TY - JOUR
T1 - Bounds for moments of ℓ-torsion in class groups
AU - Koymans, Peter
AU - Thorner, Jesse
N1 - Publisher Copyright:
© The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2024.
PY - 2024/10
Y1 - 2024/10
N2 - Fix a number field k, integers ℓ,n≥2, and a prime p. For all r≥1, we prove strong unconditional upper bounds on the rth moment of ℓ-torsion in the ideal class groups of degree p extensions of k and of degree nSn-extensions of k, improving upon results of Ellenberg, Pierce and Wood as well as GRH-conditional results of Frei and Widmer. For large r, our results are comparable with work of Heath-Brown and Pierce for imaginary quadratic extensions of Q. When r=1, our results are new even for the family of all quadratic extensions of Q, leading to an improved upper bound for the count of degree pDp-extensions over Q (where Dp is the dihedral group of order 2p).
AB - Fix a number field k, integers ℓ,n≥2, and a prime p. For all r≥1, we prove strong unconditional upper bounds on the rth moment of ℓ-torsion in the ideal class groups of degree p extensions of k and of degree nSn-extensions of k, improving upon results of Ellenberg, Pierce and Wood as well as GRH-conditional results of Frei and Widmer. For large r, our results are comparable with work of Heath-Brown and Pierce for imaginary quadratic extensions of Q. When r=1, our results are new even for the family of all quadratic extensions of Q, leading to an improved upper bound for the count of degree pDp-extensions over Q (where Dp is the dihedral group of order 2p).
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U2 - 10.1007/s00208-024-02839-3
DO - 10.1007/s00208-024-02839-3
M3 - Article
AN - SCOPUS:85188352103
SN - 0025-5831
VL - 390
SP - 3221
EP - 3237
JO - Mathematische Annalen
JF - Mathematische Annalen
IS - 2
ER -