## Towers of 2-covers of hyperelliptic curves

HTML articles powered by AMS MathViewer

- by Nils Bruin and E. Victor Flynn PDF
- Trans. Amer. Math. Soc.
**357**(2005), 4329-4347 Request permission

## Abstract:

In this article, we give a way of constructing an unramified Galois-cover of a hyperelliptic curve. The geometric Galois-group is an elementary abelian $2$-group. The construction does not make use of the embedding of the curve in its Jacobian, and it readily displays all subcovers. We show that the cover we construct is isomorphic to the pullback along the multiplication-by-$2$ map of an embedding of the curve in its Jacobian. We show that the constructed cover has an abundance of elliptic and hyperelliptic subcovers. This makes this cover especially suited for covering techniques employed for determining the rational points on curves. In particular the hyperelliptic subcovers give a chance for applying the method iteratively, thus creating towers of elementary abelian 2-covers of hyperelliptic curves. As an application, we determine the rational points on the genus $2$ curve arising from the question of whether the sum of the first $n$ fourth powers can ever be a square. For this curve, a simple covering step fails, but a second step succeeds.## References

- B. J. Birch,
*Cyclotomic fields and Kummer extensions*, Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), Thompson, Washington, D.C., 1967, pp. 85–93. MR**0219507** - Siegfried Bosch, Werner Lütkebohmert, and Michel Raynaud,
*Néron models*, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 21, Springer-Verlag, Berlin, 1990. MR**1045822**, DOI 10.1007/978-3-642-51438-8 - Nils Bruin,
*Chabauty methods and covering techniques applied to generalised Fermat equations*, Ph.D. thesis, Universiteit Leiden, 1999. - Nils Bruin,
*Chabauty methods using elliptic curves*, Tech. Report W99–14, Leiden, 1999. - Nils Bruin and Victor Flynn,
*Transcript of computations*, available from ftp://ftp.liv.ac.uk/pub/genus2/bruinflynn/tow2cov or http://www.cecm.sfu.ca/~bruin/tow2cov, 2001. - J. W. S. Cassels and E. V. Flynn,
*Prolegomena to a middlebrow arithmetic of curves of genus $2$*, London Mathematical Society Lecture Note Series, vol. 230, Cambridge University Press, Cambridge, 1996. MR**1406090**, DOI 10.1017/CBO9780511526084 - Claude Chabauty,
*Sur les points rationnels des variétés algébriques dont l’irrégularité est supérieure à la dimension*, C. R. Acad. Sci. Paris**212**(1941), 1022–1024 (French). MR**11005** - Robert F. Coleman,
*Effective Chabauty*, Duke Math. J.**52**(1985), no. 3, 765–770. MR**808103**, DOI 10.1215/S0012-7094-85-05240-8 - M. Daberkow, C. Fieker, J. Klüners, M. Pohst, K. Roegner, M. Schörnig, and K. Wildanger,
*KANT V4*, J. Symbolic Comput.**24**(1997), no. 3-4, 267–283. Computational algebra and number theory (London, 1993). MR**1484479**, DOI 10.1006/jsco.1996.0126 - E. V. Flynn,
*A flexible method for applying Chabauty’s theorem*, Compositio Math.**105**(1997), no. 1, 79–94. MR**1436746**, DOI 10.1023/A:1000111601294 - J. S. Milne,
*Jacobian varieties*, Arithmetic geometry (Storrs, Conn., 1984) Springer, New York, 1986, pp. 167–212. MR**861976** - Bjorn Poonen and Edward F. Schaefer,
*Explicit descent for Jacobians of cyclic covers of the projective line*, J. Reine Angew. Math.**488**(1997), 141–188. MR**1465369** - Edward F. Schaefer,
*$2$-descent on the Jacobians of hyperelliptic curves*, J. Number Theory**51**(1995), no. 2, 219–232. MR**1326746**, DOI 10.1006/jnth.1995.1044 - Edward F. Schaefer,
*Computing a Selmer group of a Jacobian using functions on the curve*, Math. Ann.**310**(1998), no. 3, 447–471. MR**1612262**, DOI 10.1007/s002080050156 - Juan J. Schäffer,
*The equation $1^p+2^p+3^p+\cdots +n^p=m^q$*, Acta Math.**95**(1956), 155–189. MR**78395**, DOI 10.1007/BF02401100 - Joseph H. Silverman,
*The arithmetic of elliptic curves*, Graduate Texts in Mathematics, vol. 106, Springer-Verlag, New York, 1986. MR**817210**, DOI 10.1007/978-1-4757-1920-8 - Michael Stoll,
*Implementing 2-descent for Jacobians of hyperelliptic curves*, Acta Arith.**98**(2001), no. 3, 245–277. MR**1829626**, DOI 10.4064/aa98-3-4 - Joseph L. Wetherell,
*Bounding the number of rational points on certain curves of high rank*, Ph.D. thesis, U.C. Berkeley, 1997.

## Additional Information

**Nils Bruin**- Affiliation: Department of Mathematics, Simon Fraser University, Burnaby, British Columbia, Canada V5A 1S6
- MR Author ID: 653028
- Email: bruin@member.ams.org
**E. Victor Flynn**- Affiliation: Mathematical Institute, University of Oxford, Oxford OX1 3LB, United Kingdom
- Email: flynn@maths.ox.ac.uk
- Received by editor(s): July 9, 2001
- Received by editor(s) in revised form: September 22, 2002
- Published electronically: June 22, 2005
- Additional Notes: The first author was supported by the Pacific Institute for the Mathematical Sciences, Simon Fraser University and the University of British Columbia

The second author received financial support from EPSRC Grant Number GR/R82975/01 - © Copyright 2005
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc.
**357**(2005), 4329-4347 - MSC (2000): Primary 11G30; Secondary 11G10, 14H40
- DOI: https://doi.org/10.1090/S0002-9947-05-03954-1
- MathSciNet review: 2156713