# Bielliptic curves of genus three and the Torelli problem for certain elliptic surfaces

## TL;DR

This paper investigates the Hodge structures of elliptic surfaces derived from bielliptic genus three curves, establishing a Torelli theorem and providing explicit examples of non-isomorphic surfaces with identical Hodge structures.

## Contribution

It proves a generic Torelli theorem for these elliptic surfaces and analyzes the degree of their period maps, advancing understanding of their Hodge-theoretic properties.

## Key findings

- The period map for second cohomology has one-dimensional fibers.
- The period map for total cohomology is of degree twelve.
- Explicit examples of non-isomorphic surfaces with identical Hodge structures.

## Abstract

We study the Hodge structure of elliptic surfaces which are canonically defined from bielliptic curves of genus three. We prove that the period map for the second cohomology has one dimensional fibers, and the period map for the total cohomology is of degree twelve, and moreover, by adding the information of the Hodge structure of the canonical divisor, we prove a generic Torelli theorem for these elliptic surfaces. Finally, we give explicit examples of the pair of non-isomorphic elliptic surfaces which have the same Hodge structure on themselves and the same Hodge structure on their canonical divisors.