Geometrical Methods in Goursat Categories

The main aim of the paper is to show that the Little Desarguesian Theorem, the Escher Cube, Closure Lemma 1 and 3, hold in any regular Mal'tsev categories. We prove that Mal'tsev categories may be characterized through variations of the Little Desarguesian Theorem, the Escher Cube, Closure Lemma 1 and 3, that is classically expressed in terms of four congruences R, S1, S2 and T, and characterizes congruence modular varieties. The proof of this result in a varietal context may be obtained exclusively through the Little Desarguesian Theorem, the Escher Cube, Closure Lemma 1 and 3. This was shown by H.P. Gumm in Geometric Methods in Congruence Modular Algebras. We prove that for any 2n+1-permutable category $\mathcal{E}$, the category Equiv$(\mathcal{E})$ of equivalence relations in $\mathcal{E}$ is also a 2n+1-permutable category.

The Plectic Weight Filtration on Cohomology of Shimura Varieties and Partial Frobenius

We prove that there is a natural plectic weight filtration on the cohomology of Hilbert modular varieties in the spirit of Nekovář and Scholl. This is achieved with the help of Morel’s work on weight t-structures and a detailed study of partial Frobenius. We prove in particular that the partial Frobenius extends to toroidal and minimal compactifications.

The effective Shafarevich conjecture for abelian varieties of -type

In this article we establish the effective Shafarevich conjecture for abelian varieties over ${\mathbb Q}$ of ${\text {GL}_2}$ -type. The proof combines Faltings’ method with Serre’s modularity conjecture, isogeny estimates and results from Arakelov theory. Our result opens the way for the effective study of integral points on certain higher dimensional moduli schemes such as, for example, Hilbert modular varieties.

Newton Polygon Stratification of the Torelli Locus in Unitary Shimura Varieties

We study the intersection of the Torelli locus with the Newton polygon stratification of the modulo $p$ reduction of certain Shimura varieties. We develop a clutching method to show that the intersection of the open Torelli locus with some Newton polygon strata is non-empty. This allows us to give a positive answer, under some compatibility conditions, to a question of Oort about smooth curves in characteristic $p$ whose Newton polygons are an amalgamate sum. As an application, we produce infinitely many new examples of Newton polygons that occur for smooth curves that are cyclic covers of the projective line. Most of these arise in inductive systems that demonstrate unlikely intersections of the open Torelli locus with the Newton polygon stratification in Siegel modular varieties. In addition, for the 20 special Shimura varieties found in Moonen’s work, we prove that all Newton polygon strata intersect the open Torelli locus (if $p>>0$ in the supersingular cases).