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On the number of elliptic curves with CM cover large algebraic fields

Gerhard Frey, Moshe Jarden (2005)

Annales de l'institut Fourier

Elliptic curves with CM unveil a new phenomenon in the theory of large algebraic fields. Rather than drawing a line between 0 and 1 or 1 and 2 they give an example where the line goes beween 2 and 3 and another one where the line goes between 3 and 4 .

On the p-rank of an abelian variety and its endomorphism algebra.

Josep González (1998)

Publicacions Matemàtiques

Let A be an abelian variety defined over a finite field. In this paper, we discuss the relationship between the p-rank of A, r(A), and its endomorphism algebra, End0(A). As is well known, End0(A) determines r(A) when A is an elliptic curve. We show that, under some conditions, the value of r(A) and the structure of End0(A) are related. For example, if the center of End0(A) is an abelian extension of Q, then A is ordinary if and only if End0(A) is a commutative field. Nevertheless, we give an example...

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