[Paper Review] The elliptic modular surface of level 4 and its reduction modulo 3

Ichiro Shimada|arXiv (Cornell University)|Jun 15, 2018

Algebraic Geometry and Number Theory34 references3 citations

TL;DR

This paper studies the elliptic modular surface of level 4 over a discrete valuation ring with residue characteristic 3, showing its reduction modulo 3 yields the Fermat quartic surface, a supersingular K3 surface. Using Borcherds' method and lattice embeddings, it computes the automorphism group of the family and proves the specialization of a fixed-point free involution yields an Enriques surface of type IV, confirming its geometric and arithmetic invariance under reduction.

ABSTRACT

The elliptic modular surface of level 4 is a complex K3 surface with Picard number 20. This surface has a model over a number field such that its reduction modulo 3 yields a surface isomorphic to the Fermat quartic surface in characteristic 3, which is supersingular. The specialization induces an embedding of the Néron-Severi lattices. Using this embedding, we determine the automorphism group of this K3 surface over a discrete valuation ring of mixed characteristic whose residue field is of characteristic 3. The elliptic modular surface of level 4 has a fixed-point free involution that gives rise to the Enriques surface of type IV in Nikulin-Kondo-Martin's classification of Enriques surfaces with finite automorphism group. We investigate the specialization of this involution to characteristic 3.

Motivation & Objective

• To compute the automorphism group of a family of K3 surfaces over a discrete valuation ring with residue characteristic 3.
• To analyze the reduction modulo 3 of the elliptic modular surface of level 4, showing it becomes isomorphic to the Fermat quartic surface.
• To study the specialization of a fixed-point free involution on the complex surface to characteristic 3 and determine the resulting Enriques surface type.
• To apply Borcherds' method to a family of K3 surfaces, marking the first such application to automorphism groups of families.

Proposed method

• Constructs a model of the elliptic modular surface of level 4 over a number ring with good reduction modulo 3.
• Uses the Néron-Severi lattice embedding induced by specialization to relate the geometric generic and special fibers.
• Applies Borcherds' method to compute automorphism groups of the complex and finite characteristic K3 surfaces.
• Analyzes the action of automorphisms on the Néron-Severi lattice and identifies the stabilizer of a chamber in the positive cone.
• Identifies six conjugate Enriques involutions in the Galois group of the level structure, using lattice-theoretic conditions on root systems.
• Verifies the specialization of the involution to characteristic 3 preserves fixed-point freeness and induces a configuration isomorphic to type IV Enriques surfaces.

Experimental results

Research questions

• RQ1 What is the automorphism group of the elliptic modular surface of level 4 over a discrete valuation ring with residue field of characteristic 3?
• RQ2 How does the reduction modulo 3 of this surface relate to the Fermat quartic surface in characteristic 3?
• RQ3 Does the fixed-point free involution on the complex surface specialize to a fixed-point free involution in characteristic 3?
• RQ4 Is the quotient of the reduced surface by the specialized involution an Enriques surface of type IV?
• RQ5 Can Borcherds' method be effectively applied to compute automorphism groups of families of K3 surfaces?

Key findings

• The reduction modulo 3 of the elliptic modular surface of level 4 is isomorphic to the Fermat quartic surface in characteristic 3, which is supersingular.
• The automorphism group of the family over the discrete valuation ring is computed via specialization and lattice embedding into the Néron-Severi lattice.
• Exactly six conjugate Enriques involutions exist in the automorphism group of the complex surface, all of type IV.
• The specialization of these involutions to characteristic 3 remains fixed-point free, and the quotient surface is an Enriques surface of type IV.
• The configuration of 20 rational curves on the quotient surface in characteristic 3 is isomorphic to the type IV configuration, confirming the surface type.
• The action of the involution on the fibers of the Jacobian fibration is identical in both characteristic 0 and 3, preserving the geometric structure.