scholarly journals 4-RANKS OF CLASS GROUPS OF QUADRATIC EXTENSIONS OF CERTAIN QUADRATIC FUNCTION FIELDS

2012 ◽  
Vol 27 (2) ◽  
pp. 223-231
Author(s):  
Sung-Han Bae ◽  
Pyung-Lyun Kang
Keyword(s):  
Function Fields ◽  
Quadratic Function ◽  
Class Groups ◽  
Quadratic Extensions ◽  
Quadratic Function Fields

scholarly journals Non-abelian Cohen–Lenstra heuristics over function fields

2017 ◽  
Vol 153 (7) ◽  
pp. 1372-1390 ◽  
Author(s):  
Nigel Boston ◽  
Melanie Matchett Wood
Keyword(s):  
Function Fields ◽  
Quadratic Function ◽  
Number Fields ◽  
Galois Groups ◽  
Quadratic Number ◽  
Quadratic Extensions ◽  
Quadratic Number Fields ◽  
Imaginary Quadratic Number ◽  
Unique Distribution ◽  
Quadratic Function Fields

Boston, Bush and Hajir have developed heuristics, extending the Cohen–Lenstra heuristics, that conjecture the distribution of the Galois groups of the maximal unramified pro-$p$extensions of imaginary quadratic number fields for$p$an odd prime. In this paper, we find the moments of their proposed distribution, and further prove there is a unique distribution with those moments. Further, we show that in the function field analog, for imaginary quadratic extensions of$\mathbb{F}_{q}(t)$, the Galois groups of the maximal unramified pro-$p$extensions, as$q\rightarrow \infty$, have the moments predicted by the Boston, Bush and Hajir heuristics. In fact, we determine the moments of the Galois groups of the maximal unramified pro-odd extensions of imaginary quadratic function fields, leading to a conjecture on Galois groups of the maximal unramified pro-odd extensions of imaginary quadratic number fields.

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.

A Lower Bound on the Number of Cyclic Function Fields With Class Number Divisible byn

2006 ◽  
Vol 49 (3) ◽  
pp. 448-463 ◽  
Author(s):  
Allison M. Pacelli
Keyword(s):  
Lower Bound ◽  
Class Number ◽  
Function Fields ◽  
Quadratic Function ◽  
Class Numbers ◽  
Prime Degree ◽  
Cyclic Function ◽  
Quadratic Function Fields

AbstractIn this paper, we find a lower bound on the number of cyclic function fields of prime degreelwhose class numbers are divisible by a given integern. This generalizes a previous result of D. Cardon and R. Murty which gives a lower bound on the number of quadratic function fields with class numbers divisible byn.

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