Abstract
Let đŽ be a non-isotrivial almost ordinary abelian surface with possibly bad reductions over a global function field of odd characteristic đ.
Suppose Î is an infinite set of positive integers such that
Funding source: National Science Foundation
Award Identifier / Grant number: DMS-2100436
Funding statement: The author is partially supported by the NSF grant DMS-2100436.
Acknowledgements
The author thanks Ananth Shankar for suggesting this problem and his help, thanks Asvin G, Qiao He, Jiaqi Hou, and Salim Tayou for valuable discussions, and thanks Jordan Ellenberg for pointing out some imprecision in the introduction. The author also thanks Keerthi Madapusi Pera for answering a question on toroidal compactifications and Martin Olsson for answering a question on log geometry.
References
[1] F. Andreatta, E.âZ. Goren, B. Howard and K. Madapusi Pera, Faltings heights of abelian varieties with complex multiplication, Ann. of Math. (2) 187 (2018), no. 2, 391â531. 10.4007/annals.2018.187.2.3Search in Google Scholar
[2] M. Baker, S.-I. Ih and R. Rumely, A finiteness property of torsion points, Algebra Number Theory 2 (2008), no. 2, 217â248. 10.2140/ant.2008.2.217Search in Google Scholar
[3] G.âA. Boxer, Torsion in the coherent cohomology of Shimura varieties and Galois representations, Ph.D. Thesis, Harvard University, 2015. Search in Google Scholar
[4] J.âH. Bruinier, Borcherds products with prescribed divisor, Bull. Lond. Math. Soc. 49 (2017), no. 6, 979â987. 10.1112/blms.12090Search in Google Scholar
[5] J.âH. Bruinier and M. Kuss, Eisenstein series attached to lattices and modular forms on orthogonal groups, Manuscripta Math. 106 (2001), no. 4, 443â459. 10.1007/s229-001-8027-1Search in Google Scholar
[6] J.âH. Bruinier and S. Zemel, Special cycles on toroidal compactifications of orthogonal Shimura varieties, Math. Ann. 384 (2022), no. 1â2, 309â371. 10.1007/s00208-021-02271-xSearch in Google Scholar
[7] C. Chai, Families of ordinary Abelian varieties: Canonical coordinates, đ-adic monodromy, Tate-linear subvarieties and Hecke orbits, (2003), https://www.math.upenn.edu/~chai/papers_pdf/fam_ord_av.pdf. Search in Google Scholar
[8] N. Chavdarov, The generic irreducibility of the numerator of the zeta function in a family of curves with large monodromy, Duke Math. J. 87 (1997), no. 1, 151â180. 10.1215/S0012-7094-97-08707-XSearch in Google Scholar
[9] A.âJ. de Jong, Crystalline DieudonnĂ© module theory via formal and rigid geometry, Publ. Math. Inst. Hautes Ătudes Sci. 82 (1995), 5â96. 10.1007/BF02698637Search in Google Scholar
[10] A.âJ. de Jong, Homomorphisms of BarsottiâTate groups and crystals in positive characteristic, Invent. Math. 134 (1998), no. 2, 301â333. 10.1007/s002220050266Search in Google Scholar
[11] A.âJ. de Jong and W. Messing, Crystalline DieudonnĂ© theory over excellent schemes, Bull. Soc. Math. France 127 (1999), no. 2, 333â348. 10.24033/bsmf.2351Search in Google Scholar
[12] G. Faltings, Integral crystalline cohomology over very ramified valuation rings, J. Amer. Math. Soc. 12 (1999), no. 1, 117â144. 10.1090/S0894-0347-99-00273-8Search in Google Scholar
[13] G. Faltings and C.-L. Chai, Degeneration of abelian varieties, Ergeb. Math. Grenzgeb. (3) 22, Springer, Berlin 1990. 10.1007/978-3-662-02632-8Search in Google Scholar
[14] A. Grothendieck, ĂlĂ©ments de gĂ©omĂ©trie algĂ©brique. IV. Ătude locale des schĂ©mas et des morphismes de schĂ©mas. II, Publ. Math. Inst. Hautes Ătudes Sci. 24 (1965),1â2231. 10.1007/BF02684322Search in Google Scholar
[15] J. Hanke, Local densities and explicit bounds for representability by a quadratric form, Duke Math. J. 124 (2004), no. 2, 351â388. 10.1215/S0012-7094-04-12424-8Search in Google Scholar
[16] B. Howard and K. Madapusi Pera, Arithmetic of Borcherds products, AstĂ©risque 421 (2020), 187â297. 10.24033/ast.1128Search in Google Scholar
[17] B. Howard and G. Pappas, RapoportâZink spaces for spinor groups, Compos. Math. 153 (2017), no. 5, 1050â1118. 10.1112/S0010437X17007011Search in Google Scholar
[18] T. Kajiwara, K. Kato and C. Nakayama, Logarithmic abelian varieties, Nagoya Math. J. 189 (2008), 63â138. 10.1017/S002776300000951XSearch in Google Scholar
[19] K. Kato, Logarithmic structures of FontaineâIllusie, Algebraic analysis, geometry, and number theory (Baltimore 1988), Johns Hopkins University, Baltimore (1989), 191â224. Search in Google Scholar
[20] K. Kato, Logarithmic Dieudonné theory, preprint (2023), https://arxiv.org/abs/2306.13943. Search in Google Scholar
[21] M. Kisin, Integral models for Shimura varieties of abelian type, J. Amer. Math. Soc. 23 (2010), no. 4, 967â1012. 10.1090/S0894-0347-10-00667-3Search in Google Scholar
[22] S.âS. Kudla and M. Rapoport, Cycles on Siegel threefolds and derivatives of Eisenstein series, Ann. Sci. Ăc. Norm. SupĂ©r. (4) 33 (2000), no. 5, 695â756. 10.1016/S0012-9593(00)01051-XSearch in Google Scholar
[23] K.-W. Lan, Arithmetic compactifications of PEL-type Shimura varieties, London Math. Soc. Monogr. Ser. 36, Princeton University, Princeton 2013. 10.23943/princeton/9780691156545.001.0001Search in Google Scholar
[24] K. Madapusi Sampath, Toroidal compactifications of integral models of Shimura varieties of Hodge type, Ph.D. Thesis, The University of Chicago, 2011, Search in Google Scholar
[25] K. Madapusi Pera, Integral canonical models for spin Shimura varieties, Compos. Math. 152 (2016), no. 4, 769â824. 10.1112/S0010437X1500740XSearch in Google Scholar
[26] K. Madapusi Pera, Toroidal compactifications of integral models of Shimura varieties of Hodge type, Ann. Sci. Ăc. Norm. SupĂ©r. (4) 52 (2019), no. 2, 393â514. 10.24033/asens.2391Search in Google Scholar
[27] D. Maulik, A.âN. Shankar and Y. Tang, Picard ranks of K3 surfaces over function fields and the Hecke orbit conjecture, Invent. Math. 228 (2022), no. 3, 1075â1143. 10.1007/s00222-022-01097-xSearch in Google Scholar
[28] D. Maulik, A.âN. Shankar and Y. Tang, Reductions of abelian surfaces over global function fields, Compos. Math. 158 (2022), no. 4, 893â950. 10.1112/S0010437X22007473Search in Google Scholar
[29] B. Moonen, Linearity properties of Shimura varieties. I, J. Algebraic Geom. 7 (1998), no. 3, 539â567. Search in Google Scholar
[30] B. Moonen, Models of Shimura varieties in mixed characteristics, Galois representations in arithmetic algebraic geometry (Durham 1996), London Math. Soc. Lecture Note Ser. 254, Cambridge University, Cambridge (1998), 267â350. 10.1017/CBO9780511662010.008Search in Google Scholar
[31] V.âK. Murty and V.âM. Patankar, Splitting of abelian varieties, Int. Math. Res. Not. IMRN 2008 (2008), no. 12, Article ID rnn033. Search in Google Scholar
[32]
A. Ogus,
Singularities of the height strata in the moduli of
[33] A.âN. Shankar and Y. Tang, Exceptional splitting of reductions of abelian surfaces, Duke Math. J. 169 (2020), no. 3, 397â434. 10.1215/00127094-2019-0046Search in Google Scholar
[34] A. Shiho, Crystalline fundamental groups. I. Isocrystals on log crystalline site and log convergent site, J. Math. Sci. Univ. Tokyo 7 (2000), no. 4, 509â656. Search in Google Scholar
[35] S. Tayou, Picard rank jumps for K3 surfaces with bad reduction, Algebra Number Theory 19 (2025), 10.2140/ant.2025.19.77. 10.2140/ant.2025.19.77Search in Google Scholar
[36] M. WĂŒrthen and H. Zhao, Log đ-divisible groups associated with log 1-motives, Canad. J. Math. 76 (2024), no. 3, 946â983. 10.4153/S0008414X23000287Search in Google Scholar
[37] T. Zink, On the slope filtration, Duke Math. J. 109 (2001), no. 1, 79â95. 10.1215/dmj/996987491Search in Google Scholar
[38] D. Zywina, The splitting of reductions of an abelian variety, Int. Math. Res. Not. IMRN 2014 (2014), no. 18, 5042â5083. 10.1093/imrn/rnt113Search in Google Scholar
[39] The Stacks project authors, Stacks project, https://stacks.math.columbia.edu, 2018. Search in Google Scholar
© 2025 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- Irreducible symplectic varieties with a large second Betti number
- Splitting of almost ordinary abelian surfaces in families and the đ-integrality conjectures
- Applications of perverse sheaves in commutative algebra
- Dualizing complexes on the moduli of parabolic bundles
- Dynamics of convex mean curvature flow
- ColdingâMinicozzi entropies in CartanâHadamard manifolds
- Asymptotic directions in the moduli space of curves
- Existence of arithmetic degrees for generic orbits and dynamical LangâSiegel problem
Articles in the same Issue
- Frontmatter
- Irreducible symplectic varieties with a large second Betti number
- Splitting of almost ordinary abelian surfaces in families and the đ-integrality conjectures
- Applications of perverse sheaves in commutative algebra
- Dualizing complexes on the moduli of parabolic bundles
- Dynamics of convex mean curvature flow
- ColdingâMinicozzi entropies in CartanâHadamard manifolds
- Asymptotic directions in the moduli space of curves
- Existence of arithmetic degrees for generic orbits and dynamical LangâSiegel problem