Endomorphism Algebras of Abelian Varieties with Special Reference to Superelliptic Jacobians

Abstract

This is (mostly) a survey article. We use an information about Galois properties of points of small order on an abelian variety in order to describe its endomorphism algebra over an algebraic closure of the ground field. We discuss in detail applications to jacobians of cyclic covers of the projective line.

Endomorphisms of ordinary superelliptic Jacobians

“…• Taking into account that the representation theory of G over Q p is "the same over F p as over Q p ([22, Sect. 15.5, Prop.43], [13]), we conclude that the F p [G]-module (F B p ) 0 is absolutely simple (see also [32,Cor. 7.5 on p. 513]).…”

“…We assume that n ≥ 5 and either q | n or p does not divide n. Remark 8.1. One may define a positive-dimensional abelian subvariety J (f,q) := P q/p (δ q )(J (f,q) ) of J(C f,q ) [28, p. 355] that is defined over K(ζ q ) and enjoys the following properties [28] (see also [32]).…”

“…(viii) If p is odd and Q[δ q ] is a maximal commutative subalgebra of End 0 (J (f,q) ) then End 0 (J (f,q) ) = Q[δ q ], End(J (f,q) ) = Z[δ q ] ([28, Th. 4.16], [32,Th. 8.3]).…”

On torsion of superelliptic Jacobians

“…There is a vast literature concerning various properties of hyper-and superelliptic Jacobians, for instance their endomorphism rings (cf. [61], [59], [60], [49], [58]), their Hodge groups (cf. [52], [50], [51], [48]), their rational points (cf.…”