Proceedings of the Japan Academy, Series A, Mathematical Sciences

On the cohomological coprimality of Galois representations associated with elliptic curves

Jerome Tomagan Dimabayao

Full-text: Open access

Abstract

Let $E$ and $E'$ be elliptic curves over an algebraic number field. We show that systems of $\ell$-adic representations associated with $E$ and $E'$ are cohomologically coprime, in the sense that the Galois cohomology groups corresponding to respective fields of division points are all trivial. This provides a generalization of some known results about the vanishing of the cohomology groups associated with the $\ell$-adic Tate module of an elliptic curve.

Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 91, Number 10 (2015), 141-146.

Dates
First available in Project Euclid: 2 December 2015

Permanent link to this document
https://projecteuclid.org/euclid.pja/1449080124

Digital Object Identifier
doi:10.3792/pjaa.91.141

Mathematical Reviews number (MathSciNet)
MR3430202

Zentralblatt MATH identifier
06554943

Subjects
Primary: 11F80: Galois representations 11G05: Elliptic curves over global fields [See also 14H52]

Keywords
Galois representations elliptic curves

Citation

Dimabayao, Jerome Tomagan. On the cohomological coprimality of Galois representations associated with elliptic curves. Proc. Japan Acad. Ser. A Math. Sci. 91 (2015), no. 10, 141--146. doi:10.3792/pjaa.91.141. https://projecteuclid.org/euclid.pja/1449080124


Export citation

References

  • J. Coates and R. Sujatha, Euler-Poincaré characteristics of abelian varieties, C. R. Acad. Sci. Paris Sér. I Math. 329 (1999), no. 4, 309–313.
  • J. Coates, R. Sujatha and J.-P. Wintenberger, On the Euler-Poincaré characteristics of finite dimensional $p$-adic Galois representations, Publ. Math. Inst. Hautes Études Sci. 93 (2001), 107–143.
  • J. T. Dimabayao, On the vanishing of cohomologies of $p$-adic Galois representations associated with elliptic curves, Kyushu J. Math. 69 (2015), no. 2, 367–386.
  • G. Faltings, Endlichkeitssätze für abelsche Varietäten über Zahlkörpern, Invent. Math. 73 (1983), no. 3, 349–366.
  • A. Greicius, Elliptic curves with surjective adelic Galois representations, Experiment. Math. 19 (2010), no. 4, 495–507.
  • S. Lang and H. Trotter, Frobenius distributions in $\mathrm{GL}_{2}$-extensions, Lecture Notes in Mathematics, 504, Springer, Berlin, 1976.
  • M. Lazard, Groupes analytiques $p$-adiques, Inst. Hautes Études Sci. Publ. Math. 26 (1965), 389–603.
  • J. Neukirch, A. Schmidt and K. Wingberg, Cohomology of number fields, 2nd ed., Grundlehren der Mathematischen Wissenschaften, 323, Springer, Berlin, 2008.
  • K. A. Ribet, Galois action on division points of Abelian varieties with real multiplications, Amer. J. Math. 98 (1976), no. 3, 751–804.
  • J.-P. Serre, Sur les groupes de congruence des variétés abéliennes. II, Izv. Akad. Nauk SSSR Ser. Mat. 35 (1971), 731–737.
  • J.-P. Serre, Propriétés galoisiennes des points d'ordre fini des courbes elliptiques, Invent. Math. 15 (1972), no. 4, 259–331.
  • J.-P. Serre, Un critère d'indépendance pour une famille de représentations $\ell$-adiques, Comment. Math. Helv. 88 (2013), no. 3, 541–554.