Related papers: Ax-Schanuel for Shimura varieties

In the context of extriangulated categories, we establish the injective version of Schanuel's lemma in homological algebra.

We suggest to look at formal sentences describing complex algebraic varieties together with their universal covers as topological invariants. We prove that for abelian varieties and Shimura varieties this is indeed a complete invariant,…

We construct explicit generating series of arithmetic extensions of Kudla's special divisors on integral models of unitary Shimura varieties over CM fields with arbitrary split levels and prove that they are modular forms valued in the…

Let $Z$ be an algebraic subvariety of a Shimura variety. We extend results of the first author to prove an effective upper bound for the degree of a non-facteur maximal special subvariety of $Z$.

We present a proof of the Chevalley-Weil Theorem that is somewhat different from the proofs appearing in the literature and with somewhat weaker hypotheses, of purely topological type. We also provide a discussion of the assumptions, and an…

We prove the Shafarevich conjecture for varieties with globally generated cotangent bundle, subject to mild numerical conditions.

We show that the compactly supported cohomology of Shimura varieties of Hodge type of infinite $\Gamma_1(p^\infty)$-level (defined with respect to a Borel subgroup) vanishes above the middle degree, under the assumption that the group of…

We prove that in the unramified case, local models of Shimura varieties with Iwahori level structure are normal and Cohen Macaulay.

We give a short and self-contained proof of the Decomposition Theorem for the non-small resolution of a Special Schubert variety. We also provide an explicit description of the perverse cohomology sheaves. As a by-product of our approach,…

We construct functorial Igusa stacks for all Hodge-type Shimura varieties, proving a conjecture of Scholze and extending earlier results of the fourth-named author for PEL-type Shimura varieties. Using the Igusa stack, we construct a sheaf…

We provide a proof of a variant of the Landau-Siegel Zeros conjecture.

We give an explicit formula for the zeroth $\mathbb{A}^1$-homology sheaf of a smooth proper variety. We also provide a simple proof of a theorem of Kahn-Sujatha which describes hom sets in the birational localization of the category of…

I give a model-theoretic setting for the modular $j$ function and its derivatives. These structures, here called $j$-fields, provide an adequate setting for interpreting the Ax-Schanuel theorem for $j$ (Pila-Tsimerman 2015). Following the…

Much of the work on Shimura varieties over the last thirty years has been devoted to constructing the theory that would follow from a good notion of motives, one incorporating the Hodge, Tate, and standard conjectures. These conjectures are…

Under simplifying hypotheses we prove a relation between the l-adic cohomology of the basic stratum of a Shimura variety of PEL-type modulo a prime of good reduction of the reflex field and the cohomology of the complex Shimura variety. In…

For a very general product $A$ of seven or more elliptic curves, every rational curve on the Kummer variety of $A$ projects trivially onto the Kummer variety of at least one of its factors. As a consequence, a very general member of certain…

In this note, we study the superspecial loci of orthogonal type Shimura varieties of signature (n-2, 2) with n>3. We prove a conjecture of Gross on the parametrizations of the superspecial locus in the special fiber of an orthogonal type…

Let X be a Mumford-Tate variety, i.e., a quotient of a Mumford-Tate domain D by a discrete subgroup. Mumford-Tate varieties are generalizations of Shimura varieties. We define the notion of a special subvariety Y in X (of Shimura type), and…

Let $X$ be the germ of a smooth complex variety at a given point $x\in {\mathbbb P}^N$ with regular osculation at order $q$ and suppose that, for any direction $v\in {\mathbbb P}T_xX$, there exists a rationnal normal curve locally contained…

Let k be a perfect field of characteristic p>0. We prove the existence of ascending and descending slope filtrations for Shimura p-divisible objects over k. We use them to classify rationally these objects over \bar k. Among geometric…