Selmer groups of hyperelliptic curves over function fields and Invariant Vinberg theory

Thời gian: 09h00, thứ năm, ngày 5/1/2023.

Địa điểm: Phòng 507, nhà A6.

Tóm tắt: Let G be a split reductive group over a field k, theta in Aut(G) be an automorphism of exact order m >1, and let ζ ∈ k be a primitive mth root of unity. We also write θ for the induced automorphism of the Lie algebra g. Then we have the associated grading of g as follows:

gi ={x∈g|θ(x)=ζix}.

We write Gθ for the fixed subgroup of θ, and G0 for its connected component. Then LieG_0=g_0, so the notation is consistent. The action of G^{theta} on g leaves each gi invariant. Vinberg theory is the study of the representation of G_0 on g_1.

It was Bhargava, the first person, who related (coregular) Vinberg representations to some arithmetic properties of genus one curves. As a result, Bhargava-Shankar were able to compute the average size of m−Selmer groups of elliptic curves over Q. Later on, Bhargava-Gross considered the family of hyperelliptic curves and obtained the average size of the 2-Selmer group of Jacobians of such curves by using a similar method.

In this talk, I will present my work on an analogous problem as above over function fields. More precisely, I will compute the average size of 2-Selmer groups of the Jacobians of hyperelliptic curves over function fields. Unlike the number fields case, the method used is geometric and similar to the one which was used by N. B. Chau in his proof in The fundamental lemma. If time permits, I would like to discuss my current consideration on the case of non-hyperelliptic curves that correspond to the exceptional Vinberg representation.