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Monogenic fields with odd class number Part II: even degree
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abstract
In 1801, Gauss proved that there were infinitely many quadratic fields with odd class number. We generalise this result by showing that there are infinitely many $S_n$-fields of any given even degree and signature that have odd class number. Also, we prove that there are infinitely many fields of any even degree at least $4$ and with at least one real embedding that have units of every signature. To do so, we bound the average number of $2$-torsion elements in the class group, narrow class group, and oriented class group of monogenised fields of even degree (and compute these averages precisely conditional on a tail estimate) using a parametrisation of Wood. These averages are the first $p$-torsion averages to be calculated for $p$ not coprime to the degree (in degree at least $3$), shedding light on the question of Cohen-Lenstra-Martinet-Malle type heuristics for class groups and narrow class groups at "bad" primes.
Forward citations
Cited by 3 Pith papers
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Geometry-of-numbers methods over global fields II: Coregular representations
Geometry-of-numbers methods are extended to count orbits in coregular spaces over arbitrary global fields, yielding bounds on average ranks and Selmer sizes for elliptic curves and hyperelliptic Jacobians.
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Counting the number of $n$-periodic $\mathbb{Z}_{p}$-and $\mathbb{F}_{p}[t]$-points of a discrete dynamical system with applications from arithmetic statistics, VI
Establishes that averages of distinct n-periodic points for maps φ_{p^ℓ,c} and φ_{(p-1)^ℓ,c} over Z_p and F_p[t] are unbounded/zero or 1/2/0 as c varies, then derives counting results for irreducibles, zeta functions,...
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Counting the number of $1_{m}$-preperiodic $\mathcal{O}_{K}$-points of a discrete dynamical system with applications from arithmetic statistics, VII
The main theorem is false: for φ_{p,c}(z)=z^p+c over F_p with p|c, every point is fixed, so the number of 1_n-preperiodic points is 0, not p.
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