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Solvable points on genus-one curves over local fields

Published online by Cambridge University Press:  16 May 2012

Ambrus Pál
Ambrus Pál*
Affiliation:
Department of Mathematics, 180 Queen’s Gate, Imperial College, London SW7 2AZ, UK([email protected])
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Abstract

Let be a field complete with respect to a discrete valuation whose residue field is perfect of characteristic . We prove that every smooth, projective, geometrically irreducible curve of genus one defined over with a non-zero divisor of degree a power of has a solvable point over .

Keywords

solvable pointsgenus one curveslocal fields14G0511G07
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 12 , Issue 1 , January 2013 , pp. 31 - 42
Copyright
Copyright © Cambridge University Press 2013

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References

1.Çiperiani, M. and Wiles, A. , Solvable points on genus one curves, Duke Math. J. 142 (2008), 381–464.Google Scholar
2.Gorenstein, D. , Finite groups, 2nd edn (American Mathematical Society, Providence, Rhode Island, 2007).Google Scholar
3.Grothendieck, A. and Raynaud, M. , Modèles de Néron et monodromie, Groupes de monodromie en Géometrie Algebrique, I, II, Lecture Notes in Math., Volume 288. pp. 313–523 (Springer, 1972).Google Scholar
4.Károlyi, Gy. and Pál, A. , The cyclomatic number of connected graphs without solvable orbits, J. Ramanujan Math. Soc. (2012), in press.Google Scholar
5.Mazur, B. and Roberts, L. , Local Euler characteristics, Invent. Math. 9 (1970), 201–223.CrossRefGoogle Scholar
6.Milne, J. , Étale cohomology (Princeton University Press, Princeton, New Jersey, 1980).Google Scholar
7.Oort, F. and Tate, J. , Group schemes of prime order, Ann. Sci. Éc. Norm . Supér. (4) 3 (1970), 1–21.Google Scholar
8.Pál, A. , Solvable points on projective algebraic curves, Canad. J. Math. 56 (2004), 612–637.Google Scholar
9.Pál, A. , Curves which do not become semi-stable after any solvable extension, Rend. Semin. Mat. Univ. Padova (2012), in press.Google Scholar
10.Roberts, L. , The flat cohomology of group schemes of rank , Amer. J. Math. 95 (1973), 688–702.CrossRefGoogle Scholar
11.Serre, J.-P. , Local fields. GTM 67 (second corrected printing, 1995) (Springer-Verlag, New York–Berlin, 1979).Google Scholar
12.Tate, J. , Finite flat group schemes, Modular forms and Fermats last theorem (Boston, MA, 1995), pp. 121–154 (Springer-Verlag, Berlin–Heidelberg–New York, 1997).Google Scholar