Functiones et Approximatio Commentarii Mathematici

Tate conjecture for some abelian surfaces over totally real or CM number fields

Cristian Virdol

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In this paper we prove Tate conjecture for abelian surfaces of the type $\operatorname{Res}_{K/F}E$ where $E$ is an elliptic curve defined over a totally real or CM number field $K$, and $F$ is a subfield of $K$ such that $[K:F]=2$.

Article information

Funct. Approx. Comment. Math., Volume 52, Number 1 (2015), 57-63.

First available in Project Euclid: 20 March 2015

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11F41: Automorphic forms on GL(2); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces [See also 14J20]
Secondary: 11F80: Galois representations 11R42: Zeta functions and $L$-functions of number fields [See also 11M41, 19F27] 11R80: Totally real fields [See also 12J15]

Tate conjecture abelian surfaces totally real fields


Virdol, Cristian. Tate conjecture for some abelian surfaces over totally real or CM number fields. Funct. Approx. Comment. Math. 52 (2015), no. 1, 57--63. doi:10.7169/facm/2015.52.1.4.

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  • T. Barnet-Lamb, T. Gee, D. Geraghty, R. Taylor, Potential automorphy and change of weight, Annals of Mathematics, to appear.
  • C.W. Curtis, I. Reiner, Methods of Representation Theory, Vol. I, Wiley, New York, 1981.
  • G. van der Geer, Hilbert modular surfaces, Springer-Verlag 1988.
  • S. Gelbart, H. Jacquet, A relation between automorphic representations of $GL(2)$ and $GL(3)$, Ann. Sci. École Norm. Sup. 11 (1979), 471–542.
  • G. Harder, R.P. Langlands, M. Rapoport, Algebraische Zycklen auf Hilbert-Blumenthal-Fl$\ddot{a}$chen, J. Reine Angew. Math. 396 (1986), 53–120.
  • A. Knightly, Tate classes on a Product of two Picard modular surfaces, J. Number Theory 107 (2004), 335–344.
  • J.S. Milne, Abelian varieties,
  • V.K. Murty, D. Prasad, Tate cycles on a product of two Hilbert modular surfaces, J. Number Theory 80(1) (2000), 25–43.
  • V.K. Murty, D. Ramakrishnan, Period relations and the Tate conjecture for Hilbert modular surfaces, Invent. Math. 89 (1987), 319–325.
  • D. Ramakrishnan, Modularity of the Rankin-Selberg L-series, and multiplicity one for SL(2), Ann. of Math. 152 (2000), 45–111.
  • D. Ramakrishnan, Modularity of solvable Artin representations of GO(4)-type, IMRN (2002), No. 1, 1–54.
  • J. Tate, Algebraic cycles and poles of zeta functions, In: Schilling, O.D.G. (ed.), Arithmetical algebraic geometry, New York: Harper and Row, 1966.