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Let $H$ be an elliptic or hyperelliptic curve over $\Q$ given by an equation $y^2 = f(x)$, and let $p$ be an odd prime number. Then the cluster picture of $H$ at $p$ associated to this particular equation for $H$, is a graphical representation of the $p$-adic distances between the roots of $f$.

The size of a cluster is the number of roots of $f(x)$ that it contains.

The depth of a cluster, is the valuation of the radius of a minimal $p$-adic disk containing all the roots in that particular cluster. The numbers that are depicted in a cluster picture represent the relative depth: the difference between the depth of a certain cluster and the smallest other cluster in which that certain cluster is contained. For the top cluster, which contains all the roots of $f(x)$, the depth is depicted instead of the relative depth (as there is no relative depth in this case).

More information about cluster pictures can be found in A user's guide to the local arithmetic of hyperelliptic curves [arXiv:2007.01749 ].

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  • Review status: beta
  • Last edited by Raymond van Bommel on 2020-08-25 08:36:33
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