4 problems

Malle's prediction. If $\operatorname{Aut}_{\overline{K}}(C)\cong D_4$, then

Let $p$ be prime and let $u\in\mathbb F_p$ be a parameter for which the 15 elliptic curves obtained from the five elliptic-curve factors and the shifts $u\mapsto u-22$ and…

Let $T\in A_5$, and let $C_T$ be the corresponding genus-two curve, with $r(T)$ denoting the rank quantity used in the source. Rational-points conjecture. For $r(T)>1$, the values…

Consider the density obtained by ordering genus-two curves of the form $y^2=f(x)$, where $f=f_6x^6+\dots+f_1x+f_0\in{\mathbb Z}[x]$, by the condition $\max\{|f_j|\}\leq N$, and let…