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 - Étale 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 -