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Brauer points on Fermat curves

Published online by Cambridge University Press:  17 April 2009

William G. McCallum
William G. McCallum
Affiliation:
Department of Mathematics, University of Arizona, Tucson, AZ 85721, United States of America
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Abstract

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In honour of George Szekeres on his 90th birthday

If X is a variety over a number field K, the set of K-rational points on X is contained in the subset of the adelic points cut out by the Brauer group; we call this set the set of Brauer points on the variety. If S is a set of valuations of K, we also define S-Brauer points in a natural way. It is natural to ask how good a bound on the rational points is provided by the Brauer (or S-Brauer) points.

Let p > 3 be a prime number, and let X be the Fermat curve of degree p, xp + yp = 1. Let K be the field of p-th roots of unity, and let r be the p-rank of the class group of K. In this paper we show that if r < (p + 3)/8, then the set of p-Brauer points on X has cardinality at most p. We construct elements of the Brauer group of X by relating it to the Weil-Chatelet group of the jacobian of X, then use the method of Coleman and Chabauty to bound the points cut out by these elements.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 63 , Issue 3 , June 2001 , pp. 393 - 406
Copyright
Copyright © Australian Mathematical Society 2001

References

[1]Bosch, S., Guntzer, U. and Remmert, R., Non-archimedean analysis (Springer-Verlag, Berlin, Heidelberg, New York, 1984).CrossRefGoogle Scholar
[2]Bourbaki, N., Lie groups and Lie algebras, Part I (Hermann, Addison-Wesley, Reading, MA, 1975).Google Scholar
[3]Buhler, J., Crandall, R., Ernvall, R., Metsänkylä, T. and Shokrollahi, M.A., ‘Irregular primes and cyclotomic invariants to twelve million’, J. Symbolic Comput. 31 (2001), 89–96.CrossRefGoogle Scholar
[4]Carlitz, L., ‘A generalization of Maillet's determinant and a bound for the first factor of the class number’, Proc. Amer, Math. Soc. 12 (1961), 256–261.Google Scholar
[5]Coleman, R.F., ‘Torsion points on curves and p-adic abelian integrals’, Ann. of Math. 121 (1983), 111–168.CrossRefGoogle Scholar
[6]Faddeev, D.K., ‘Invariants of divisor classes for the curves xk(1 − x) = yl in an l-adic cyclotomic field’, Trudy. Mat. Inst. Steklov. 64 (1961), 284–293.Google Scholar
[7]Fresnel, J. and van der Put, M., Géometrie analytique rigide et applications, Progress in Mathematics 18 (Birkhäuser, Boston, Basel, Stuttgart, 1981).Google Scholar
[8]Manin, Y.I., ‘Le groupe de Brauer-Grothendieck en géométrie diophantienne’, in Actes du Congrès International des Mathématiciens (Nice, 1970), Tome 1 (Gauthier-Villars, Paris, 1971), pp. 401–411.Google Scholar
[9]McCallum, W.G., ‘On the Shafarevich-Tate group of the Jacobian of a quotient of the Fermat curve’, Invent. Math. 93 (1988), 637–666.CrossRefGoogle Scholar
[10]McCallum, W.G., ‘The arithmetic of Fermat curves’, Math. Ann. 294 (1992), 503–511.CrossRefGoogle Scholar
[11]McCallum, W.G., ‘The method of Coleman and Chabauty’, Math. Ann. 299 (1994), 565–596.CrossRefGoogle Scholar
[12]Ribet, K., ‘On modular representations of Gal (/ℚ) arising from modular forms’ Invent. Math. 100 (1990), 431–476.CrossRefGoogle Scholar
[13]Scharaschkin, V., ‘The Brauer-Manin obstruction for curves’, (preprint).Google Scholar
[14]Washington, L.C., Introduction to cyclotomic fields, Graduate Texts in Mathematics 83 (Springer-Verlag, Berlin, Heidelberg, New York, 1982).CrossRefGoogle Scholar