The exponent of an abelian subvariety (Q1184154)

The aim of this paper is to contribute to the study of Prym-Tyurin varieties. We recall that to a correspondence of a curve \(C\), satisfying the equation \(\varphi^ 2+(m-2)\varphi-(m-1)=0\), \(m\) a positive integer, it is associated an abelian subvariety of the Jacobian \(J(C)\). The authors prove that, given a polarized abelian variety \(X\), to any of its abelian subvarieties can be associated an equation as above, this gives a geometric interpretation of the exponent \(m\). -- In fact let \((X,L)\) be a polarized abelian variety, \(\varepsilon\) an idempotent of \(\text{End}_ \mathbb{Q}(X)\) symmetric with respect to the Rosati involution, and \(n\) the minimal positive integer such that \(n\varepsilon\) belongs to \(\text{End}(X)\), we denote by \(X^ \varepsilon\) the image of the endomorphism, then the correspondence \(\varepsilon\to X^ \varepsilon\) is a bijection between the set of symmetric idempotents of \(\text{End}_ \mathbb{Q}(X)\) and the set of abelian subvarieties of \(X\). When \(X\) is principally polarized, \(n\) can be interpreted as the exponent of \(X^ \varepsilon\) and once we set \(\varphi=1-n\varepsilon\) we obtain a solution for the equation. Moreover some properties of the exponents are studied, in particular it is obtained, as a consequence, de Franchis' theorem on the finite number of morphisms onto curves of genus \(g\geq 2\).