A geometric approach to the Cohen-Lenstra heuristics
We give a new geometric description of when an element of the class group of a quadratic field, thought of as a quadratic form $q$, is $n$-torsion. We show that $q$ corresponds to an $n$-torsion element if and only if there exists a degree $n$ polynomial whose resultant with $q$ is $\pm 1$. This is...
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| Main Author | |
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| Format | Journal Article |
| Language | English |
| Published |
18.06.2021
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| Subjects | |
| Online Access | Get full text |
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| Summary: | We give a new geometric description of when an element of the class group of
a quadratic field, thought of as a quadratic form $q$, is $n$-torsion. We show
that $q$ corresponds to an $n$-torsion element if and only if there exists a
degree $n$ polynomial whose resultant with $q$ is $\pm 1$. This is motivated by
a more precise geometric parameterization which directly connects torsion in
class groups of quadratic fields to Selmer groups of singular genus $1$ curves. |
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| DOI: | 10.48550/arxiv.2106.10357 |