Abelian surfaces, Siegel modular forms, and the Paramodularity Conjecture

Abstract

[en] This master’s thesis studies the modularity of elliptic curves over the rationals and two generalizations. The first is a theorem of Ribet based on Serre’s modularity conjecture, asserting that all abelian varieties of $\mathrm{G} \mathrm{L}_{2}$-type come from the Eichler-Shimura construction. The second is the Paramodularity Conjecture, which says that all abelian surfaces with trivial endomorphism ring have an associated Siegel paramodular form with coinciding $L$-function. We give background on abelian varieties, Galois representations and classical modular forms, all necessary to state modularity. Further, we explain the Eichler-Shimura construction and relation. We then study the basic theory of Siegel modular forms with respect to the paramodular group. The final chapter gives the statement of the Paramodularity Conjecture, along with a commentary of what a generalization to $\mathrm{GL}_{4}$-type abelian varieties could look like. An important part of this project is centered on explicit computation of Fourier-Siegel coefficients, and special care has been taken to present computational principles which are scattered across the literature. We also provide the first public implementation of the specialization method that was used to prove the first instance of the Paramodularity Conjecture.