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Towers of 2-covers of hyperelliptic curves. (English) Zbl 1145.11317
Summary: 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.

11G30 Curves of arbitrary genus or genus \(\ne 1\) over global fields
11G10 Abelian varieties of dimension \(> 1\)
14H40 Jacobians, Prym varieties
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[1] B. J. Birch, Cyclotomic fields and Kummer extensions, Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), Thompson, Washington, D.C., 1967, pp. 85 – 93.
[2] 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. · Zbl 0705.14001
[3] Nils Bruin, Chabauty methods and covering techniques applied to generalised Fermat equations, Ph.D. thesis, Universiteit Leiden, 1999.
[4] Nils Bruin, Chabauty methods using elliptic curves, Tech. Report W99-14, Leiden, 1999. · Zbl 1135.11320
[5] 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. · Zbl 1145.11317
[6] 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. · Zbl 0857.14018
[7] 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). · Zbl 0025.24903
[8] Robert F. Coleman, Effective Chabauty, Duke Math. J. 52 (1985), no. 3, 765 – 770. · Zbl 0588.14015 · doi:10.1215/S0012-7094-85-05240-8 · doi.org
[9] 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). · Zbl 0886.11070 · doi:10.1006/jsco.1996.0126 · doi.org
[10] E. V. Flynn, A flexible method for applying Chabauty’s theorem, Compositio Math. 105 (1997), no. 1, 79 – 94. · Zbl 0882.14009 · doi:10.1023/A:1000111601294 · doi.org
[11] J. S. Milne, Jacobian varieties, Arithmetic geometry (Storrs, Conn., 1984) Springer, New York, 1986, pp. 167 – 212.
[12] 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. · Zbl 0888.11023
[13] Edward F. Schaefer, 2-descent on the Jacobians of hyperelliptic curves, J. Number Theory 51 (1995), no. 2, 219 – 232. · Zbl 0832.14016 · doi:10.1006/jnth.1995.1044 · doi.org
[14] Edward F. Schaefer, Computing a Selmer group of a Jacobian using functions on the curve, Math. Ann. 310 (1998), no. 3, 447 – 471. · Zbl 0889.11021 · doi:10.1007/s002080050156 · doi.org
[15] Juan J. Schäffer, The equation 1^\?+2^\?+3^\?+\cdots+\?^\?=\?^\?, Acta Math. 95 (1956), 155 – 189. · Zbl 0071.03702 · doi:10.1007/BF02401100 · doi.org
[16] Joseph H. Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 106, Springer-Verlag, New York, 1986. · Zbl 0585.14026
[17] Michael Stoll, Implementing 2-descent for Jacobians of hyperelliptic curves, Acta Arith. 98 (2001), no. 3, 245 – 277. · Zbl 0972.11058 · doi:10.4064/aa98-3-4 · doi.org
[18] Joseph L. Wetherell, Bounding the number of rational points on certain curves of high rank, Ph.D. thesis, U.C. Berkeley, 1997.
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