(Update: cancelled) Explicit uniform bounds for the Brauer groups of CM Kummer surfaces and Hilbert 10th problem
Francesca Balestrieri (IST Austria, Vienna)
Wednesday 4th March, 2020 16:00-17:00 Maths 311B
The following is joint work with Alexis Johnson and Rachel Newton. Let k be a number field and let E_1 and E_2 be elliptic curves over k with CM by orders O_1 and O_2 in some imaginary quadratic fields K_1 and K_2. Let X be the (twisted) Kummer surface over k associated to E_1 x E_2. Note that X is a K3 surface. Recent work by Skorobogatov and Orr shows that there is a uniform bound for the size of the Brauer group for K3 surfaces, even though their work is not effective. In talk, I will give explicit uniform bounds for the size of the Brauer group Br X/Br k in terms of the degree [k: Q] only. Using work by Kresch and Tschinkel, this implies that the Brauer-Manin set X(A_k)^Br is effectively computable, thus giving a positive answer to Hilbert 10th problem for the class of (twisted) Kummer surfaces associated to the product of two CM elliptic curves.