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I would like references or a result about the computation of the picard number of the jacobian of an algebraic curve.

What about the special case when the picard number of the Jacobian is one (is any classification possible)?

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  • $\begingroup$ Can we classify the case when picard number of Jacobian is one ? $\endgroup$
    – Sun
    Commented Nov 20, 2013 at 14:22
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    $\begingroup$ Dear Sun, I think what some people (including myself) didn't like about this question is that you didn't give any context for your question. In particular, you should perhaps have said what you already knew, what specific question you were wondering about (instead of just please give me references on a particular topic). Was there a particular curve whose Jacobian you wanted? (Or curves in some family?) $\endgroup$
    – Karl Schwede
    Commented Nov 20, 2013 at 18:24

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Do you know about the embedding of $\hbox{NS}(A)\otimes\mathbb{R}$ into $\hbox{End}(A)\otimes\mathbb{R}$ once you choose a polarization? The image is the subset (actually, a Jordan algebra) fixed by the Rosati involution. This is all in Mumford's Abelian Varieties. In some sense, it reduces the computation of $\hbox{NS}(A)\otimes\mathbb{R}$ to linear algebra, although in practice it may not be so easy to compute the endomorphism ring.

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  • $\begingroup$ Yeah I know this. In some cases picard number of jacobian of curve is one. Can you tell me exact result about it. $\endgroup$
    – Sun
    Commented Nov 20, 2013 at 13:45
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The preprint http://arxiv.org/pdf/1310.3402.pdf may interest you (look at §3 and 4).

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  • $\begingroup$ Thanks. I can find some important result here. In some cases picard number of jacobian of curve is one. Can you tell me exact result about it? $\endgroup$
    – Sun
    Commented Nov 20, 2013 at 14:21
  • $\begingroup$ What kind of result? There are examples (due to Mori) of Jacobians of hyperelliptic curves in any genus with Jacobian of Picard number 1. What else can you ask? $\endgroup$
    – abx
    Commented Nov 20, 2013 at 14:36
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    $\begingroup$ Thanks, I think Mori's paper has answer of my question. $\endgroup$
    – Sun
    Commented Nov 20, 2013 at 17:12

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