@article{Representations:2989,
      recid = {2989},
      author = {Chidambaram, Shiva},
      title = {Moduli Spaces of Abelian Varieties Associated to mod-p  Galois Representations},
      publisher = {University of Chicago},
      school = {Ph.D.},
      address = {2021-06},
      pages = {88},
      abstract = {This thesis consists of four research papers stapled  together. In this work, we study moduli spaces of  principally polarised abelian varieties of dimension~$g >  1$ with~$p$-torsion structure for prime~$p$. In particular,  given a Galois representation~$\overline{\rho}:  G_{\mathbb{Q}} \rightarrow \mathrm{GSp}(2g,\mathbb{F}_p)$  with cyclotomic similitude character, we study various  rationality aspects of the  twist~$\mathcal{A}_g(\overline{\rho})$ of the Siegel  modular variety~$\mathcal{A}_g(p)$ of genus~$g$ and  level~$p$.

Using a description of the cohomology of the  compactification~$\mathcal{A}_2^*(3)$ given by Hoffman and  Weintraub, we show that the  variety~$\mathcal{A}_2(\overline{\rho})$ is not rational in  general. When~$\overline{\rho}$ is surjective, the minimal  degree of a rational cover is~$6$. Boxer, Calegari, Gee,  and Pilloni have shown the existence of a rational  cover~$\mathcal{A}_2^w(\overline{\rho})$ of degree~$6$. We  find explicit formulae parametrizing the pullback  $\mathcal{M}_2^w(\overline{\rho})$ of  $\mathcal{A}_2^w(\overline{\rho})$ under the Torelli map  $\mathcal{M}_2 \rightarrow \mathcal{A}_2$. This describes  the universal family of genus~$2$ curves with a rational  Weierstrass point, having fixed~$3$-torsion of Jacobian.  This exploits Shioda's work on Mordell-Weil lattices and  the invariant theory of the complex reflection group~$C_3  \times \mathrm{Sp}_4(\mathbb{F}_3)$. We also outline how  similar results can be obtained  for~$(g,p)=(2,2),(3,2),(4,2)$.

By making use of the  modularity lifting theorem for abelian surfaces proved by  Boxer, Calegari, Gee. and Pilloni, we produce some explicit  examples of modular abelian surfaces~$A$  with~$\mathrm{End}_{\mathbb{C}}(A) = \mathbb{Z}$. Using the  explicit formulae describing families of abelian surfaces  with fixed~$3$-torsion, and transferring modularity in the  family yields infinitely many such examples.

When~$g = 1$  and~$p > 5$, the existence of mod-$p$ Galois  representations not arising from elliptic curves  over~$\mathbb{Q}$ is known. For~$g > 1$ and~$(g,p) \ne  (2,2), (2,3), (3,2)$, we investigate a local obstruction to  the existence of rational points  on~$\mathcal{A}_g(\overline{\rho})$, and thus construct  Galois representations~$\overline{\rho}: G_{\mathbb{Q}}  \rightarrow \mathrm{GSp}(2g,\mathbb{F}_p)$ with cyclotomic  similitude character, that do not arise from  the~$p$-torsion of any~$g$-dimensional abelian variety  over~$\mathbb{Q}$. This is accomplished by solving  embedding problems with local conditions at suitably chosen  auxiliary primes~$l \ne p$, with the help of Galois  cohomological machinery.},
      url = {http://knowledge.uchicago.edu/record/2989},
      doi = {https://doi.org/10.6082/uchicago.2989},
}