Questions

A list of questions that I am interested in. I have no idea how hard they are. If you have an idea, please send me a message.

Are all geometrically rational Hilbert modular surfaces birational to P2 over Q?

This seems to be true for those computed explicitly by Elkies and Kumar. Is there a reason forcing them to be rational?

Do K3 surfaces with exactly one genus 1 fibration satisfy the potential Hilbert Property?

This is expected. Maybe some of these admit covers by K3 surfaces with multiple elliptic fibrations, being amenable to a descent argument.

Can the Hilbert Property for non-unirational varieties be proven by counting?

For Pn, Cohen proved an upper bound counting points in thin sets, which leads to an analytic proof of Hilbert's Irreducibility Theorem.

Does M23 act faithfully on a family of K3 surfaces over Q?

The automorphism groups of K3s are all subgroups of M23 with at least 5 orbits.

What is the middle l-adic cohomology of diagonal hypersurfaces as a Galois module over Q?

For diagonal equations of degree n, the Galois action over the n-th cyclotomic field Q(mu_n) can be computed à la Weil. What is missing is the cyclotomic action (except when Q(mu_n) is quadratic).

Are the projective diagonal quartic surfaces x^4+y^4=2z^4-2w^4 and x^4+y^4=8z^4-8w^4 isomorphic over Q?

These are the only diagonal quartic equations with a non-trivial transcendental Brauer class of order 2. They become isomorphic over Q(i). Their Picard groups are isomorphic as Galois modules. Their middle cohomologies are isomorphic with Q_2 coefficients and with Z_l coefficients for all odd primes l.

Is there an explicit description of universal torsors of del Pezzo surfaces as intersections of flag varieties?

The universal torsor can be described as an intersection of scaled copies of G/P under the Segre embedding. However, the construction of the locus of scaling factors is inductive.

Is the obstruction to the local-global principle for Apollonian Circle Packings explained by a Brauer class?

The prima facie answer is no but maybe there is a different interpretation.