## Functiones et Approximatio Commentarii Mathematici

### Arithmetic local constants for abelian varieties with extra endomorphisms

Sunil Chetty

#### Abstract

This work generalizes the theory of arithmetic local constants, introduced by Mazur and Rubin, to better address abelian varieties with a larger endomorphism ring than $\mathbb{Z}$. We then study the growth of the $p^\infty$-Selmer rank of our abelian variety, and we address the problem of extending the results of Mazur and Rubin to dihedral towers $k\subset K\subset F$ in which $[F:K]$ is not a $p$-power extension.

#### Article information

Source
Funct. Approx. Comment. Math. Volume 55, Number 1 (2016), 59-81.

Dates
First available in Project Euclid: 19 September 2016

http://projecteuclid.org/euclid.facm/1474301230

Digital Object Identifier
doi:10.7169/facm/2016.55.1.5

Mathematical Reviews number (MathSciNet)
MR3549013

#### Citation

Chetty, Sunil. Arithmetic local constants for abelian varieties with extra endomorphisms. Funct. Approx. Comment. Math. 55 (2016), no. 1, 59--81. doi:10.7169/facm/2016.55.1.5. http://projecteuclid.org/euclid.facm/1474301230.

#### References

• K. Brown, Cohomology of Groups, volume 87 of Graduate Texts in Mathematics, Springer, 1982.
• J.W.S. Cassels, Diophantine equations with special reference to elliptic curves, J. London Math. 41 (1966), 193–291.
• S. Chetty, Comparing local constants of elliptic curves in dihedral extensions, submitted, draft: arXiv:1002.2671.
• S. Chetty and L. Li, Computing local constants for cm elliptic curves, Rocky Mountain J. Math. 44(3) (2014), 853–863.
• M. Flach, A generalisation of the Cassels-Tate pairing, J. reine angew. Math. 412 (1990), 113–127.
• E. Howe, Isogeny classes of abelian varieties with no principal polarizations, in Moduli of abelian varieties (Texel Island, 1999), volume 195 of Progress in Mathematics, pages 203–216. Birkhäuser, 2001.
• S. Lang, Algebra, volume 211 of Graduate Texts in Mathematics, Springer, revised third edition, 2002.
• J. Lubin and M. Rosen, The norm map for ordinary abelian varieties, Journal of Algebra 52 (1978), 236–240.
• B. Mazur and K. Rubin, Finding large Selmer rank via an arithmetic theory of local constants, Annals of Mathematics 166(2) (2007), 581–614.
• B. Mazur, K. Rubin, and A. Silverberg, Twisting Commutative Algebraic Groups, Journal of Algebra 314(1) (2007), 419–438.
• J.S. Milne, Abelian Varieties, in G. Cornell and J. Silverman, editors, Arithmetic Geometry, Springer-Verlag, 1986, available at http://www.jmilne.org/math/.
• J. Rotman, Advanced Modern Algebra, Prentice Hall, 2002.
• K. Rubin, Euler Systems, Hermann Weyl Lectures - The Institute for Advanced Study, Princeton University Press, 2000.
• J-P. Serre, Local Fields, volume 42 of Graduate Texts in Mathematics, Springer, 1977.
• M. Seveso, The arithmetic theory of local constants for abelian varieties, Rend. Semin. Mat. Univ. Padova 127 (2012), 17–39.
• J. Silverman, Arithmetic of Elliptic Curves, volume 106 of Graduate Texts in Mathematics, Springer, 1986.