TY - CHAP

T1 - Galois Action on the Homology of Fermat Curves

AU - Davis, Rachel

AU - Pries, Rachel

AU - Stojanoska, Vesna

AU - Wickelgren, Kirsten

N1 - Funding Information:
Acknowledgements We would like to thank BIRS for hosting the WIN3 conference where we began this project. Some of this work was done while the third and fourth authors were in residence at MSRI during the spring 2014 Algebraic topology semester, supported by NSF grant 0932078 000. The second author was supported by NSF grant DMS-1101712. The third author was supported by NSF grant DMS-1307390. The fourth author was supported by an American Institute of Mathematics 5 year fellowship and NSF grant DMS-1406380. We would also like to thank Sharifi and the referee for helpful remarks.

PY - 2016

Y1 - 2016

N2 - In his paper titled “Torsion points on Fermat Jacobians, roots of circular units and relative singular homology,” Anderson determines the homology of the degree n Fermat curve as a Galois module for the action of the absolute Galois group GQ(ζn). In particular, when n is an odd prime p, he shows that the action of GQ(ζp) on a more powerful relative homology group factors through the Galois group of the splitting field of the polynomial 1−(1−xp)p. If p satisfies Vandiver’s conjecture, we give a proof that the Galois group G of this splitting field over Q(ζp) is an elementary abelian p-group of rank (p+1)/2. Using an explicit basis for G, we completely compute the relative homology, the homology, and the homology of an open subset of the degree 3 Fermat curve as Galois modules. We then compute several Galois cohomology groups which arise in connection with obstructions to rational points. In Anderson (Duke Math J 54(2):501 – 561, 1987), the author determines the homology of the degree n Fermat curve as a Galois module for the action of the absolute Galois group GQ(ζn). In particular, when n is an odd prime p, he shows that the action of GQ(ζp) on a more powerful relative homology group factors through the Galois group of the splitting field of the polynomial 1−(1−xp)p. If p satisfies Vandiver’s conjecture, we give a proof that the Galois group G of this splitting field over Q(ζp) is an elementary abelian p-group of rank (p+1)/2. Using an explicit basis for G, we completely compute the relative homology, the homology, and the homology of an open subset of the degree 3 Fermat curve as Galois modules. We then compute several Galois cohomology groups which arise in connection with obstructions to rational points.

AB - In his paper titled “Torsion points on Fermat Jacobians, roots of circular units and relative singular homology,” Anderson determines the homology of the degree n Fermat curve as a Galois module for the action of the absolute Galois group GQ(ζn). In particular, when n is an odd prime p, he shows that the action of GQ(ζp) on a more powerful relative homology group factors through the Galois group of the splitting field of the polynomial 1−(1−xp)p. If p satisfies Vandiver’s conjecture, we give a proof that the Galois group G of this splitting field over Q(ζp) is an elementary abelian p-group of rank (p+1)/2. Using an explicit basis for G, we completely compute the relative homology, the homology, and the homology of an open subset of the degree 3 Fermat curve as Galois modules. We then compute several Galois cohomology groups which arise in connection with obstructions to rational points. In Anderson (Duke Math J 54(2):501 – 561, 1987), the author determines the homology of the degree n Fermat curve as a Galois module for the action of the absolute Galois group GQ(ζn). In particular, when n is an odd prime p, he shows that the action of GQ(ζp) on a more powerful relative homology group factors through the Galois group of the splitting field of the polynomial 1−(1−xp)p. If p satisfies Vandiver’s conjecture, we give a proof that the Galois group G of this splitting field over Q(ζp) is an elementary abelian p-group of rank (p+1)/2. Using an explicit basis for G, we completely compute the relative homology, the homology, and the homology of an open subset of the degree 3 Fermat curve as Galois modules. We then compute several Galois cohomology groups which arise in connection with obstructions to rational points.

KW - Cohomology

KW - Cyclotomic field

KW - Fermat curve

KW - Galois module

KW - Homology

KW - Ã‰tale fundamental group

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U2 - 10.1007/978-3-319-30976-7_3

DO - 10.1007/978-3-319-30976-7_3

M3 - Chapter

VL - 3

T3 - Association for Women in Mathematics Series

SP - 57

EP - 86

BT - Directions in Number Theory

PB - Springer

ER -