On Hilbert modular threefolds of discriminant 49

TL;DR

This paper explores the geometry of Hilbert modular threefolds associated with a specific cubic field of discriminant 49, revealing an octic surface with numerous singular points, advancing understanding of their geometric properties.

Contribution

It introduces the geometric structure of Hilbert modular threefolds for discriminant 49 and identifies a special octic surface with 84 A_2 singularities.

Findings

Discovery of an octic in P^3 with 84 A_2 singular points

Detailed analysis of the geometry of Hilbert modular threefolds for discriminant 49

Identification of related varieties and their singularities

Abstract

Let K be the totally real cubic field of discriminant 49, let O be its ring of integers, and let p be the prime over 7. Let Gamma (p)\subset Gamma = SL_2(O) be the principal congruence subgroup of level p. This paper investigates the geometry of the Hilbert modular threefold attached to Gamma (p) and some related varieties. In particular, we discover an octic in P^3 with 84 isolated singular points of type A_2.