## Algebra & Number Theory

### Tate cycles on some unitary Shimura varieties mod $p$

#### Abstract

Let $F$ be a real quadratic field in which a fixed prime $p$ is inert, and $E 0$ be an imaginary quadratic field in which $p$ splits; put $E = E 0 F$. Let $X$ be the fiber over $F p 2$ of the Shimura variety for $G ( U ( 1 , n − 1 ) × U ( n − 1 , 1 ) )$ with hyperspecial level structure at $p$ for some integer $n ≥ 2$. We show that under some genericity conditions the middle-dimensional Tate classes of $X$ are generated by the irreducible components of its supersingular locus. We also discuss a general conjecture regarding special cycles on the special fibers of unitary Shimura varieties, and on their relation to Newton stratification.

#### Article information

Source
Algebra Number Theory, Volume 11, Number 10 (2017), 2213-2288.

Dates
Revised: 24 August 2017
Accepted: 28 September 2017
First available in Project Euclid: 1 February 2018

https://projecteuclid.org/euclid.ant/1517454184

Digital Object Identifier
doi:10.2140/ant.2017.11.2213

Mathematical Reviews number (MathSciNet)
MR3744356

Zentralblatt MATH identifier
06825450

#### Citation

Helm, David; Tian, Yichao; Xiao, Liang. Tate cycles on some unitary Shimura varieties mod $p$. Algebra Number Theory 11 (2017), no. 10, 2213--2288. doi:10.2140/ant.2017.11.2213. https://projecteuclid.org/euclid.ant/1517454184

#### References

• A. Caraiani, “Local-global compatibility and the action of monodromy on nearby cycles”, Duke Math. J. 161:12 (2012), 2311–2413.
• P. Di Francesco, “${\rm SU}(N)$ meander determinants”, J. Math. Phys. 38:11 (1997), 5905–5943.
• B. Fontaine, J. Kamnitzer, and G. Kuperberg, “Buildings, spiders, and geometric Satake”, Compos. Math. 149:11 (2013), 1871–1912.
• W. Fulton, Intersection theory, Second ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics 2, Springer, 1998.
• U. Görtz and X. He, “Basic loci of Coxeter type in Shimura varieties”, Camb. J. Math. 3:3 (2015), 323–353.
• B. H. Gross, “On the Satake isomorphism”, pp. 223–237 in Galois representations in arithmetic algebraic geometry (Durham, 1996), edited by A. J. Scholl and R. L. Taylor, London Math. Soc. Lecture Note Ser. 254, Cambridge Univ. Press, 1998.
• A. Grothendieck, Groupes de Barsotti–Tate et cristaux de Dieudonné (Séminaire de Mathématiques Supérieures, No. 45 Été, 1970), Les Presses de l'Université de Montréal, Montreal, Quebec, 1974.
• M. Harris and R. Taylor, The geometry and cohomology of some simple Shimura varieties, Annals of Mathematics Studies 151, Princeton University Press, Princeton, NJ, 2001. With an appendix by Vladimir G. Berkovich.
• D. Helm, “Towards a geometric Jacquet–Langlands correspondence for unitary Shimura varieties”, Duke Math. J. 155:3 (2010), 483–518.
• D. Helm, “A geometric Jacquet–Langlands correspondence for $U(2)$ Shimura varieties”, Israel J. Math. 187 (2012), 37–80.
• B. Howard and G. Pappas, “On the supersingular locus of the ${\rm GU}(2,2)$ Shimura variety”, Algebra Number Theory 8:7 (2014), 1659–1699.
• H. Jacquet and J. A. Shalika, “On Euler products and the classification of automorphic representations. I”, Amer. J. Math. 103:3 (1981), 499–558.
• M. Kisin, “Integral models for Shimura varieties of abelian type”, J. Amer. Math. Soc. 23:4 (2010), 967–1012.
• R. E. Kottwitz, “On the $\lambda$-adic representations associated to some simple Shimura varieties”, Invent. Math. 108:3 (1992), 653–665.
• R. E. Kottwitz, “Points on some Shimura varieties over finite fields”, J. Amer. Math. Soc. 5:2 (1992), 373–444.
• K.-W. Lan, Arithmetic compactifications of PEL-type Shimura varieties, London Mathematical Society Monographs Series 36, Princeton University Press, Princeton, NJ, 2013.
• B. Mazur and W. Messing, Universal extensions and one dimensional crystalline cohomology, Lecture Notes in Mathematics 370, Springer, 1974.
• J. S. Milne, “Canonical models of (mixed) Shimura varieties and automorphic vector bundles”, pp. 283–414 in Automorphic forms, Shimura varieties, and $L$-functions (Ann Arbor, MI, 1988), vol. I, edited by L. Clozel and J. S. Milne, Perspect. Math. 10, Academic Press, Boston, 1990.
• D. Mumford, Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics 5, Hindustan Book Agency, New Delhi, 2008.
• G. Pappas and M. Rapoport, “Local models in the ramified case. II. Splitting models”, Duke Math. J. 127:2 (2005), 193–250.
• M. Rapoport, U. Terstiege, and S. Wilson, “The supersingular locus of the Shimura variety for ${\rm GU}(1,n-1)$ over a ramified prime”, Math. Z. 276:3-4 (2014), 1165–1188.
• D. A. Reduzzi and L. Xiao, “Partial Hasse invariants on splitting models of Hilbert modular varieties”, Ann. Sci. Éc. Norm. Supér. $(4)$ 50:3 (2017), 579–607.
• J.-P. Serre, “Two letters on quaternions and modular forms (mod $p$)”, Israel J. Math. 95 (1996), 281–299. With introduction, appendix and references by R. Livné.
• M. Artin, A. Grothendieck, and J. Verdier, Théorie des topos et cohomologie étale des schémas. Tome 2 (Séminaire de Géométrie Algébrique du Bois-Marie 1963–1964), Lecture Notes in Mathematics 270, Springer, 1972.
• S.-W. Shin, “On the cohomological base change for unitary similitude groups”, (2014). appendix to W. Goldring, “Galois representations associated to holomorphic limits of discrete series I: Unitary Groups”, Compositio Math. 150:2 (2014), 191–228.
• J. Tate, “Endomorphisms of abelian varieties over finite fields”, Invent. Math. 2 (1966), 134–144.
• Y. Tian and L. Xiao, “Tate cycles on some quaternionic Shimura varieties mod $p$”, 2014.
• Y. Tian and L. Xiao, “On Goren–Oort stratification for quaternionic Shimura varieties”, Compos. Math. 152:10 (2016), 2134–2220.
• I. Vollaard and T. Wedhorn, “The supersingular locus of the Shimura variety of ${\rm GU}(1,n-1)$ II”, Invent. Math. 184:3 (2011), 591–627.
• P.-J. White, “Tempered automorphic representation of the unitary group”, preprint, 2012, http://preprints.ihes.fr/2011/M/M-11-17.pdf.
• L. Xiao and X. Zhu, “Cycles on Shimura varieties via geometric Satake”, 2017.
• X. Zhu, “Affine Grassmannians and the geometric Satake in mixed characteristic”, Ann. of Math. $(2)$ 185:2 (2017), 403–492.