Noncommutative geometry of elliptic surfaces

TL;DR

This paper explores the noncommutative geometric framework of elliptic surfaces over the projective line, linking noncommutative tori and continued fractions to analyze their algebraic properties.

Contribution

It introduces a novel approach connecting noncommutative geometry with elliptic surface theory, enabling new insights into their Picard numbers and ranks.

Findings

Calculated Picard numbers for elliptic surfaces with complex multiplication fibers

Established a correspondence between noncommutative tori and elliptic surfaces

Studied minimal models using noncommutative geometric methods

Abstract

We recast elliptic surfaces over the projective line in terms of the non-commutative tori and one-parameter families of the periodic continued fractions. The correspondence is used to study the Picard numbers, the ranks and the minimal models of such surfaces. As an example, we calculate the Picard numbers of elliptic surfaces with fibers having complex multiplication.