Abstract
This thesis is devoted to proving the following: For all (u1, u2, u3, u4) in a Zariski dense open subset of C4 there is a genus 3 curve X(u1, u2, u3, u4) with the following properties: 1. X(u1, u2, u3, u4) is not hyperelliptic. 2. End(Jac((X(u1, u2, u3, u4))) ⊗Q contains the real cubic field Q(ζ7+ζ7-1) where ζ7 is a primitive 7th root of unity. 3. These curves X(u1, u2, u3, u4) define a three-dimensional subvariety of the moduli space of genus 3 curves M3. 4. The curve X(u1, u2, u3, u4) is defined over the field Q(u1, u2, u3, u4), and the endomorphisms are defined over Q(ζ7, u1, u2, u3, u4). This theorem is a joint result of J. W. Hoffman, Ryotaro Okazaki,Yukiko Sakai, Haohao Wang and Zhibin Liang. My contribution to this project is the following: (1) Verification of property 3 above. This is accomplished in two ways. One utilizes the Igusa invariants of genus 2 curves. The other uses deformation theory, especially variations of Hodge structures of smooth hypersurfaces. (2) We also give an application to the zeta function of the curve X(u1, u2, u3, u4) when (u1, u2, u3, u4) ∈ Q4. We calculate an example that shows that the corresponding representation of Gal(Q/Q) is of GL2-type, as is expected for curves with real multiplications by cubic number fields.
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