Let X be a smooth scheme over a nite eld of characteristic p. Consider the coecient objects
of locally constant rank on X in l-adic Weil cohomology: these are lisse Weil sheaves in étale …

Let $ U $ be a smooth affine curve over a number field $ K $ with a compactification $ X $
and let ${\mathbb {L}} $ be a rank $2 $, geometrically irreducible lisse $\overline {{\mathbb …

Let $ X/\mathbb {C} $ be a smooth projective variety and let $ L $ be an irreducible
$\overline {\mathbb {Q}} _ {\ell} $-local system on $ X $ with torsion determinant. Suppose …

We show that only finitely many complex genus two curves and four punctured spheres
admit rank two local systems of geometric origin, and moreover each carries finitely many …

For each punctured curve over a finite field, we construct local systems which do not come
from a family of abelian varieties. We do so by proving a criterion which must be satisfied by …

Let X/F q be a smooth, geometrically connected variety. For X projective, we prove a
Lefschetz-style theorem for abelian schemes of GL 2-type on X, modeled after a theorem of …

We study the formal properties of correspondences of curves without a core, focusing on the
case of étale correspondences. The motivating examples come from Hecke …

In the study of algebraic varieties over number fields or finite fields, there are many
properties of these varieties that can be difficult to compute. This thesis discusses variants of …