Integrality of Seshadri constants and irreducibility of principal polarizations on products of two isogenous elliptic curves

Abstract

AbstractIn this paper we consider the question of when all Seshadri constants on a product of two isogenous elliptic curves $$E_1\times E_2$$ E 1 × E 2 without complex multiplication are integers. By studying elliptic curves on $$E_1\times E_2$$ E 1 × E 2 we translate this question into a purely numerical problem expressed by quadratic forms. By solving that problem, we show that all Seshadri constants on $$E_1\times E_2$$ E 1 × E 2 are integers if and only if the minimal degree of an isogeny $$E_1\rightarrow E_2$$ E 1 → E 2 equals 1 or 2. Furthermore, this method enables a characterization of irreducible principal polarizations on $$E_1\times E_2$$ E 1 × E 2 .