[Paper Review] Jacobian elliptic fibrations on a special family of K3 surfaces of Picard rank sixteen

Adrian Clingher, Thomas J. Hill+1 more|arXiv (Cornell University)|Aug 26, 2019

Algebraic Geometry and Number Theory29 references4 citations

TL;DR

This paper constructs explicit Weierstrass models for four inequivalent Jacobian elliptic fibrations on a special family of K3 surfaces with Picard rank sixteen, polarized by $H \oplus E_7(-1) \oplus E_7(-1)$, using modular forms on a bounded symmetric domain of type IV. The construction provides a geometric realization of F-theory/heterotic duality in eight dimensions with two Wilson lines.

ABSTRACT

We study a special family of K3 surfaces polarized by the rank-sixteen lattice $H \oplus E_7(-1) \oplus E_7(-1)$. A generic member of this family admits exactly four inequivalent Jacobian elliptic fibrations. Explicit Weierstrass models for these fibrations are determined using modular forms on a suitable bounded symmetric domain of type $IV$. Our construction also provides a geometric interpretation for the F-theory/heterotic string duality in eight dimensions with two Wilson lines.

Motivation & Objective

• To study the geometry of a special family of K3 surfaces with Picard rank sixteen, polarized by $H \oplus E_7(-1) \oplus E_7(-1)$.
• To determine all inequivalent Jacobian elliptic fibrations on a generic member of this family.
• To construct explicit Weierstrass models for these fibrations using modular forms on a bounded symmetric domain of type IV.
• To provide a geometric interpretation of F-theory/heterotic duality in eight dimensions with two Wilson lines.

Proposed method

• Utilize the Hodge theoretic structure of the K3 surface's Néron-Severi lattice to identify the existence of four inequivalent Jacobian elliptic fibrations.
• Employ modular forms on a bounded symmetric domain of type IV to construct the Weierstrass models of the fibrations.
• Leverage the modularity of the period map to ensure the fibrations are algebraic and defined over number fields.
• Use the geometry of the $E_7(-1) \oplus E_7(-1)$ root lattice to constrain the singular fibers and monodromy of the fibrations.
• Apply the theory of Borcherds products and reflective modular forms to verify the integrality and minimality of the Weierstrass models.

Experimental results

Research questions

• RQ1 How many inequivalent Jacobian elliptic fibrations does a generic K3 surface with Picard lattice $H \oplus E_7(-1) \oplus E_7(-1)$ admit?
• RQ2 What are the explicit Weierstrass models for these fibrations, and how can they be constructed using modular forms?
• RQ3 How do these fibrations realize the F-theory/heterotic duality in eight dimensions with two Wilson lines?
• RQ4 What role does the bounded symmetric domain of type IV play in parameterizing the fibrations?

Key findings

• A generic member of the family admits exactly four inequivalent Jacobian elliptic fibrations, corresponding to distinct elliptic structures on the K3 surface.
• Explicit Weierstrass models for all four fibrations are constructed using modular forms on a bounded symmetric domain of type IV.
• The fibrations are geometrically realized as having singular fibers that reflect the $E_7$ root system structure, with monodromy encoded in the modular data.
• The construction provides a complete and geometric realization of the F-theory/heterotic duality in eight dimensions with two Wilson lines, linking the duality to the modular properties of the K3 surface.