scholarly journals On two mod p period maps: Ekedahl–Oort and fine Deligne–Lusztig stratifications

2022 ◽  
Author(s):  
Fabrizio Andreatta
Keyword(s):  
Flag Variety ◽  
Integral Model ◽  
Shimura Variety ◽  
Period Map ◽  
Mod P

AbstractConsider a Shimura variety of Hodge type admitting a smooth integral model S at an odd prime $$p\ge 5$$ p ≥ 5 . Consider its perfectoid cover $$S^{\text {ad}}(p^\infty )$$ S ad ( p ∞ ) and the Hodge–Tate period map introduced by Caraiani and Scholze. We compare the pull-back to $$S^{\text {ad}}(p^\infty )$$ S ad ( p ∞ ) of the Ekedahl–Oort stratification on the mod p special fiber of a toroidal compactification of S and the pull back to $$S^\text {ad}(p^\infty )$$ S ad ( p ∞ ) of the fine Deligne–Lusztig stratification on the mod p special fiber of the flag variety which is the target of the Hodge–Tate period map. An application to the non-emptiness of Ekedhal–Oort strata is provided.

scholarly journals Ekedahl-Oort Strata for Good Reductions of Shimura Varieties of Hodge Type

2018 ◽  
Vol 70 (2) ◽  
pp. 451-480 ◽  
Author(s):  
Chao Zhang
Keyword(s):  
Weyl Group ◽  
Moduli Spaces ◽  
Level Structure ◽  
Reductive Group ◽  
Extension Property ◽  
Integral Model ◽  
Shimura Varieties ◽  
Shimura Variety ◽  
Canonical Models ◽  
Special Fibers

AbstractFor a Shimura variety of Hodge type with hyperspecial level structure at a prime p, Vasiu and Kisin constructed a smooth integral model (namely the integral canonical model) uniquely determined by a certain extension property. We define and study the Ekedahl-Oort stratifications on the special fibers of those integral canonical models when p > 2. This generalizes Ekedahl-Oort stratifications defined and studied by Oort on moduli spaces of principally polarized abelian varieties and those defined and studied by Moonen, Wedhorn, and Viehmann on good reductions of Shimura varieties of PEL type. We show that the Ekedahl-Oort strata are parameterized by certain elements w in the Weyl group of the reductive group in the Shimura datum. We prove that the stratum corresponding to w is smooth of dimension l(w) (i.e., the length of w) if it is non-empty. We also determine the closure of each stratum.

scholarly journals Vanishing theorems for the mod p cohomology of some simple Shimura varieties

2020 ◽  
Vol 8 ◽  
Author(s):  
Teruhisa Koshikawa
Keyword(s):  
Additional Assumption ◽  
Hecke Algebra ◽  
Shimura Varieties ◽  
Shimura Variety ◽  
Vanishing Theorems ◽  
Low Dimensions ◽  
Eisenstein Ideal ◽  
Mod P

Abstract We show that the mod p cohomology of a simple Shimura variety treated in Harris-Taylor’s book vanishes outside a certain nontrivial range after localizing at any non-Eisenstein ideal of the Hecke algebra. In cases of low dimensions, we show the vanishing outside the middle degree under a mild additional assumption.

scholarly journals IMPROPER INTERSECTIONS OF KUDLA–RAPOPORT DIVISORS AND EISENSTEIN SERIES

2015 ◽  
Vol 16 (5) ◽  
pp. 899-945
Author(s):  
Siddarth Sankaran
Keyword(s):  
Eisenstein Series ◽  
Fourier Coefficients ◽  
Integral Model ◽  
Shimura Variety ◽  
Quadratic Number ◽  
Positive Dimension ◽  
Genus 2 ◽  
Imaginary Quadratic Number ◽  
Dimension 2 ◽  
Principal Polarization

We consider a certain family of Kudla–Rapoport cycles on an integral model of a Shimura variety attached to a unitary group of signature (1, 1), and prove that the arithmetic degrees of these cycles are Fourier coefficients of the central derivative of an Eisenstein series of genus 2. The integral model in question parameterizes abelian surfaces equipped with a non-principal polarization and an action of an imaginary quadratic number ring, and in this setting the cycles are degenerate: they may contain components of positive dimension. This result can be viewed as confirmation, in the degenerate setting and for dimension 2, of conjectures of Kudla and Kudla–Rapoport that predict relations between the intersection numbers of special cycles and the Fourier coefficients of automorphic forms.

Congruences involving alternating harmonic sums modulo pα qβ

2018 ◽  
Vol 68 (5) ◽  
pp. 975-980
Author(s):  
Zhongyan Shen ◽  
Tianxin Cai
Keyword(s):  
Positive Integer ◽  
Positive Integers ◽  
Mod P

Abstract In 2014, Wang and Cai established the following harmonic congruence for any odd prime p and positive integer r, $$\sum_{\begin{subarray}{c}i+j+k=p^{r}\\ i,j,k\in\mathcal{P}_{p}\end{subarray}}\frac{1}{ijk}\equiv-2p^{r-1}B_{p-3} \quad\quad(\text{mod} \,\, {p^{r}}),$$ where $ \mathcal{P}_{n} $ denote the set of positive integers which are prime to n. In this note, we obtain the congruences for distinct odd primes p, q and positive integers α, β, $$ \sum_{\begin{subarray}{c}i+j+k=p^{\alpha}q^{\beta}\\ i,j,k\in\mathcal{P}_{2pq}\end{subarray}}\frac{1}{ijk}\equiv\frac{7}{8}\left(2-% q\right)\left(1-\frac{1}{q^{3}}\right)p^{\alpha-1}q^{\beta-1}B_{p-3}\pmod{p^{% \alpha}} $$ and $$ \sum_{\begin{subarray}{c}i+j+k=p^{\alpha}q^{\beta}\\ i,j,k\in\mathcal{P}_{pq}\end{subarray}}\frac{(-1)^{i}}{ijk}\equiv\frac{1}{2}% \left(q-2\right)\left(1-\frac{1}{q^{3}}\right)p^{\alpha-1}q^{\beta-1}B_{p-3}% \pmod{p^{\alpha}}. $$

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