## The Galois action on the lower central series of the fundamental group of the Fermat curve

Publication
, Journal Article

Davis, R; Pries, R; Wickelgren, K

Published in: Israel Journal of Mathematics

January 1, 2023

Information about the absolute Galois group GK of a number field K is encoded in how it acts on the étale fundamental group π of a curve X defined over K. In the case that K = ℚ(ζn) is the cyclotomic field and X is the Fermat curve of degree n ≥ 3, Anderson determined the action of GK on the étale homology with coefficients in ℤ/nℤ. The étale homology is the first quotient in the lower central series of the étale 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 GK on all degrees of the associated graded quotient of the lower central series of the étale fundamental group of the Fermat curve of degree n, with coefficients in ℤ/nℤ.

### Duke Scholars

## Published In

Israel Journal of Mathematics

## DOI

## EISSN

1565-8511

## ISSN

0021-2172

## Publication Date

January 1, 2023

## Related Subject Headings

- General Mathematics
- 4904 Pure mathematics
- 4902 Mathematical physics
- 4901 Applied mathematics
- 0102 Applied Mathematics
- 0101 Pure Mathematics

### Citation

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MLA

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Davis, R., Pries, R., & Wickelgren, K. (2023). The Galois action on the lower central series of the fundamental group of the Fermat curve.

*Israel Journal of Mathematics*. https://doi.org/10.1007/s11856-023-2571-zDavis, R., R. Pries, and K. Wickelgren. “The Galois action on the lower central series of the fundamental group of the Fermat curve.”

*Israel Journal of Mathematics*, January 1, 2023. https://doi.org/10.1007/s11856-023-2571-z.Davis R, Pries R, Wickelgren K. The Galois action on the lower central series of the fundamental group of the Fermat curve. Israel Journal of Mathematics. 2023 Jan 1;

Davis, R., et al. “The Galois action on the lower central series of the fundamental group of the Fermat curve.”

*Israel Journal of Mathematics*, Jan. 2023.*Scopus*, doi:10.1007/s11856-023-2571-z.Davis R, Pries R, Wickelgren K. The Galois action on the lower central series of the fundamental group of the Fermat curve. Israel Journal of Mathematics. 2023 Jan 1;

## Published In

Israel Journal of Mathematics

## DOI

## EISSN

1565-8511

## ISSN

0021-2172

## Publication Date

January 1, 2023

## Related Subject Headings

- General Mathematics
- 4904 Pure mathematics
- 4902 Mathematical physics
- 4901 Applied mathematics
- 0102 Applied Mathematics
- 0101 Pure Mathematics