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We study the Hodge structure of elliptic surfaces which are canonicallydefined from bielliptic curves of genus three. We prove that the period map forthe second cohomology has one dimensional fibers, and the period map for thetotal cohomology is of degree twelve, and moreover, by adding the informationof the Hodge structure of the canonical divisor, we prove a generic Torellitheorem for these elliptic surfaces. Finally, we give explicit examples of thepair of non-isomorphic elliptic surfaces which have the same Hodge structure onthemselves and the same Hodge structure on their canonical divisors.

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