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UNLIKELY INTERSECTIONS IN FINITE CHARACTERISTIC

  • ANANTH N. SHANKAR (a1) and JACOB TSIMERMAN (a2)

Abstract

We present a heuristic argument based on Honda–Tate theory against many conjectures in ‘unlikely intersections’ over the algebraic closure of a finite field; notably, we conjecture that every abelian variety of dimension 4 is isogenous to a Jacobian. Using methods of additive combinatorics, we answer a related question of Chai and Oort where the ambient Shimura variety is a power of the modular curve.

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Copyright

This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.

References

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[1] Chai, C.-L. and Oort, F., ‘Abelian varieties isogenous to a Jacobian’, Ann. of Math. (2) 176(1) (2012), 589635.
[2] Deligne, P., ‘Variétés abéliennes ordinaires sur un corps fini’, Invent. Math. 8 (1969), 238243.
[3] DiPippo, S. and Howe, E., ‘Real polynomials with all roots on the unit circle and abelian varieties over finite fields’, J. Number Theory 73(2) (1998), 426450.
[4] Howe, E., ‘Principally polarized ordinary abelian varieties over finite fields’, Trans. Amer. Math. Soc. 347(7) (1995), 23612401.
[5] Katz, N. H. and Shen, C.-Y., ‘Garaev’s inequality in finite fields not of prime order’, http://www.math.rochester.edu/ojac/vol3/Katz_2008.pdf.
[6] Lenstra, H. W. Jr, ‘Factoring integers with elliptic curves’, Ann. of Math. (2) 126 (1987), 649673.
[7] Li, L. and Roche-Newton, O., ‘An improved sum-product estimate for general finite fields’, SIAM J. Discrete Math. 25(3) (2011), 12851296.
[8] Oort, F., ‘Foliations in moduli spaces of abelian varieties’, J. Amer. Soc. 17(2) (2004), 267296.
[9] Oort, F., ‘Foliations in moduli spaces of abelian varieties and dimensions of leaves’, in Algebra, Arithmetic, and Geometry: in Honor of Yu. I. Manin, Vol. II, Progress in Mathematics, 270 (Birkhauser Boston, Inc., Boston, MA, 2009), 465501.
[10] Roche-Newton, O., ‘On sum-product estimates and related problems in discrete geometry’, PhD Thesis.
[11] Ruzsa, I. Z., ‘Sumsets and structure’, in Combinatorial Number Theory and Additive Group Theory, Advanced Courses in Mathematics. CRM Barcelona (Birkhäuser, Basel, 2009), 87210.
[12] Shimura, G., Introduction to the Arithmetic Theory of Automorphic Functions, Vol. 1, Publications of the Mathematica Society of Japan, 11 , (Princeton University Press, 1971).
[13] Shparlinski, I. E., ‘On the additive energy of the distance set in finite fields’, Finite Fields Appl. 42 (2016), 187199.
[14] Tsimerman, J., ‘The existence of an abelian varietiy over isogenous to no Jacobian’, Ann. of Math. (2) 176(1) (2012), 637650.
[15] Tsimerman, J., ‘A proof of the Andre–Oort conjecture for A g ’, Ann. of Math. (2) 187(2) (2018), 379390.
[16] Zarhin, Y., ‘Eigenvalues of Frobenius endomorphisms of abelian varieties of low dimension’, J. Pure Appl. Algebra 219 (2015), 20762098.
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Forum of Mathematics, Sigma
  • ISSN: -
  • EISSN: 2050-5094
  • URL: /core/journals/forum-of-mathematics-sigma
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