Can anyone give me a precise information or formulation of Birch and Swinnerton-Dyer conjecture for Jacobians -- I mean for Albanese varieties. Any reference to useful links or expository articles, or any material is appreciated.
On Albanese varieties
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number-theory
reference-request
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8"any reference to useful links or expository articles,or any material is appreciated thanks a lot" - you have tried searching on Google Scholar, I presume? – 2011-05-09
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1What's the reason for the downvotes? Please explain. – 2012-08-13
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2@MakotoKato : Its common in MO and Math.SE , to down-vote without any reason. I have shouted, requested , begged and did everything , to explain the reason for down voting , but no one cared. Apart from reducing the reputation, if users post the reason its useful for constructing good questions next time. But I don't know why everyone is not that CIVIC. Thank you sir. – 2012-08-13
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1@Iyengar I agree with you. I don't think your question deserves downvotes. The only reason I can think of is that they have grudges or jealousy on you. That's a despicable thing to do if that is the case, IMO. – 2012-08-13
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0@MakotoKato : Yes sir, it happened to me many times. Many people here are filled with grudges and I think you too know it and experienced it . But Thank you for your response. We never care about the reputation, and we should make it explicit. Either they must change or we must. I think the latter is better. – 2012-08-13
1 Answers
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Every abelian variety is an Albanese variety: in fact, every abelian variety is its own Albanese variety. Thus asking about BSD "for Albanese varieties" is equivalent to asking about BSD for all abelian varieties: i.e., the general case.
The story might change if you want to restrict the class of varieties $V$ you want to take the Albanese variety of. In particular one is probably in slightly better shape looking at Jacobians -- i.e., Albanese varieties of curves -- than arbitrary abelian varieties, although in any case very little is known about BSD for anything but elliptic curves over $\mathbb{Q}$ of analytic rank at most $1$.