Duke Mathematical Journal

Exceptional splitting of reductions of abelian surfaces

Ananth N. Shankar and Yunqing Tang

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Abstract

Heuristics based on the Sato–Tate conjecture and the Lang–Trotter philosophy suggest that an abelian surface defined over a number field has infinitely many places of split reduction. We prove this result for abelian surfaces with real multiplication. As in previous work by Charles and Elkies, this shows that a density 0 set of primes pertaining to the reduction of abelian varieties is infinite. The proof relies on the Arakelov intersection theory on Hilbert modular surfaces.

Article information

Source
Duke Math. J., Volume 169, Number 3 (2020), 397-434.

Dates
Received: 5 March 2018
Revised: 5 June 2019
First available in Project Euclid: 11 December 2019

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1576033298

Digital Object Identifier
doi:10.1215/00127094-2019-0046

Mathematical Reviews number (MathSciNet)
MR4065146

Subjects
Primary: 11G05: Elliptic curves over global fields [See also 14H52]
Secondary: 14G40: Arithmetic varieties and schemes; Arakelov theory; heights [See also 11G50, 37P30] 11G18: Arithmetic aspects of modular and Shimura varieties [See also 14G35]

Keywords
Arakelov intersection theory Hilbert modular surfaces Borcherds theory

Citation

Shankar, Ananth N.; Tang, Yunqing. Exceptional splitting of reductions of abelian surfaces. Duke Math. J. 169 (2020), no. 3, 397--434. doi:10.1215/00127094-2019-0046. https://projecteuclid.org/euclid.dmj/1576033298


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References

  • [1] J. D. Achter, Explicit bounds for split reductions of simple abelian varieties, J. Théor. Nombres Bordeaux 24 (2012), no. 1, 41–55.
  • [2] J. D. Achter and E. W. Howe, Split abelian surfaces over finite fields and reductions of genus-$2$ curves, Algebra Number Theory 11 (2017), no. 1, 39–76.
  • [3] P. Autissier, Hauteur moyenne de variétés abéliennes isogènes, Manuscripta Math. 117 (2005), no. 1, 85–92.
  • [4] R. E. Borcherds, Automorphic forms with singularities on Grassmannians, Invent. Math. 132 (1998), no. 3, 491–562.
  • [5] J. H. Bruinier, Borcherds products with prescribed divisor, Bull. Lond. Math. Soc. 49 (2017), no. 6, 979–987.
  • [6] J. H. Bruinier, J. I. Burgos Gil, and U. Kühn, Borcherds products and arithmetic intersection theory on Hilbert modular surfaces, Duke Math. J. 139 (2007), no. 1, 1–88.
  • [7] J. I. Burgos Gil, J. Kramer, and U. Kühn, Cohomological arithmetic Chow rings, J. Inst. Math. Jussieu 6 (2007), no. 1, 1–172.
  • [8] H. Carayol, Sur la mauvaise réduction des courbes de Shimura, Compos. Math. 59 (1986), no. 2, 151–230.
  • [9] C.-L. Chai, Arithmetic minimal compactification of the Hilbert-Blumenthal moduli spaces, Ann. of Math. (2) 131 (1990), no. 3, 541–554.
  • [10] F. Charles, Exceptional isogenies between reductions of pairs of elliptic curves, Duke Math. J. 167 (2018), no. 11, 2039–2072.
  • [11] 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, 151–180.
  • [12] W. C. Chi, $l$-adic and $\lambda $-adic representations associated to abelian varieties defined over number fields, Amer. J. Math. 114 (1992), no. 2, 315–353.
  • [13] L. Clozel, H. Oh, and E. Ullmo, Hecke operators and equidistribution of Hecke points, Invent. Math. 144 (2001), no. 2, 327–351.
  • [14] P. Deligne, “Variétés de Shimura: interprétation modulaire, et techniques de construction de modèles canoniques” in Automorphic Forms, Representations and $L$-Functions, Part 2 (Corvallis, OR, 1977), Proc. Sympos. Pure Math. 33, Amer. Math. Soc., Providence, 1979, 247–289.
  • [15] P. Deligne, J. S. Milne, A. Ogus, and K.-Y. Shih, Hodge Cycles, Motives, and Shimura Varieties, Lecture Notes in Math. 900, Springer, Berlin, 1982.
  • [16] N. D. Elkies, The existence of infinitely many supersingular primes for every elliptic curve over $\mathbf{Q}$, Invent. Math. 89 (1987), no. 3, 561–567.
  • [17] N. D. Elkies, Supersingular primes for elliptic curves over real number fields, Compos. Math. 72 (1989), no. 2, 165–172.
  • [18] A. Eskin and Y. R. Katznelson, Singular symmetric matrices, Duke Math. J. 79 (1995), no. 2, 515–547.
  • [19] G. Faltings, Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, Invent. Math. 73 (1983), no. 3, 349–366. Erratum, Invent. Math. 75 (1984), no. 2, 381.
  • [20] G. Faltings and C.-L. Chai, Degeneration of Abelian Varieties, Ergeb. Math. Grenzgeb. (3) 22, Springer, Berlin, 1990.
  • [21] F. Fité, K. S. Kedlaya, V. Rotger, and A. V. Sutherland, Sato-Tate distributions and Galois endomorphism modules in genus 2, Compos. Math. 148 (2012), no. 5, 1390–1442.
  • [22] E. Z. Goren, Lectures on Hilbert Modular Varieties and Modular Forms, CRM Monogr. Ser. 14, Amer. Math. Soc., Providence, 2002.
  • [23] A. Grothendieck and J. Dieudonné, Éléments de géométrie algébrique, IV: Étude locale des schémas et des morphismes de schémas. I, Publ. Math. Inst. Hautes Études Sci. 20 (1964).
  • [24] F. Hirzebruch and D. Zagier, Intersection numbers of curves on Hilbert modular surfaces and modular forms of Nebentypus, Invent. Math. 36 (1976), 57–113.
  • [25] F. Hörmann, The Geometric and Arithmetic Volume of Shimura Varieties of Orthogonal Type, CRM Monogr. Ser. 35, Amer. Math. Soc., Providence, 2014.
  • [26] B. Howard and T. Yang, Intersections of Hirzebruch-Zagier Divisors and CM Cycles, Lecture Notes in Math. 2041, Springer, Heidelberg, 2012.
  • [27] N. M. Katz and P. Sarnak, Random Matrices, Frobenius Eigenvalues, and Monodromy, Amer. Math. Soc. Colloq. Publ. 45, Amer. Math. Soc., Providence, 1999.
  • [28] K. S. Kedlaya, “Sato-Tate groups of genus $2$ curves” in Advances on Superelliptic Curves and Their Applications, NATO Sci. Peace Secur. Ser. D Inf. Commun. Secur. 41, IOS, Amsterdam, 2015, 117–136.
  • [29] M. Kisin, Integral models for Shimura varieties of abelian type, J. Amer. Math. Soc. 23 (2010), no. 4, 967–1012.
  • [30] S. S. Kudla and M. Rapoport, Arithmetic Hirzebruch-Zagier cycles, J. Reine Angew. Math. 515 (1999), 155–244.
  • [31] S. Lang and H. Trotter, Frobenius Distributions in $\mathrm{GL}_{2}$-Extensions: Distribution of Frobenius Automorphisms in $\mathrm{GL}_{2}$-Extensions of the Rational Numbers, Lecture Notes in Math. 504, Springer, Berlin, 1976.
  • [32] W. Messing, The Crystals Associated to Barsotti-Tate Groups: With Applications to Abelian Schemes, Lecture Notes in Math. 264, Springer, Berlin, 1972.
  • [33] L. Mocz, A new Northcott property for Faltings height, Ph.D. dissertation, Princeton University, Princeton, 2017.
  • [34] V. K. Murty and V. M. Patankar, Splitting of abelian varieties, Int. Math. Res. Not. IMRN 2008, no. 12, art. ID 033.
  • [35] G. Pappas, Arithmetic models for Hilbert modular varieties, Compos. Math. 98 (1995), no. 1, 43–76.
  • [36] M. Rapoport, Compactifications de l’espace de modules de Hilbert-Blumenthal, Compos. Math. 36 (1978), no. 3, 255–335.
  • [37] W. F. Sawin, Ordinary primes for Abelian surfaces, C. R. Math. Acad. Sci. Paris 354 (2016), no. 6, 566–568.
  • [38] W. M. Schmidt, Asymptotic formulae for point lattices of bounded determinant and subspaces of bounded height, Duke Math. J. 35 (1968), 327–339.
  • [39] J.-P. Serre, Lectures on $N_{X}(p)$, Chapman and Hall/CRC Res. Notes Math. 11, CRC Press, Boca Raton, FL, 2012.
  • [40] J. Thorner and A. Zaman, A Chebotarev variant of the Brun-Titchmarsh theorem and bounds for the Lang-Trotter conjectures, Int. Math. Res. Not. IMRN 2018, no. 16, 4991–5027.
  • [41] G. van der Geer, Hilbert Modular Surfaces, Ergeb. Math. Grenzgeb. (3) 16, Springer, Berlin, 1988.
  • [42] T. Yang, An arithmetic intersection formula on Hilbert modular surfaces, Amer. J. Math. 132 (2010), no. 5, 1275–1309.
  • [43] D. Zywina, The splitting of reductions of an abelian variety, Int. Math. Res. Not. IMRN 2014, no. 18, 5042–5083.