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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...

On the principle of real moduli flexibility: perfect parametrizations

Edoardo Ballico, Riccardo Ghiloni (2014)

Annales Polonici Mathematici

Let V be a real algebraic manifold of positive dimension. The aim of this paper is to show that, for every integer b (arbitrarily large), there exists a trivial Nash family = V y y R b of real algebraic manifolds such that V₀ = V, is an algebraic family of real algebraic manifolds over y R b 0 (possibly singular over y = 0) and is perfectly parametrized by R b in the sense that V y is birationally nonisomorphic to V z for every y , z R b with y ≠ z. A similar result continues to hold if V is a singular real algebraic set.

On the Pythagoras numbers of real analytic set germs

José F. Fernando, Jesús M. Ruiz (2005)

Bulletin de la Société Mathématique de France

We show that (i) the Pythagoras number of a real analytic set germ is the supremum of the Pythagoras numbers of the curve germs it contains, and (ii) every real analytic curve germ is contained in a real analytic surface germ with the same Pythagoras number (or Pythagoras number 2 if the curve is Pythagorean). This gives new examples and counterexamples concerning sums of squares and positive semidefinite analytic function germs.

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