TY - JOUR
T1 - Bounds for moments of ℓ-torsion in class groups
AU - Koymans, Peter
AU - Thorner, Jesse
N1 - We thank Roger Heath-Brown and Lillian Pierce for their encouragement and the anonymous referee for their helpful comments. PK would like to thank Carlo Pagano and Martin Widmer for fruitful discussions. PK gratefully acknowledges the support of Dr. Max Rössler, the Walter Haefner Foundation and the ETH Zürich Foundation. JT gratefully acknowledges the support of the Simons Foundation (MP-TSM-00002484).
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 -