety possesses a Galois embedding into some projective space, then it must be isogenous to the self product of an elliptic curve. We prove moreover that the self product of an elliptic curve always has infinitely many Galois embeddings.

Erratum to: A note on Galois embeddings of abelian varieties

“…Proof. Since the quotient of A by G is smooth (and not an abelian variety), A must contain an elliptic curve E. This statement was proved in [3] and also in [1] by other means. Let…”

“…In particular, there is a subgroup of automorphisms G of A such that the quotient variety A/G is isomorphic to P d . The Main Theorem of [1] states the following in the case that A is an abelian variety:…”

“…Consider the following example which was brought to my attention by Bronson Lim: Let A = E 2 for an elliptic curve E, let p ∈ E[2] be a 2-torsion point on E, and let G be generated by (−x, y) and (x, y) → (x, −y). It is easy to see that A/G is singular (if t ∈ E is such that 2t = p, then (t, t) is fixed only by σ 1 σ 2 , which is not a pseudoreflection), but A/H P 1 × P 1 .…”

“…If X = A, then A is isogenous to E d and we are done. Assume then that X is a proper abelian subvariety, and consider the norm endomorphism N X ∈ End(A) of X with respect to the polarization L := π * O P d (1). Since X is H -invariant and the polarization is fixed by each element of H , we have that the complementary abelian subvariety of X with respect to L is also H -invariant.…”

Galois subspaces for the rational normal curve

“…The calculation of Galois subspaces for different varieties has been an object of interest in the past few years, and especially in the case of curves and abelian varieties. See, for instance, [Auf17], [Cuk99], [PS05], [Tak16], [Yos07] and the references therein. The purpose of this article is to classify all linear subspaces of P n that induce a Galois morphism for an embedding of P 1 into P n , thereby answering a question asked by Yoshihara in his list of open problems [Yos18].…”