An official website of the United States government
Suppose C is a cyclic Galois cover of the projective line branched at the three points 0, 1, and ∞. Under a mild condition on the ramification, we determine the structure of the graded Lie algebra of the lower central series of the fundamental group of C in terms of a basis which is well-suited to studying the action of the absolute Galois group of Q.
more »
« less
The Galois action on the lower central series of the fundamental group of the Fermat curve
Abstract. Information about the absolute Galois group G K of a number field K is encoded in how it acts on the ´etale fundamental group π of a curve X defined over K. In the case that K = Q ( ζ n ) is the cyclotomic field and X is the Fermat curve of degree n ≥ 3, Anderson determined the action of G K on the ´etale homology with coefficients in Z/nZ. The ´etale homology is the first quotient in the lower central series of the ´etale fundamental group. In this paper, we determine the Galois module structure of the graded Lie algebra for π. As a consequence, this determines the action of G K on all degrees of the associated graded quotient of the lower central series of the ´etale fundamental group of the Fermat curve of degree n, with coefficients in Z/nZ.
more »
« less
- Award ID(s):
- 2200418
- PAR ID:
- 10520705
- Publisher / Repository:
- Magnes Press
- Date Published:
- Journal Name:
- Israel Journal of Mathematics
- ISSN:
- 0021-2172
- Subject(s) / Keyword(s):
- Fermat curve, cyclotomic field, ´etale fundamental group, homology, lower central series, 2nilpotent quotient, Galois module.
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
We compute the RO(G)‐graded equivariant algebraic K‐groups of a finite field with an action by its Galois group G. Specifically, we show these K‐groups split as the sum of an explicitly computable term and the well‐studied RO(G)‐graded coefficient groups of the equivariant Eilenberg–MacLane spectrum HZ. Our comparison between the equivariant K‐theory spectrum and HZ further shows they share the same Tate spectra and geometric fixed point spectra. In the case where G has prime order, we provide an explicit presentation of the equivariant K‐groups.more » « less
-
null (Ed.)Abstract The duality principle for group representations developed in Dutkay et al. (J Funct Anal 257:1133–1143, 2009), Han and Larson (Bull Lond Math Soc 40:685–695, 2008) exhibits a fact that the well-known duality principle in Gabor analysis is not an isolated incident but a more general phenomenon residing in the context of group representation theory. There are two other well-known fundamental properties in Gabor analysis: the biorthogonality and the fundamental identity of Gabor analysis. The main purpose of this this paper is to show that these two fundamental properties remain to be true for general projective unitary group representations. Moreover, we also present a general duality theorem which shows that that muti-frame generators meet super-frame generators through a dual commutant pair of group representations. Applying it to the Gabor representations, we obtain that $$\{\pi _{\Lambda }(m, n)g_{1} \oplus \cdots \oplus \pi _{\Lambda }(m, n)g_{k}\}_{m, n \in {\mathbb {Z}}^{d}}$$ { π Λ ( m , n ) g 1 ⊕ ⋯ ⊕ π Λ ( m , n ) g k } m , n ∈ Z d is a frame for $$L^{2}({\mathbb {R}}\,^{d})\oplus \cdots \oplus L^{2}({\mathbb {R}}\,^{d})$$ L 2 ( R d ) ⊕ ⋯ ⊕ L 2 ( R d ) if and only if $$\cup _{i=1}^{k}\{\pi _{\Lambda ^{o}}(m, n)g_{i}\}_{m, n\in {\mathbb {Z}}^{d}}$$ ∪ i = 1 k { π Λ o ( m , n ) g i } m , n ∈ Z d is a Riesz sequence, and $$\cup _{i=1}^{k} \{\pi _{\Lambda }(m, n)g_{i}\}_{m, n\in {\mathbb {Z}}^{d}}$$ ∪ i = 1 k { π Λ ( m , n ) g i } m , n ∈ Z d is a frame for $$L^{2}({\mathbb {R}}\,^{d})$$ L 2 ( R d ) if and only if $$\{\pi _{\Lambda ^{o}}(m, n)g_{1} \oplus \cdots \oplus \pi _{\Lambda ^{o}}(m, n)g_{k}\}_{m, n \in {\mathbb {Z}}^{d}}$$ { π Λ o ( m , n ) g 1 ⊕ ⋯ ⊕ π Λ o ( m , n ) g k } m , n ∈ Z d is a Riesz sequence, where $$\pi _{\Lambda }$$ π Λ and $$\pi _{\Lambda ^{o}}$$ π Λ o is a pair of Gabor representations restricted to a time–frequency lattice $$\Lambda $$ Λ and its adjoint lattice $$\Lambda ^{o}$$ Λ o in $${\mathbb {R}}\,^{d}\times {\mathbb {R}}\,^{d}$$ R d × R d .more » « less
-
Abstract Given a profinite group G of finite p -cohomological dimension and a pro- p quotient H of G by a closed normal subgroup N , we study the filtration on the Iwasawa cohomology of N by powers of the augmentation ideal in the group algebra of H . We show that the graded pieces are related to the cohomology of G via analogues of Bockstein maps for the powers of the augmentation ideal. For certain groups H , we relate the values of these generalized Bockstein maps to Massey products relative to a restricted class of defining systems depending on H . We apply our study to prove lower bounds on the p -ranks of class groups of certain nonabelian extensions of $$\mathbb {Q}$$ and to give a new proof of the vanishing of Massey triple products in Galois cohomology.more » « less
-
null (Ed.)We construct a stable homotopy refinement of quantum annular homology, a link homology theory introduced by Beliakova, Putyra and Wehrli. For each $$r\geq ~2$$ we associate to an annular link $$L$$ a naive $$\mathbb {Z}/r\mathbb {Z}$$ -equivariant spectrum whose cohomology is isomorphic to the quantum annular homology of $$L$$ as modules over $$\mathbb {Z}[\mathbb {Z}/r\mathbb {Z}]$$ . The construction relies on an equivariant version of the Burnside category approach of Lawson, Lipshitz and Sarkar. The quotient under the cyclic group action is shown to recover the stable homotopy refinement of annular Khovanov homology. We study spectrum level lifts of structural properties of quantum annular homology.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1007/s11856-023-2571-z
