Decompositions of Abelian surface and quadratic forms

TL;DR

This paper derives arithmetic formulas to determine the number of ways a complex Abelian surface can be decomposed into a product of two elliptic curves, enhancing understanding of its structural decompositions.

Contribution

It introduces explicit arithmetic formulas for counting decompositions of complex Abelian surfaces into elliptic curves, a novel approach in the field.

Findings

Derived formulas for counting decompositions

Quantified the number of decompositions for specific surfaces

Enhanced understanding of Abelian surface structures

Abstract

When a complex Abelian surface can be decomposed into a product of two elliptic curves, how many decompositions does the Abelian surface admit ? We provide arithmetic formulae for the number of decompositions of a complex Abelian surface.