Class groups under relative quadratic extensions

2011 ◽  
Vol 150 (4) ◽  
pp. 399-414
Author(s):  
Qin Yue
Keyword(s):  
Class Groups ◽  
Quadratic Extensions ◽  
Relative Quadratic Extensions

scholarly journals Selmer groups and class groups

2014 ◽  
Vol 151 (3) ◽  
pp. 416-434 ◽  
Author(s):  
Kęstutis Česnavičius
Keyword(s):  
Number Fields ◽  
Torsion Subgroup ◽  
Selmer Groups ◽  
Ideal Class Group ◽  
Class Groups ◽  
New Approach ◽  
Quadratic Extensions ◽  
Iwasawa Invariants ◽  
Cohomological Interpretation ◽  
The Ideal

AbstractLet $A$ be an abelian variety over a global field $K$ of characteristic $p\geqslant 0$. If $A$ has nontrivial (respectively full) $K$-rational $l$-torsion for a prime $l\neq p$, we exploit the fppf cohomological interpretation of the $l$-Selmer group $\text{Sel}_{l}\,A$ to bound $\#\text{Sel}_{l}\,A$ from below (respectively above) in terms of the cardinality of the $l$-torsion subgroup of the ideal class group of $K$. Applied over families of finite extensions of $K$, the bounds relate the growth of Selmer groups and class groups. For function fields, this technique proves the unboundedness of $l$-ranks of class groups of quadratic extensions of every $K$ containing a fixed finite field $\mathbb{F}_{p^{n}}$ (depending on $l$). For number fields, it suggests a new approach to the Iwasawa ${\it\mu}=0$ conjecture through inequalities, valid when $A(K)[l]\neq 0$, between Iwasawa invariants governing the growth of Selmer groups and class groups in a $\mathbb{Z}_{l}$-extension.

scholarly journals ON THE STRONGLY AMBIGUOUS CLASSES OF 𝕜/ℚ(i) WHERE $\mathds{k} = {\mathbb Q}(\sqrt{2p_{1}p_{2}, i})$

2014 ◽  
Vol 07 (01) ◽  
pp. 1450021 ◽  
Author(s):  
Abdelmalek Azizi ◽  
Abdelkader Zekhnini ◽  
Mohammed Taous
Keyword(s):  
Infinite Family ◽  
Number Fields ◽  
Class Groups ◽  
Quadratic Extensions ◽  
Genus Field ◽  
The Absolute

We construct an infinite family of imaginary bicyclic biquadratic number fields 𝕜 with the 2-ranks of their 2-class groups are ≥ 3, whose strongly ambiguous classes of 𝕜/ℚ(i) capitulate in the absolute genus field 𝕜(*), which is strictly included in the relative genus field (𝕜/ℚ(i))* and we study the capitulation of the 2-ideal classes of 𝕜 in its quadratic extensions included in 𝕜(*).

A Sequence of Results on Class Number Congruences

1993 ◽  
Vol 36 (2) ◽  
pp. 139-143
Author(s):  
Antone Costa
Keyword(s):  
Class Number ◽  
Positive Density ◽  
Quadratic Extensions ◽  
Totally Real ◽  
Relative Quadratic Extensions

AbstractLet p ≡ 1 mod 8 be a rational prime and let h(—p) be the class number of . In [1], Barrucand and Cohn show that h(-p) = 0 mod 8 iff p = x2 + 32y2 for some x,y € Z. In this article, we generalize their result to a family of relative quadratic extensions K/F, where Fk is the maximum totally real subfield of Q(ζ2k+2 ), and a power of a prime of Fk from a family of positive density.

scholarly journals Exponents of Class Groups of Quadratic Function Fields over Finite Fields

2001 ◽  
Vol 44 (4) ◽  
pp. 398-407 ◽  
Author(s):  
David A. Cardon ◽  
M. Ram Murty
Keyword(s):  
Lower Bound ◽  
Positive Integer ◽  
Finite Fields ◽  
Function Field ◽  
Quadratic Function ◽  
Class Groups ◽  
Quadratic Extensions ◽  
Fixed Order ◽  
Fixed Positive Integer ◽  
Quadratic Function Fields

AbstractWe find a lower bound on the number of imaginary quadratic extensions of the function field whose class groups have an element of a fixed order.More precisely, let q ≥ 5 be a power of an odd prime and let g be a fixed positive integer ≥ 3. There are polynomials D ∈ with deg(D) ≤ ℓ such that the class groups of the quadratic extensions have an element of order g.

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