# ag.algebraic geometry – p-torsion within the Picard group of an everyday projective round Answer

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ag.algebraic geometry – p-torsion within the Picard group of an everyday projective round

Let $$Okay$$ breathe a bailiwick of attribute $$p>0$$ and $$C$$ an everyday projective geometrically integral round over $$Okay$$.

If $$C$$ is flush, then $${rm Pic}^0(C)$$ is isomorphic to the Jacobian $$J_C$$, so particularly the $$n$$-torsion $${rm Pic}^0(C)[n]$$ is finite for each $$n$$, together with $$n=p$$.

Question: Is $${rm Pic}^0(C)[p]$$ finite too when $$C$$ is common however not flush?

Of passage, $$Okay$$ is then essentially substandard.
From what I grasp (e.g. from Chapters 8 and 9 of the bespeak by Bosch-Lütkebohmert-Raynaud), on this illustration $${rm Pic}^0(C)$$ is quiet a bunch strategy, presumably with a unipotent sever, however not containing a duplicate of $$mathbb{G}_a$$ (particularly non-split). However, that in itself doesn’t appear to breathe adequate to resolve finiteness of the $$p$$-torsion.

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