A twisted class number formula and Gross's special units over an imaginary quadratic field

Saad El Boukhari

Czechoslovak Mathematical Journal (2023)

  • Volume: 73, Issue: 4, page 1333-1347
  • ISSN: 0011-4642

Abstract

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Let F / k be a finite abelian extension of number fields with k imaginary quadratic. Let O F be the ring of integers of F and n ≥ 2 a rational integer. We construct a submodule in the higher odd-degree algebraic K -groups of O F using corresponding Gross’s special elements. We show that this submodule is of finite index and prove that this index can be computed using the higher “twisted” class number of F , which is the cardinal of the finite algebraic K -group K 2 n - 2 ( O F ) .

How to cite

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El Boukhari, Saad. "A twisted class number formula and Gross's special units over an imaginary quadratic field." Czechoslovak Mathematical Journal 73.4 (2023): 1333-1347. <http://eudml.org/doc/299417>.

@article{ElBoukhari2023,
abstract = {Let $F/k$ be a finite abelian extension of number fields with $k$ imaginary quadratic. Let $O_F$ be the ring of integers of $F$ and $n\ge 2$ a rational integer. We construct a submodule in the higher odd-degree algebraic $K$-groups of $O_F$ using corresponding Gross’s special elements. We show that this submodule is of finite index and prove that this index can be computed using the higher “twisted” class number of $F$, which is the cardinal of the finite algebraic $K$-group $K_\{2n-2\}(O_F)$.},
author = {El Boukhari, Saad},
journal = {Czechoslovak Mathematical Journal},
keywords = {algebraic $K$-theory; Dedekind zeta function; Artin $L$-function; Beilinson regulator; generalized index; Lichtenbaum conjecture},
language = {eng},
number = {4},
pages = {1333-1347},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A twisted class number formula and Gross's special units over an imaginary quadratic field},
url = {http://eudml.org/doc/299417},
volume = {73},
year = {2023},
}

TY - JOUR
AU - El Boukhari, Saad
TI - A twisted class number formula and Gross's special units over an imaginary quadratic field
JO - Czechoslovak Mathematical Journal
PY - 2023
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 73
IS - 4
SP - 1333
EP - 1347
AB - Let $F/k$ be a finite abelian extension of number fields with $k$ imaginary quadratic. Let $O_F$ be the ring of integers of $F$ and $n\ge 2$ a rational integer. We construct a submodule in the higher odd-degree algebraic $K$-groups of $O_F$ using corresponding Gross’s special elements. We show that this submodule is of finite index and prove that this index can be computed using the higher “twisted” class number of $F$, which is the cardinal of the finite algebraic $K$-group $K_{2n-2}(O_F)$.
LA - eng
KW - algebraic $K$-theory; Dedekind zeta function; Artin $L$-function; Beilinson regulator; generalized index; Lichtenbaum conjecture
UR - http://eudml.org/doc/299417
ER -

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