Explicit Equations of Non-Hyperelliptic Genus 3 Curves with Real Multiplication by <strong>Q</strong>(&zeta;₇+&zeta;₇^-1)

• Main Author: Liang, Dun
• Other Authors: Lee, Hwang
• Format: Others
• Language: en
• Published: LSU 2014
• Subjects: Mathematics
• Online Access: http://etd.lsu.edu/docs/available/etd-11052014-135432/

<p>This thesis is devoted to proving the following:</p> <p>For all (u₁, u₂, u₃, u₄) in a Zariski dense open subset of <strong>C</strong>⁴ there is a genus 3 curve X(u₁, u₂, u₃, u₄) with the following properties:</p> <p>1. X(u₁, u₂, u₃, u₄) is not hyperelliptic.<br> 2. End(Jac((X(u₁, u₂, u₃, u₄))) &otimes;<strong>Q</strong> contains the real cubic field <strong>Q</strong>(&zeta;₇+&zeta;₇^-1) where &zeta;₇ is a primitive 7th root of unity.<br> 3. These curves X(u₁, u₂, u₃, u₄) define a three-dimensional subvariety of the moduli space of genus 3 curves M₃.<br> 4. The curve X(u₁, u₂, u₃, u₄) is defined over the field <strong>Q</strong>(u₁, u₂, u₃, u₄), and the endomorphisms are defined over <strong>Q</strong>(&zeta;₇, u₁, u₂, u₃, u₄).</p> <p>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) &isin; <strong>Q</strong>⁴. We calculate an example that shows that the corresponding representation of Gal(<span style="text-decoration: overline"><strong>Q</strong></span>/<strong>Q</strong>) is of GL₂-type, as is expected for curves with real multiplications by cubic number fields.</p>