INTEGRAL AND ADELIC ASPECTS OF THE MUMFORD–TATE CONJECTURE

Let $Y$ be an abelian variety over a subfield $k\subset \mathbb{C}$ that is of finite type over $\mathbb{Q}$. We prove that if the Mumford–Tate conjecture for $Y$ is true, then also some refined integral and adelic conjectures due to Serre are true for $Y$. In particular, if a certain Hodge-maximality condition is satisfied, we obtain an adelic open image theorem for the Galois representation on the (full) Tate module of $Y$. We also obtain an (unconditional) adelic open image theorem for K3 surfaces. These results are special cases of a more general statement for the image of a natural adelic representation of the fundamental group of a Shimura variety.

A note on a theorem of Cadoret and Tamagawa

We prove Cadoret and Tamagawa’s open image theorem for curves defined over number fields using their arguments and the machinery of Ellenberg–Hall–Kowalski employed in their paper on expander graphs.

TWISTED GALOIS STRATIFICATION

We prove a direct image theorem stating that the direct image of a Galois formula by a morphism of difference schemes is equivalent to a Galois formula over fields with powers of Frobenius. As a consequence, we obtain an effective quantifier elimination procedure and a precise algebraic–geometric description of definable sets over fields with Frobenii in terms of twisted Galois formulas associated with finite Galois covers of difference schemes.

Uniform bounds for bounded geodesic image theorems

We give a universal bound for the bounded geodesic image theorem of Masur–Minsky. The proof uses elementary techniques. We also give a universal bound for a stronger version of subsurface projection, this demonstrates good control over many standard subsurface projections simultaneously.