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arXiv · 1911.10843

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

Maximilian Schmidt
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Abstract

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

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Maximilian Schmidt. 2019-11-25. Integrality of Seshadri constants and irreducibility of principal polarizations on products of two isogenous elliptic curves. https://arxiv.org/abs/1911.10843

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