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Non-abelian Cohen–Lenstra heuristics over function fields
- Part of
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- Journal:
- Compositio Mathematica / Volume 153 / Issue 7 / July 2017
- Published online by Cambridge University Press:
- 12 May 2017, pp. 1372-1390
- Print publication:
- July 2017
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- Article
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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.
