Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userOn the Tate and Langlands--Rapoport conjectures for special fibres of integral canonical models of Shimura varieties of abelian type
369
0
0.0
(
0
)
Authors
Adrian Vasiu
Ask ChatGPT about the research
No Arabic abstract
We prove the isogeny property for special fibres of integral canonical models of compact Shimura varieties of $A_n$, $B_n$, $C_n$, and $D_n^{dbR}$ type. The approach used also shows that many crystalline cycles on abelian varieties over finite fields which are specializations of Hodge cycles, are algebraic. These two results have many applications. First, we prove a variant of the conditional Langlands--Rapoport conjecture for these special fibres. Second, for certain isogeny sets we prove a variant of the unconditional Langlands--Rapoport conjecture (like for many basic loci). Third, we prove that integral canonical models of compact Shimura varieties of Hodge type that are of $A_n$, $B_n$, $C_n$, and $D_n^{dbR}$ type, are closed subschemes of integral canonical models of Siegel modular varieties.
rate research
Read More
This is a survey of the three main methods developed in the last 15 years to prove the existence of integral canonical models of Shimura varieties of Hodge type. The only new part is formed by corrections to results of Kisin.
Let $(G,X)$ be a Shimura datum of Hodge type, and $mathscr{S}_K(G,X)$ its integral model with hyperspecial level structure. We prove that $mathscr{S}_K(G,X)$ admits a closed embedding, which is compatible with moduli interpretations, into the integral model $mathscr{S}_{K}(mathrm{GSp},S^{pm})$ for a Siegel modular variety. In particular, the normalization step in the construction of $mathscr{S}_K(G,X)$ is redundant. In particular, our results apply to the earlier integral models constructed by Rapoport and Kottwitz, as those models agree with the Hodge type integral models for appropriately chosen Shimura data.
Let $F$ be a totally real field in which a fixed prime $p$ is inert, and let $E$ be a CM extension of $F$ in which $p$ splits. We fix two positive integers $r,s in mathbb N$. We investigate the Tate conjecture on the special fiber of $G(U(r,s) times U(s,r))$-Shimura variety. We construct cycles which we conjecture to generate the Tate classes and verify our conjecture in the case of $G(U(1,s) times U(s,1))$. We also discuss the general conjecture regarding special cycles on the special fibers of unitary Shimura varieties.
We prove the Mumford--Tate conjecture for those abelian varieties over number fields whose extensions to C have attached adjoint Shimura varieties that are products of simple, adjoint Shimura varieties of certain Shimura types. In particular, we prove the conjecture for the orthogonal case (i.e., for the $B_n$ and $D_n^R$ Shimura types). As a main tool, we construct embeddings of Shimura varieties (whose adjoints are) of prescribed abelian type into unitary Shimura varieties of PEL type. These constructions implicitly classify the adjoints of Shimura varieties of PEL type.
We determine the behavior of automorphic Green functions along the boundary components of toroidal compactifications of orthogonal Shimura varieties. We use this analysis to define boundary components of special divisors and prove that the generating series of the resulting special divisors on a toroidal compactification is modular.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
