Event Details

Speaker: Thais Gomes Ribeiro (University of Birmingham) Title: Hypergeometric structures of a family of K3 surfaces

Abstract: Let k be a field and ψ ∈ k. Consider the family of K3 surfaces in ℙₖ³: X_ψ: x³y+y^4+z³w+zw³-12ψxyzw=0.

We study this family from both arithmetic and complex geometric point of view. On the arithmetic side, that is, when k=𝔽_q, for q a power of a prime, we give explicit formulas for the number of points #X(𝔽_q) in terms of finite field hypergeometric sums.

On the complex side, that is, when k=ℂ, we compute some solutions of the Picard-Fuchs equation of the family and show that they are expressed in terms of hypergeometric series whose parameters are closely related to the parameters in the finite field hypergeometric sums mentioned before.

In the talk, I will introduce the concepts mentioned in the previous paragraphs and explore the results we obtained. This is a work (in progress) in collaboration with Adriana Salerno, Eli Orvis, Jessamin Dukes, Leah Sturman, Rachel Davis and Ursula Whitcher.