## 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 obvious answer is no but maybe there is a different interpretation.