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What are some examples of finite dimensional irreducible complex representations of $SL_2(\mathbb{Q}_p)$? One knows such a representations cannot be smooth, so probably the examples will be ...

The Newton Method over $\mathbb{R}$ has the property that the precision is doubled (under some continuous differentiable assumption) in each iteration. For the ring $\mathbb{Z}_p$ of $p$-adic integers,...