Open Access
March 2015 Tate conjecture for some abelian surfaces over totally real or CM number fields
Cristian Virdol
Funct. Approx. Comment. Math. 52(1): 57-63 (March 2015). DOI: 10.7169/facm/2015.52.1.4
Abstract

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 $[K:F]=2$.

References

1.

T. Barnet-Lamb, T. Gee, D. Geraghty, R. Taylor, Potential automorphy and change of weight, Annals of Mathematics, to appear. MR3152941 10.4007/annals.2014.179.2.3 T. Barnet-Lamb, T. Gee, D. Geraghty, R. Taylor, Potential automorphy and change of weight, Annals of Mathematics, to appear. MR3152941 10.4007/annals.2014.179.2.3

2.

C.W. Curtis, I. Reiner, Methods of Representation Theory, Vol. I, Wiley, New York, 1981. MR632548 C.W. Curtis, I. Reiner, Methods of Representation Theory, Vol. I, Wiley, New York, 1981. MR632548

3.

G. van der Geer, Hilbert modular surfaces, Springer-Verlag 1988. MR930101 G. van der Geer, Hilbert modular surfaces, Springer-Verlag 1988. MR930101

4.

S. Gelbart, H. Jacquet, A relation between automorphic representations of $GL(2)$ and $GL(3)$, Ann. Sci. École Norm. Sup. 11 (1979), 471–542. MR533066 S. Gelbart, H. Jacquet, A relation between automorphic representations of $GL(2)$ and $GL(3)$, Ann. Sci. École Norm. Sup. 11 (1979), 471–542. MR533066

5.

G. Harder, R.P. Langlands, M. Rapoport, Algebraische Zycklen auf Hilbert-Blumenthal-Fl$\ddot{a}$chen, J. Reine Angew. Math. 396 (1986), 53–120. MR833013 G. Harder, R.P. Langlands, M. Rapoport, Algebraische Zycklen auf Hilbert-Blumenthal-Fl$\ddot{a}$chen, J. Reine Angew. Math. 396 (1986), 53–120. MR833013

6.

A. Knightly, Tate classes on a Product of two Picard modular surfaces, J. Number Theory 107 (2004), 335–344. MR2072393 10.1016/j.jnt.2004.04.004 A. Knightly, Tate classes on a Product of two Picard modular surfaces, J. Number Theory 107 (2004), 335–344. MR2072393 10.1016/j.jnt.2004.04.004

7.

J.S. Milne, Abelian varieties, www.jmilne.org/math/CourseNotes/av.html J.S. Milne, Abelian varieties, www.jmilne.org/math/CourseNotes/av.html

8.

V.K. Murty, D. Prasad, Tate cycles on a product of two Hilbert modular surfaces, J. Number Theory 80(1) (2000), 25–43. MR1735646 10.1006/jnth.1999.2446 V.K. Murty, D. Prasad, Tate cycles on a product of two Hilbert modular surfaces, J. Number Theory 80(1) (2000), 25–43. MR1735646 10.1006/jnth.1999.2446

9.

V.K. Murty, D. Ramakrishnan, Period relations and the Tate conjecture for Hilbert modular surfaces, Invent. Math. 89 (1987), 319–325. MR894382 10.1007/BF01389081 V.K. Murty, D. Ramakrishnan, Period relations and the Tate conjecture for Hilbert modular surfaces, Invent. Math. 89 (1987), 319–325. MR894382 10.1007/BF01389081

10.

D. Ramakrishnan, Modularity of the Rankin-Selberg L-series, and multiplicity one for SL(2), Ann. of Math. 152 (2000), 45–111. MR1792292 10.2307/2661379 D. Ramakrishnan, Modularity of the Rankin-Selberg L-series, and multiplicity one for SL(2), Ann. of Math. 152 (2000), 45–111. MR1792292 10.2307/2661379

11.

D. Ramakrishnan, Modularity of solvable Artin representations of GO(4)-type, IMRN (2002), No. 1, 1–54. MR1874921 10.1155/S1073792802000016 D. Ramakrishnan, Modularity of solvable Artin representations of GO(4)-type, IMRN (2002), No. 1, 1–54. MR1874921 10.1155/S1073792802000016

12.

J. Tate, Algebraic cycles and poles of zeta functions, In: Schilling, O.D.G. (ed.), Arithmetical algebraic geometry, New York: Harper and Row, 1966.  MR225778 J. Tate, Algebraic cycles and poles of zeta functions, In: Schilling, O.D.G. (ed.), Arithmetical algebraic geometry, New York: Harper and Row, 1966.  MR225778
Copyright © 2015 Adam Mickiewicz University
Cristian Virdol "Tate conjecture for some abelian surfaces over totally real or CM number fields," Functiones et Approximatio Commentarii Mathematici 52(1), 57-63, (March 2015). https://doi.org/10.7169/facm/2015.52.1.4
Published: March 2015
Vol.52 • No. 1 • March 2015
Back to Top