On endomorphisms of hypersurfaces

Abstract

For any prime $p\ge 5$, we show that generic hypersurface $X_p\subset {\mathbb{P}}^p$ defined over $\mathbb{Q}$ admits a non-trivial rational dominant self-map of degree $>1$, defined over $\bar{\mathbb{Q}}$. A simple arithmetic application of this fact is also given.

11 pages; the text has been revised following the referee's suggestions