Related papers: Finiteness of reductions of Hecke orbits

The purpose of this paper is to prove a local p-adic monodromy theorem for ordinary abelian surfaces and K3 surfaces with bad reduction in characteristic p. As an application, we get a finiteness result for the reduction of their Hecke…

We study abelian varieties and K3 surfaces with complex multiplication defined over number fields of fixed degree. We show that these varieties fall into finitely many isomorphism classes over an algebraic closure of the field of rational…

We prove the ordinary Hecke orbit conjecture for Shimura varieties of Hodge type at primes of good reduction. We make use of the global Serre-Tate coordinates of Chai as well as recent results of D'Addezio about the $p$-adic monodromy of…

For a point $x_0$ in a Shimura variety attached to a Shimura datum of Hodge type $(G,X)$, we have an associated abelian scheme $A_0$. Fixing a non-empty finite set $\mathcal{S}$ of primes, we consider the simultaneous supersingular…

Let $C/k$ be a smooth curve over a finite field of characteristic $p>0$. We prove that there are finitely many principally polarized abelian schemes of given dimension $g$ over $C$ up to $p$-power isogeny. For curves over $\overline{k}$, we…

We show that up to potential isogeny, there are only finitely many abelian varieties of dimension $d$ defined over a number field $K$, such that for any finite place $v$ outside a fixed finite set $S$ of places of $K$ containing the…

Faltings proved that there are finitely many abelian varieties of genus $g$ over a number field $K$, with good reduction outside a finite set of primes $S$. Fixing one of these abelian varieties $A$, we prove that there are finitely many…

We prove a uniform version of a finiteness conjecture due to Rasmussen and Tamagawa in the case of CM abelian varieties. This extends recent results concerning CM elliptic curves to CM abelian varieties of arbitrary dimension.

In this paper we prove that the cohomology groups with compact support of stacks of shtukas are modules of finite type over a Hecke algebra. As an application, we extend the construction of excursion operators, defined by V. Lafforgue on…

Using integral $p$-adic Hodge theory, Kato and Koshikawa define a generalization of the Faltings height of an abelian variety to motives defined over a number field. Assuming the adelic Mumford-Tate conjecture, we prove a finiteness…

In this paper, we study a conjecture of Rasmussen and Tamagawa, on the finiteness of the set of isomorphism classes of abelian varieties with constrained prime power torsion. Our result is related with abelian varieties which have complex…

We prove the Hecke orbit conjecture of Chai--Oort for Shimura varieties of Hodge type at odd primes of good reduction. We use a novel result for the local monodromy groups of $F$-isocrystals "coming from geometry", which refines Crew's…

We show that there are uncountably many algebraic extensions of $\mathbb{Q}$ containing at most finitely many moduli of CM simple principally polarized abelian varieties of any fixed dimension $g\geqslant1$, generalizing a result of…

Recently S. Patrikis, J.F. Voloch and Y. Zarhin have proven, assuming several well known conjectures, that the finite descent obstruction holds on the moduli space of principally polarised abelian varieties. We show an analogous result for…

Let k be an algebraically closed field and A be a finitely generated, centrally finite, non- negatively graded (not necessarily commutative) k-algebra. In this note we construct a representation scheme for graded maximal Cohen-Macaulay A…

For a semi-stable abelian variety A_K over a complete discrete valuation field K, we show that every finite subgroup scheme of A_K extends to a log finite flat group scheme over the valuation ring of K endowed with the canonical log…

The finite monodromy groups of abelian varieties over number fields have been introduced by Grothendieck. They represent the local obstruction to semi-stable reduction. In this paper we prove a criteria for finite groups to be realized as…

Over an algebraically closed field, various finiteness results are known regarding the automorphism group of a K3 surface and the action of the automorphisms on the Picard lattice. We formulate and prove versions of these results over…

We extend to large contexts pertaining to Shimura varieties of Hodge type a result of Zink on the existence of lifts to characteristic 0 of suitable representatives of certain isogeny classes of abelian varieties endowed with Frobenius and…

Let A be an abelian variety defined over a number field F. For a prime number $\ell$, we consider the field extension of F generated by the $\ell$-powered torsion points of A. According to a conjecture made by Rasmussen and Tamagawa, if we…