Ordinary primes for some varieties with extra endomorphisms

• Autores: Francesc Fité
• Localización: Publicacions matematiques, ISSN 0214-1493, Vol. 68, Nº. 1, 2024, págs. 27-40
• Idioma: inglés
• DOI: 10.5565/PUBLMAT6812402
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• Resumen

  • Let A be an abelian variety defined over a number field and of dimension g. When g ≤ 2, by the recent work of Sawin, we know the exact (nonzero) value of the density of the set of primes which are ordinary for A. In higher dimension very little is known. We show that if g = 3 and A has multiplication by an imaginary quadratic field E, then there exists a nonzero density set of ordinary primes for A. We reach the same conclusion if g = 4 and the pair (A, E) has signature (2, 2). We also obtain partial results when g = 3 and A has multiplication by a totally real cubic field. We showthat our methods also apply to certain abelian varieties of Albert type IV of higher dimension. These results are derived from an extended version of the `-adic methods of Katz, Ogus, and Serre in the presence of extra endomorphisms.

• Referencias bibliográficas
  • S. Asif, F. Fite, and D. Pentland ´ , Computing L-polynomials of Picard curves from Cartier– Manin matrices, With an appendix by A. V. Sutherland,...
  • G. Boxer, F. Calegari, T. Gee, and V. Pilloni, Abelian surfaces over totally real fields are potentially modular, Publ. Math. Inst. Hautes...
  • F. Fite and X. Guitart ´ , Tate module tensor decompositions and the Sato–Tate conjecture for certain abelian varieties potentially of GL2-type,...
  • X. Guitart, Abelian varieties with many endomorphisms and their absolutely simple factors, Rev. Mat. Iberoam. 28(2) (2012), 591–601. DOI:...
  • D. Mumford, A note of Shimura’s paper “Discontinuous groups and abelian varieties”, Math. Ann. 181 (1969), 345–351. DOI: 10.1007/BF01350672
  • D. Mumford, Abelian Varieties, With appendices by C. P. Ramanujam and Yuri Manin, Corrected reprint of the second (1974) edition, Tata Institute...
  • R. Noot, Abelian varieties-Galois representation and properties of ordinary reduction, Special issue in honour of Frans Oort, Compositio Math....
  • A. Ogus, Hodge cycles and crystalline cohomology, in: Hodge Cycles, Motives, and Shimura Varieties, Lecture Notes in Mathematics 900, Springer-Verlag,...
  • R. Pink, l-adic algebraic monodromy groups, cocharacters, and the Mumford–Tate conjecture, J. Reine Angew. Math. 495 (1998), 187–237. DOI:...
  • K. A. Ribet, Galois action on division points of Abelian varieties with real multiplications, Amer. J. Math. 98(3) (1976), 751–804. DOI: 10.2307/2373815
  • K. A. Ribet, Galois representations attached to eigenforms with Nebentypus, in: Modular Functions of One Variable V (Proc. Second Internat....
  • K. A. Ribet, Abelian varieties over Q and modular forms, in: Algebra and Topology 1992 (Taej˘on), Korea Adv. Inst. Sci. Tech., Taej˘on, 1992,...
  • W. F. Sawin, Ordinary primes for Abelian surfaces, C. R. Math. Acad. Sci. Paris 354(6) (2016), 566–568. DOI: 10.1016/j.crma.2016.01.025.
  • J.-P. Serre, Quelques applications du th´eor`eme de densit´e de Chebotarev, Inst. Hautes Etudes ´ Sci. Publ. Math. 54 (1981), 323–401....
  • J.-P. Serre, Abelian l-Adic Representations and Elliptic Curves, With the collaboration of Willem Kuyk and John Labute, Second edition, Advanced...
  • G. Shimura, On analytic families of polarized abelian varieties and automorphic functions, Ann. of Math. (2) 78(1) (1963), 149–192. DOI: 10.2307/1970507
  • J. Suh, Ordinary primes in Hilbert modular varieties, Compos. Math. 156(4) (2020), 647–678. DOI: 10.1112/s0010437x19007826.
  • J. T. Tate, p-divisible groups, in: Proceedings of a Conference on Local Fields (Driebergen, 1966), Springer, Berlin, 1967, pp. 158–183. DOI:...
  • A. Weil, On a certain type of characters of the id`ele-class group of an algebraic number-field, in: Proceedings of the International...
  • A. Wiles, On ordinary λ-adic representations associated to modular forms, Invent. Math. 94(3) (1988), 529–573. DOI: 10.1007/BF01394275
  • Y. Zarhin, Torsion of abelian varieties over GL(2)-extensions of number fields, Math. Ann. 284(4) (1989), 631–646. DOI: 10.1007/BF01443356