AVERAGE SIZE OF 2-SELMER GROUPS OF JACOBIANS OF HYPERELLIPTIC CURVES OVER FUNCTION FIELDS

Abstract

Recently, the average size of Selmer groups of Jacobians of hyperelliptic curves over the rational field was studied extensively by M. Bhargava, B. Gross, Arul Shankar, Ananth Shankar, and X. Wang. In this thesis, I try to complete the picture by considering the similar problems in the function fields setting. Significantly, we develop a different method (compare to the one in the rational field setting) that could be seen as a generalization of the method used in [Ho.P. Quoc, Le Hung V. Bao, and Ngo B. Chau, Average size of 2-Selmer groups of elliptic curves over function fields, Math. Res. Lett. 21 (2014), no. 6, 1305–1339]. More precisely, we construct two moduli spaces M and A whose points represent elements in 2-Selmer groups and hyperelliptic curves respectively. Hence, the main remaining problem is to count points on fibers of the Hitchin map M --> A. This counting problem eventually can be solved by using Vinberg invariant theory and the parabolic canonical reduction theory of principal bundles.