# Hodge structure of K3 type with real multiplication and Simple Abelian Fourfolds with Definite Quaternionic Multiplication

## TL;DR

This paper constructs transcendental lattices for K3 surfaces with real multiplication up to degree 6 and demonstrates that certain Abelian fourfolds with quaternionic multiplication are Kuga-Satake varieties of these K3 surfaces.

## Contribution

It provides a general method for constructing transcendental lattices of K3 surfaces with real multiplication and links Abelian fourfolds with quaternionic multiplication to K3 surfaces via Kuga-Satake correspondence.

## Key findings

- Explicit formulas for discriminants of transcendental lattices.
- Realization of Abelian fourfolds as Kuga-Satake varieties of K3 surfaces.
- Connection between quaternionic multiplication and real multiplication in K3 surfaces.

## Abstract

In this paper we give a general construction of transcendental lattices for K3 surfaces with real multiplication by arbitrary field up to degree 6 along with formula for their discriminants. We also show that all simple Abelian fourfolds with definite quaternionic multiplication can be realized as Kuga-Satake varieties of K3 surfaces with Picard rank 16 and real multiplication by a quadratic field by keeping track of the arithmetic input on both sides.