On endomorphisms of hypersurfaces

TL;DR

This paper proves that for primes p ≥ 5, generic hypersurfaces in projective p-space over Q have non-trivial rational self-maps of degree greater than one, with implications for arithmetic properties.

Contribution

It establishes the existence of non-trivial rational self-maps on generic hypersurfaces in high-dimensional projective spaces over Q, a new result in algebraic geometry.

Findings

Existence of non-trivial rational self-maps for generic hypersurfaces in P^p over Q

Applicable arithmetic consequences derived from these maps

Results hold for primes p ≥ 5

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.