Fermat quotients with CM Jacobians and trivial automorphism group
Abstract
Let Fn be the Fermat curve of degree n over ℚ, and let Dn=Fn/S3 be its quotient by coordinate permutations. We prove that Dn has trivial geometric automorphism group for every n≥ 11 with gcd(n,6)=1. Its Jacobian has complex multiplication. For n=11, this Jacobian is absolutely simple of dimension five. We construct an isogeny over ℚ of degree 115 from Jac(D11) to the Jacobian of y11=x(1−x)2. For primes p≡ 1pmod 3, we also consider the metacyclic quotients Cp of Fp over ℚ. By a theorem of Carocca, Lange and Rodríguez, their Jacobians are absolutely simple and have complex multiplication. We prove that Cp has trivial geometric automorphism group if and only if p≥ 19. Write p=r2+rs+s2 with r,s positive integers. The natural degree-three cover Yp→ Cp satisfies gon(Yp)=r+s, so gon(Cp)≥ ⌈(r+s)/3⌉. Since g(Cp)=(p−1)/6, both the genus and the gonality of Cp tend to infinity with p.
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BibTeX
@misc{tafazolian2026fermat,
title = {{Fermat quotients with CM Jacobians and trivial automorphism group}},
author = {Saeed Tafazolian},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.08176},
eprint = {2611.08176},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.08176},
}
RIS
TY - UNPB
TI - Fermat quotients with CM Jacobians and trivial automorphism group
AU - Saeed Tafazolian
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2611.08176
UR - https://arxiv.org/abs/2611.08176
AB - Let F_n be the Fermat curve of degree n over ℚ, and let D_n=F_n/S_3 be its quotient by coordinate permutations. We prove that D_n has trivial geometric automorphism group for every n≥ 11 with gcd(n,6)=1. Its Jacobian has complex multiplication. For n=11, this Jacobian is absolutely simple of dimension five. We construct an isogeny over ℚ of degree 11^5 from Jac(D_11) to the Jacobian of y^11=x(1−x)^2. For primes p≡ 1pmod 3, we also consider the metacyclic quotients C_p of F_p over ℚ. By a theorem of Carocca, Lange and Rodríguez, their Jacobians are absolutely simple and have complex multiplication. We prove that C_p has trivial geometric automorphism group if and only if p≥ 19. Write p=r^2+rs+s^2 with r,s positive integers. The natural degree-three cover Y_p→ C_p satisfies gon(Y_p)=r+s, so gon(C_p)≥ ⌈(r+s)/3⌉. Since g(C_p)=(p−1)/6, both the genus and the gonality of C_p tend to infinity with p.
ER -