#### Identifier

etd-11052014-135432

#### Degree

Doctor of Philosophy (PhD)

#### Department

Mathematics

#### Document Type

Dissertation

#### Abstract

This thesis is devoted to proving the following:

For all (u_{1}, u_{2}, u_{3}, u_{4}) in a Zariski dense open subset of **C**^{4} there is a genus 3 curve X(u_{1}, u_{2}, u_{3}, u_{4}) with the following properties:

1. X(u_{1}, u_{2}, u_{3}, u_{4}) is not hyperelliptic.

2. End(Jac((X(u_{1}, u_{2}, u_{3}, u_{4}))) ⊗**Q** contains the real cubic field **Q**(ζ_{7}+ζ_{7}^{-1}) where ζ_{7} is a primitive 7th root of unity.

3. These curves X(u_{1}, u_{2}, u_{3}, u_{4}) define a three-dimensional subvariety of the moduli space of genus 3 curves M_{3}.

4. The curve X(u_{1}, u_{2}, u_{3}, u_{4}) is defined over the field **Q**(u_{1}, u_{2}, u_{3}, u_{4}), and the endomorphisms are defined over **Q**(ζ_{7}, u_{1}, u_{2}, u_{3}, u_{4}).

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(u_{1}, u_{2}, u_{3}, u_{4}) when (u_{1}, u_{2}, u_{3}, u_{4}) ∈ **Q**^{4}. We calculate an example that shows that the corresponding representation of Gal(**Q**/**Q**) is of GL_{2}-type, as is expected for curves with real multiplications by cubic number fields.

#### Date

2014

#### Document Availability at the Time of Submission

Release the entire work immediately for access worldwide.

#### Recommended Citation

Liang, Dun, "Explicit Equations of Non-Hyperelliptic Genus 3 Curves with Real Multiplication by **Q**(Î¶_{7}+Î¶_{7}^{-1})" (2014). *LSU Doctoral Dissertations*. 719.

http://digitalcommons.lsu.edu/gradschool_dissertations/719

#### Committee Chair

Hoffman, Jerome W