Arithmetic of a fake projective plane and related elliptic surfaces

TL;DR

This paper explores the arithmetic properties of a specific fake projective plane and related elliptic surfaces, analyzing their construction via ball quotients and p-adic uniformization.

Contribution

It provides a new arithmetic perspective on Keum's fake projective plane and examines related elliptic surfaces through ball quotient and p-adic uniformization techniques.

Findings

Arithmetic description of Keum's fake projective plane

Analysis of two related elliptic surfaces

Connection between ball quotients and elliptic surface constructions

Abstract

The purpose of the present paper is to explain the fake projective plane constructed by J.H. Keum from the point of view of arithmetic ball quotients. Beside the ball quotient associated with the fake projective plane, we also analize two further naturally related ball quotients whose minimal desingularizations lead to two elliptic surfaces, one already considered by J.H. Keum as well as the one constructed by M.N. Ishida in terms of p-adic uniformization.